# Mathematics — Full Content

> A searchable reference for the math formulas you actually use. Algebra, geometry, trigonometry, and calculus, rendered with KaTeX.

Concatenated MDX body of every formula page, in canonical order. JSX shortcodes (<Math>, <Diagram>, <Notice>) are preserved as-is — LaTeX inside <Math math="..." /> is the inline LaTeX source.

---

# Basics

## Powers

URL: http://math.mkabumattar.com/basics/powers

Exponent rules: products, quotients, powers of powers, negative and fractional exponents, and identities.

Powers record how many times a base multiplies by itself. The laws below combine, split, and rewrite exponential expressions without expanding them, and extend the operation consistently to negative and fractional exponents. The same rules underpin polynomial, radical, and logarithmic manipulation.

Multiplying two powers of the same base adds the exponents. Writing *a<sup>n</sup>* as *n* factors of *a* and *a<sup>m</sup>* as *m* factors and concatenating gives *n + m* factors. The rule extends to any real exponents once the operation is defined consistently for them.

<Math math="a^{n}\times a^{m}=a^{n+m}" display />

Raising a power to another power multiplies the exponents. Apply the product rule *m* times to *a<sup>n</sup>* to get *a<sup>n</sup>* repeated *m* times, which collapses to *a<sup>n·m</sup>*. Use this to flatten nested exponentials.

<Math math="\left(a^{n}\right)^{m}=a^{n\times m}" display />

A negative exponent inverts the base. The product rule forces this: *a<sup>n</sup>·a<sup>-n</sup> = a<sup>0</sup> = 1*, so *a<sup>-n</sup> = 1/a<sup>n</sup>*. With this definition the laws extend from natural numbers to all integers.

<Math math="a^{-n}=\frac{1}{a^{n}}" display />

The exponent distributes over a product of bases. Each factor in *(ab)<sup>n</sup>* contributes *n* copies of *a* and *n* copies of *b*, which regroup as *a<sup>n</sup>b<sup>n</sup>*. Distribution fails over sums: *(a + b)<sup>n</sup>* is not *a<sup>n</sup> + b<sup>n</sup>*. That case needs the binomial theorem.

<Math math="\left(a\times b\right)^{n} = \left(a\right)^{n}\times\left( b\right)^{n}" display />

A negative exponent on a fraction flips the fraction and removes the sign. This is a corollary of the negative-exponent rule combined with quotient distribution. The flipped form gives cleaner arithmetic when the original numerator is larger than the denominator.

<Math math="\left(\frac{a}{b}\right)^{-n} = \left(\frac{b}{a}\right)^{n}=\frac{b^{n}}{a^{n}}" display />

Dividing two powers of the same base subtracts the exponents. Cancelling common factors removes *min(n, m)* copies of *a* and leaves the difference. The second equality keeps the exponent positive when *n - m* is negative.

<Math math="\frac{a^{n}}{a^{m}}=a^{n-m}=\frac{{1}}{{a^{m-n}}}" display />

The exponent also distributes over a quotient. Combine the product-distribution rule with the negative-exponent rule applied to *b*. The identity collapses ratios of powers into a single fraction with separately-exponentiated parts.

<Math math="\left(\frac{a}{b}\right)^{n}=\frac{a^{n}}{b^{n}}" display />

Any non-zero base raised to the zero power equals one. The quotient rule forces this: *a<sup>n</sup>/a<sup>n</sup> = a<sup>0</sup>* and the left side equals 1. The base zero is excluded; *0<sup>0</sup>* is treated separately.

<Math math="a^{0}=1 \text{ }\left(if\text{ }a\neq 0\right)" display />

A fractional exponent denotes a root combined with a power. The denominator selects the root and the numerator the power. The power-of-a-power rule guarantees the two orderings agree, so you can pick whichever keeps intermediate values smaller or radical-free.

<Math math="a^{\frac{a}{m}}=\left(a^{\frac{1}{m}}\right)^{n}=\left(a^{n}\right)^{\frac{1}{m}}" display />

<Notice type="warning">
`0^{0}` is an indeterminate form in analytic contexts, not 1. Combinatorics often defines it as 1 for convenience, but the choice is conventional, not algebraic.
</Notice>

---

## Radicals

URL: http://math.mkabumattar.com/basics/radicals

Radical and root identities: products, quotients, nested roots, and absolute value rules.

Radicals express *n*-th roots, the inverses of *n*-th powers. The identities below convert between radical and exponent notation, split roots across products and quotients, and handle the parity-dependent case when an even root is taken of an even power. Working through fractional exponents pushes the parity question out of the calculation and back to the final principal-root step.

A radical is a unit-fractional exponent. Every exponent rule transfers directly to radicals through this identification, so there is no separate set of root laws. Most identities below are corollaries of the exponent rules read through this bridge.

<Math math="\sqrt[n]{a}=a^{\frac{1}{n}}" display />

The *n*-th root distributes over a product of non-negative factors. This is the radical form of *(ab)<sup>1/n</sup> = a<sup>1/n</sup>b<sup>1/n</sup>* and is how perfect *n*-th powers get pulled out from under the root. Non-negativity matters: applying the rule to two negatives can produce sign errors.

<Math math="\sqrt[n]{a\times b}=\sqrt[n]{a}\times\sqrt[n]{b}" display />

The *n*-th root distributes over a quotient. Read in reverse, it combines two separate radicals into one for rationalising denominators. It also requires non-negative operands for real-valued safety.

<Math math="\sqrt[n]{\frac{a}{b}}=\frac{\sqrt[n]{a}}{\sqrt[n]{b}}" display />

A nested radical collapses by multiplying the indices. The identity follows from *(a<sup>1/n</sup>)<sup>1/m</sup> = a<sup>1/(mn)</sup>* via the power-of-a-power rule. Iterated roots become a single root with a larger index, which is usually easier to handle symbolically.

<Math math="\sqrt[m]{\sqrt[n]{a}}=\sqrt[m\times n]{a}" display />

For an odd index the root undoes the power and preserves the sign. Odd roots of negatives stay real because an odd power of a negative remains negative, so the inverse map is defined for all real inputs. This is the only case where the radical and the power cancel cleanly.

<Math math="\sqrt[n]{a^{n}}=a\text{ }\left(if\text{ }n\text{ }is\text{ }odd\right)" display />

For an even index the principal root is non-negative, so the result is the absolute value. Even powers erase sign information: *(-3)<sup>2</sup> = 9* matches *3<sup>2</sup> = 9*. The inverse cannot recover the original sign and picks the non-negative branch by convention. This is where the *|x|* in simplified even roots of even powers comes from.

<Math math="\sqrt[n]{a^{n}}=\left|a\right|\text{ }\left(if\text{ }n\text{ }is\text{ }even\right)" display />

<Notice type="warning">
The product and quotient rules assume real, non-negative radicands. Applying them blindly to negatives can produce false equalities such as `sqrt(-1) * sqrt(-1) = sqrt(1) = 1`.
</Notice>

---

## Logarithms

URL: http://math.mkabumattar.com/basics/logarithms

Logarithm definition, natural and common logs, and key properties.

Logarithms are the inverses of exponentials: *log<sub>b</sub>(x)* is the power to which *b* must be raised to produce *x*. They turn products into sums and powers into products, which is why they are the standard tool for solving exponential equations. They also rescale multiplicative phenomena into additive ones, producing the log scales used in decibels, magnitudes, pH, and information theory.

## Definition

A logarithm is the inverse of an exponential with the same base. Left to right, the equivalence extracts the exponent that produced a known value. Right to left, it recovers the value from a known exponent. The base *b* must be positive and not equal to one for the inverse to be a single-valued real function.

The defining equivalence between log and exponential form, used freely in both directions:

<Math math="y=\log_{b}{x}\rightarrow x=b^{y}" display />

## Example

A concrete instance, read in both directions. The exponential form is usually the easier check when verifying a logarithm by hand.

<Math math="\log_{5}{125}=3\rightarrow 5^{3}=125" display />

## Special Logarithms

Two bases get dedicated notation: the natural log (base *e*) and the common log (base 10). The natural log is the calculus default because its derivative is *1/x*. The common log dominates engineering because it matches the decimal place-value system.

The natural logarithm uses Euler's number as its base. Its derivative is the reciprocal function *1/x*. It is the standard log in analysis, probability, and any setting built on continuous growth.

<Math math="\ln\left(x\right)=\log_{e}{x}\text{ }\left(Natural \text{ }log\right)" display />

The common logarithm uses base 10 and is standard for engineering and pH-style scales. Its integer part counts the digits before the decimal point in the input, which is why it underlies decibels, Richter magnitudes, and significant-figure scaling.

<Math math="\log\left(x\right)=\log_{10}{x}\text{ }\left(Common \text{ }log\right)" display />

The constant *e* is the irrational base for which the exponential function is its own derivative. It equals the limit *(1 + 1/n)<sup>n</sup>* as *n* tends to infinity, which is why it surfaces in every continuous growth or decay model.

<Math math="Whare \text{ }e=2.7182818..." display />

## Logarithms Properties

These identities simplify and expand logarithmic expressions algebraically. Each translates an exponent rule under the log-exp inverse pairing. Products become sums, powers become products, quotients become differences.

The log of the base itself is one, since *b<sup>1</sup> = b*. This is the smallest non-trivial value any logarithm takes and is a useful reference point in change-of-base computations.

<Math math="\log_{b}{b}=1" display />

The log of one is zero for every base, since *b<sup>0</sup> = 1*. Every logarithm curve crosses the *x*-axis at this single point.

<Math math="\log_{b}{1}=0" display />

Log and exponential of the same base cancel when the exponential is inside the log. This is the inverse-function identity read left to right. It collapses any *log<sub>b</sub>(b<sup>x</sup>)* to *x* directly.

<Math math="\log_{b}{b^{x}}=x" display />

Inverse property with the exponent outside: the same cancellation in the opposite order. Together the two identities say *log<sub>b</sub>* and *b<sup>x</sup>* compose to the identity in both directions.

<Math math="b^{\log_{b}{x}}=x" display />

The power rule brings an exponent out as a coefficient. It is the most-used logarithm identity for solving exponential equations: take the log of both sides of *b<sup>x</sup> = c* and apply this rule to get *x = log c / log b*. The rule holds for any real exponent.

<Math math="\log_{b}{x^{r}}=r\times\log_{b}{x}" display />

The product rule turns multiplication inside the log into addition. Logarithm tables once reduced tedious multiplications to lookups and additions using this identity.

<Math math="\log_{b}\left({x\times y}\right)=\log_{b}{x}+\log_{b}{y}" display />

The quotient rule turns division inside the log into subtraction. Read in reverse, it consolidates a difference of two same-base logs into a single log of a ratio.

<Math math="\log_{b}\left({\frac{b}{d}}\right)=\log_{b}{x}-\log_{b}{y}" display />

<Notice type="warning">
The real-valued logarithm is defined only for positive arguments. `log_b(0)` diverges to negative infinity, and `log_b(x)` for `x < 0` has no real value; it requires the complex logarithm.
</Notice>

---

## Quadratic Equations

URL: http://math.mkabumattar.com/basics/quadratic-equations

Standard form, the quadratic formula, and the role of the discriminant.

A quadratic equation is a degree-two polynomial equation in a single variable. Every quadratic can be solved by the formula below. The sign of its discriminant (the expression under the radical) tells whether the roots are real or complex without computing them. The formula is the general result of completing the square on the standard form.

The standard form fixes coefficients *a*, *b*, *c* with a non-zero leading coefficient. The condition *a ≠ 0* makes the equation genuinely quadratic; with *a = 0* it collapses to a linear equation and the formula's denominator vanishes.

<Math math="ax^{2}+bx+c=0 \text{ }\left(and\text{ }a\neq 0\right)\text{ }then" display />

The quadratic formula gives both roots from the coefficients directly. The *±* captures the parabola's symmetry about its axis *x = -b/(2a)*: the two roots lie equidistant from this vertical line, offset by the radical term. When the discriminant is computable, the formula is the most direct route to the roots.

<Math math="x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}" display />

The discriminant *b<sup>2</sup> - 4ac* controls how many times the parabola crosses the *x*-axis. The three cases below give two crossings, one tangent touch, or no real crossing.

<Diagram alt="Parabola with two real roots and labeled vertex" viewBox="0 0 220 160" caption="When the discriminant is positive, the parabola crosses the x-axis at two distinct points; the vertex lies below the axis.">
  <line x1="10" y1="110" x2="210" y2="110" />
  <line x1="110" y1="10" x2="110" y2="150" />
  <path d="M 30 30 Q 110 200 190 30" />
  <circle cx="65" cy="110" r="2.5" fill="currentColor" />
  <circle cx="155" cy="110" r="2.5" fill="currentColor" />
  <circle cx="110" cy="140" r="2.5" fill="currentColor" />
  <text x="60" y="124" font-size="10" fill="currentColor" stroke="none">x₁</text>
  <text x="150" y="124" font-size="10" fill="currentColor" stroke="none">x₂</text>
  <text x="116" y="146" font-size="10" fill="currentColor" stroke="none">vertex</text>
  <text x="200" y="106" font-size="10" fill="currentColor" stroke="none">x</text>
  <text x="114" y="18" font-size="10" fill="currentColor" stroke="none">y</text>
</Diagram>

If the discriminant is positive, the parabola crosses the *x*-axis at two distinct points. The radical evaluates to a positive real, so the *±* produces two distinct values, matching the diagram.

<Math math="if\text{ }b^{2}-4ac>0\text{ }then\text{ }two\text{ }real\text{ }solution" display />

If the discriminant is zero, the parabola is tangent to the *x*-axis at one point: a repeated root. The radical vanishes and the *±* collapses to a single value *x = -b/(2a)*, the *x*-coordinate of the vertex.

<Math math="if\text{ }b^{2}-4ac=0\text{ }then\text{ }reqeated\text{ }real\text{ }solution" display />

If the discriminant is negative, the roots are a complex-conjugate pair and the parabola does not meet the *x*-axis. The radical becomes imaginary, and the *±* gives two complex roots that mirror one another across the real axis.

<Math math="if\text{ }b^{2}-4ac<0\text{ }then\text{ }two\text{ }complex\text{ }solution" display />

<Notice type="tip">
The discriminant *b<sup>2</sup> - 4ac* alone determines the nature of the roots, so you can classify solutions without computing them. The sum of the roots is *-b/a* and the product is *c/a* (Vieta's formulas); this is often faster than the full formula when only those quantities are needed.
</Notice>

---

## Polynomials

URL: http://math.mkabumattar.com/basics/polynomials

Common polynomial identities and factorizations including squares, cubes, and difference of powers.

Polynomial identities are the standard expansions and factorisations of sums, differences, and powers of two terms. Recognising them on sight replaces a lot of multiplying out or trial factoring with template matching. The cube and higher-power identities all share one shape: a linear factor *(x ± a)* times a longer polynomial whose coefficients alternate or stay constant based on the sign.

Difference of squares factors into conjugate binomials. The cross terms cancel because the *+y* and *-y* contributions to the *x* coefficient annihilate, leaving only the difference of pure squares. This is the standard identity for rationalising denominators and spotting hidden factorisations.

<Math math="x^{2}-y^{2}=\left(x+y\right)\left(x-y\right)" display />

Product of two linear binomials sharing a leading variable (the FOIL pattern). The middle coefficient is the sum of the constants and the trailing coefficient is their product. This is how you factor quadratics by inspecting sum-product pairs of the constant term.

<Math math="\left(x+a\right)\left(x+b\right)=x^{2}+\left(a+b\right)x+ab" display />

Square of a sum. The cross term *2xy* appears because the product is counted twice (once as *xy* and once as *yx*). It is the most commonly dropped piece when squaring a binomial by hand.

<Math math="\left(a+y\right)^{2}=x^{2}+2xy+y^{2}" display />

Square of a difference. Identical to the sum case except the cross term picks up a negative sign from the negative second term. Squaring kills the sign on the *y<sup>2</sup>* term but not on the cross term.

<Math math="\left(a-y\right)^{2}=x^{2}-2xy+y^{2}" display />

Cube of a sum. The coefficients 1, 3, 3, 1 are the fourth row of Pascal's triangle, generalised by the binomial theorem to *(x + y)<sup>n</sup>*. Memorise the cube directly; re-deriving it each time is slower.

<Math math="\left(a+y\right)^{3}=x^{3}+3xy^{2}+3x^{2}y+y{3}" display />

Cube of a difference. Signs alternate because each occurrence of the negative term flips the sign. The *x<sup>3</sup>* term stays positive because *(-y)<sup>0</sup>* is positive; the *y<sup>3</sup>* term is negative because *(-y)<sup>3</sup>* is negative.

<Math math="\left(a-y\right)^{3}=x^{3}-3xy^{2}-3x^{2}y-y{3}" display />

Sum of cubes factors into a linear and a quadratic factor. The quadratic *x<sup>2</sup> - xy + y<sup>2</sup>* is irreducible over the reals (its discriminant is *-3y<sup>2</sup>*), so this is as far as the factorisation goes without complex numbers.

<Math math="x^{3}+y^{3}=\left(x+y\right)\left(x^{2}-xy+y^{2}\right)" display />

Difference of cubes factors the same way, with the sign of the middle term of the quadratic reversed. Together with the sum-of-cubes formula, this is the *n = 3* case of the general *x<sup>n</sup> ± a<sup>n</sup>* identities below.

<Math math="x^{3}-y^{3}=\left(x-y\right)\left(x^{2}+xy+y^{2}\right)" display />

Difference of even powers factors by treating *x<sup>n</sup>* and *y<sup>n</sup>* as a difference of squares. Iterate this halving until the exponent is odd or one to get a complete real factorisation, alternating sum and difference factors at each level.

<Math math="x^{2x}-y^{2x}=\left(x^{n}-y^{n}\right)\left(x^{n}+y^{n}\right)" display />

For odd *n*, both the difference and the sum of *n*-th powers factor cleanly. The difference identity holds for all *n*. The sum identity is restricted to odd *n* because *x<sup>n</sup> + a<sup>n</sup>* has *x = -a* as a root only when raising *-a* to *n* recovers *-a<sup>n</sup>*.

<Math math="if\text{ }n \text{ }is \text{ }odd \text{ }then" display />

Difference of *n*-th powers (any positive integer *n*). The long factor is a geometric sum *Σ x<sup>n-1-k</sup>a<sup>k</sup>* and reduces to *(x<sup>n</sup> - a<sup>n</sup>)/(x - a)* when *x ≠ a*. The geometric series sum follows from this identity.

<Math math="x^{n}-a^{n}=\left(x-a\right)\left(x^{n-1}+ax^{n-2}+...+a^{n-1}\right)" display />

Sum of *n*-th powers, for odd *n*, factors with alternating signs in the long factor. Substituting *-a* for *a* in the difference formula flips the sign on every other term, producing the alternation.

<Math math="x^{n}+a^{n}=\left(x+a\right)\left(x^{n-1}-ax^{n-2}+a^{2}x^{n-2}+...+a^{n-1}\right)" display />

<Notice type="tip">
For sum of *n*-th powers the linear factor *(x + a)* only exists when *n* is odd. When *n* is even, *x<sup>n</sup> + a<sup>n</sup>* has no real linear factors and is irreducible over the reals (for *n = 2*) or factors only into quadratics.
</Notice>

---

## Progression

URL: http://math.mkabumattar.com/basics/progression

Arithmetic, geometric, and harmonic progressions with their means and key relationships.

A progression is a sequence whose terms follow a fixed rule. The three classical progressions (arithmetic, geometric, harmonic) each generate a *mean* (AM, GM, HM) and a closed-form expression for the *n*-th term and partial sum, so you do not have to compute term by term. HP is the reciprocal of an AP, and for positive numbers the three means satisfy *AM ≥ GM ≥ HM*.

## Arithmetic Progression (AP)

An arithmetic progression is a sequence in which each term after the first is obtained by adding a constant *d* (the common difference) to the preceding term. The generic AP is *a, a + d, a + 2d, a + 3d, …* where *a* is the first term. Plotting the terms on a number line gives equally-spaced points.

<Diagram alt="Arithmetic progression on a number line with equal gaps of d" viewBox="0 0 240 90" caption="Successive terms of an AP are equally spaced; each gap equals the common difference d.">
  <line x1="10" y1="55" x2="230" y2="55" />
  <circle cx="30" cy="55" r="2.5" fill="currentColor" />
  <circle cx="80" cy="55" r="2.5" fill="currentColor" />
  <circle cx="130" cy="55" r="2.5" fill="currentColor" />
  <circle cx="180" cy="55" r="2.5" fill="currentColor" />
  <text x="24" y="72" font-size="10" fill="currentColor" stroke="none">a₁</text>
  <text x="74" y="72" font-size="10" fill="currentColor" stroke="none">a₂</text>
  <text x="124" y="72" font-size="10" fill="currentColor" stroke="none">a₃</text>
  <text x="174" y="72" font-size="10" fill="currentColor" stroke="none">a₄</text>
  <path d="M 32 42 L 78 42" />
  <path d="M 78 42 L 74 39" />
  <path d="M 78 42 L 74 45" />
  <path d="M 82 42 L 128 42" />
  <path d="M 128 42 L 124 39" />
  <path d="M 128 42 L 124 45" />
  <path d="M 132 42 L 178 42" />
  <path d="M 178 42 L 174 39" />
  <path d="M 178 42 L 174 45" />
  <text x="51" y="36" font-size="10" fill="currentColor" stroke="none">d</text>
  <text x="101" y="36" font-size="10" fill="currentColor" stroke="none">d</text>
  <text x="151" y="36" font-size="10" fill="currentColor" stroke="none">d</text>
</Diagram>

**• n<sup>th</sup> term of AP**

The *n*-th term adds the common difference *n - 1* times to the first term. The shift of one reflects that the first term contributes zero copies of *d*.

<Math math="t_{n}=a+(n-1)d" display />

(Where t<sub>n</sub> = n<sup>th</sup> term, a = first term, 1 = last term and d = common difference)

**• Number of term in an AP**

Solving the *n*-th term formula for *n* counts the terms between a known first and last term. Use it when the endpoints of a finite AP are visible but the length is not (for instance, counting multiples of three between two given values).

<Math math="n=\frac{\left(1+a\right)}{d}+1" display />

(Where n = number of terms, a = first term, 1 = last term and d = common difference)

**• Sum of first n terms in AP**

The partial sum is *n* times the average of the first and last terms. Gauss's pairing trick gives the shortcut: pair the first term with the last, the second with the second-to-last, and so on. Each pair sums to *a + l*, and there are *n/2* such pairs. The two forms below differ only in whether the last term is given directly or expressed through *a* and *d*.

<Math math="S_{n}=\left(\frac{n}{2}\right)\left[2a+\left(n+1\right)d\right]" display />

Equivalent form using the last term directly. This is the most compact way to write the partial sum when both endpoints are known:

<Math math="\text{ }\text{ }\text{ }\text{ }=\left(\frac{n}{2}\right)\left[a+1\right]" display />

(Whare a = first term, d = commn difference and 1 = n<sup>th</sup> term = a +(n-1)d)

**• Arithmtic Mean**

If *a, b, c* are in AP, then *b* is the Arithmetic Mean (A.M.) of *a* and *c*. The AM is the centre value that two endpoints determine to form a three-term AP.

The middle term of three numbers in AP is the average of the outer two. The constant-difference condition *b - a = c - b* rearranges to:

<Math math="b=\frac{a+c}{2}" display />

A.M between two numbers *a* and *b*. Any two numbers can be treated as the outer terms of a three-term AP whose centre is their average:

<Math math="\text{ }\text{ }=\frac{a+b}{2}" display />

If a, a<sub>1</sub>, a<sub>2</sub>, ... a<sub>n</sub>, b are in AP we can say that a<sub>1</sub>, a<sub>2</sub>, ... a<sub>n</sub> are the n Arithmetic Means berween a and b.

**• Nots**

Most AP problems simplify when the terms are taken symmetrically around the central term. The common-difference contributions cancel pairwise on summing: for odd counts the *d* terms cancel directly, and for even counts they cancel in symmetric pairs of *±d*, *±3d*, and so on.

<Math math="3\text{ }terms\text{ }: \left(a-d\right), a,\left(a+d\right)" display />

<Math math="4\text{ }terms\text{ }: \left(a-3d\right), \left(a-d\right),\left(a+d\right),\left(a+3d\right)" display />

<Math math="5\text{ }terms\text{ }: \left(a-2d\right), \left(a-d\right),a,\left(a+d\right),\left(a+2d\right)" display />

T<sub>n</sub> = S<sub>n</sub> - S<sub>n-1</sub>

## Geometric Progression (GP)

A geometric progression is a sequence of non-zero numbers in which the ratio of consecutive terms is a fixed constant *r*. AP is built on addition of a fixed *d*; GP is built on multiplication by a fixed *r*. The parallel structure carries through to the *n*-th term, sum, and mean formulas below. Plotting GP terms gives exponential growth or decay.

**• n<sup>th</sup> term of GP**

The *n*-th term multiplies the first term by the common ratio *n - 1* times. The exponent shift mirrors the *(n - 1)d* in the AP case: the first term contributes the ratio zero times.

<Math math="t_{n}=ar^{\left(n-1\right)}" display />

(Where t<sub>n</sub> = n<sup>th</sup>  term, a = first term, r = common ratio and n = number of terms)

**• Sum of first n Terms in GP**

The partial sum is a closed-form geometric series. The two branches differ only in sign arrangement so the numerator stays positive in each regime; they are algebraically equivalent. The derivation is the standard *S - rS* telescoping trick, which collapses all interior terms.

<Math math="Sn = \left\{ \begin{array}{l l} \frac{a\left(r^{n}-1\right)}{r-1} & \quad \text{ }if\text{ }\left(r > 1\right)\\  \frac{a\left(1-r^{n}\right)}{1-r} & \quad \text{ }if\text{ }\left(r < 1\right)\ \end{array} \right." display />

(Where a = the first term, r = common ratio, n = number of terms)

Note: When n = ∞, then

For a convergent infinite GP (ratio strictly between 0 and 1), the series sums to a finite limit. The term *r<sup>n</sup>* goes to zero as *n* grows, so the partial-sum formula collapses to the ratio shown. This is the formula behind repeating-decimal-to-fraction conversion.

<Math math="S_{\infty}=\frac{a}{1-r} \text{ }\left(if\text{ } 0<r<1\right)" display />

**• Geometic Mean**

If three non-zero numbers *a, b, c* are in GP, then *b* is the Geometric Mean (GM) of *a* and *c*. The GM plays the multiplicative role the AM plays additively: it is the value for which the ratios *b/a* and *c/b* match.

The geometric mean of two numbers is the square root of their product. The constant-ratio condition gives *b<sup>2</sup> = ac*:

<Math math="b=\sqrt{ac}" display />

The GM of two arbitrary numbers *a* and *b* follows the same template:

<Math math="b=\sqrt{ab}" display />

(Note that if a and b are of opposite sign, their GM is not defined)

**• Note**

If *a, b, c* are in GP, the following ratio identity holds. It follows from *b/a = c/b*:

<Math math="\frac{a-b}{b-c}=\frac{a}{b}" display />

Most GP problems simplify when the terms are taken in a symmetric multiplicative form. The central term is unaffected, and ratio terms pair as *r* and *1/r* so products simplify.

<Math math="3\text{ }terms\text{ }:\left(\frac{b}{r}\right),a,ar" display />

<Math math="5\text{ }terms\text{ }:\left(\frac{a}{r^{2}}\right),\left(\frac{a}{r}\right),a,ar,ar^{2}" display />

## Harmonic Progression (HP)

A harmonic progression is a sequence of non-zero numbers whose reciprocals form an AP. HP arises in averaging rates (speeds over a fixed distance, parallel resistances, capacitances in series) where the linear quantity is the reciprocal of what you measure directly.

A harmonic progression is defined by the reciprocals of its terms forming an AP:

<Math math="\text{ }\frac{1}{a1},\frac{1}{a2},\frac{1}{a3}, ... \frac{1}{an}\text{ }" display />

are in AP

• If *a, (a+d), (a+2d), …* are in AP, the *n*-th term of the AP is *a + (n-1)d*. So if

<Math math="\text{ }\frac{1}{a},\frac{1}{\left(a+d\right)},\frac{1}{\left(a+2d\right)},...\text{ }" display />

are in HP, the *n*-th term of the HP is the reciprocal of the corresponding AP term. HP has no independent closed form for the partial sum; unlike AP and GP, partial sums of HPs are not generally expressible in elementary closed form.

<Math math="\text{ }HP=\frac{1}{\left[a+\left(n-1\right)d\right]}\text{ }" display />

• If *a, b, c* are in HP, then *b* is the Harmonic Mean (HM) of *a* and *c*. By definition, the HM is the reciprocal of the AM of the reciprocals. For positive inputs it is always less than or equal to both the AM and the GM.

The harmonic mean is twice the product divided by the sum, from inverting *2/b = 1/a + 1/c*:

<Math math="\text{ }b=\frac{\left(2ac\right)}{\left(a+c\right)}\text{ }" display />

• The HM of two arbitrary numbers follows the same template:

<Math math="\text{ }b=\frac{\left(2ab\right)}{\left(a+b\right)}\text{ }" display />

• If *a, b, c* are in HP, the relation reads more cleanly in reciprocals. *1/b* is the average of *1/a* and *1/c*, which is the AP condition on the reciprocals:

<Math math="\text{ }, \text{ }\frac{2}{b}=\frac{1}{a}+\frac{1}{c}" display />

## Relationship between AM, GM and HM of Two Numbers

For any two positive numbers the three means stand in a fixed relationship. The identity below links the three: knowing any two of *AM, GM, HM* determines the third. The chain *AM ≥ GM ≥ HM* (with equality only when the inputs are equal) is the two-number case of the classical inequality of means.

If GM, AM, and HM are the Geometric, Arithmetic, and Harmonic means of two positive numbers, then the GM is the geometric mean of the AM and HM. Squaring gives the symmetric form: *GM<sup>2</sup>* equals their product.

<Math math="GM^{2}= AM \times HM" display />

<Notice type="tip">
For any two positive numbers, *AM &ge; GM &ge; HM* with equality only when the numbers are equal. This ordering generalizes to *n* numbers and is the basis for many classical inequalities.
</Notice>

---

## Complex Numbers

URL: http://math.mkabumattar.com/basics/complex-numbers

The imaginary unit, arithmetic operations, modulus, and complex conjugate.

A complex number has the form *a + ib*, where *a* and *b* are real and *i* is the imaginary unit. Treating *i* as an algebraic symbol with *i<sup>2</sup> = -1* lets arithmetic proceed like ordinary polynomial arithmetic, with the modulus and conjugate adding extra structure. Geometrically, every complex number is a point in the Argand plane: real part on the horizontal axis, imaginary part on the vertical.

<Diagram alt="Argand diagram showing point z equals a plus i b with projections" viewBox="0 0 220 160" caption="A complex number z = a + ib plotted on the Argand plane; the real and imaginary parts are the horizontal and vertical projections from the origin.">
  <line x1="10" y1="120" x2="210" y2="120" />
  <line x1="40" y1="10" x2="40" y2="150" />
  <line x1="40" y1="120" x2="150" y2="50" />
  <line x1="150" y1="120" x2="150" y2="50" stroke-dasharray="3 3" />
  <line x1="40" y1="50" x2="150" y2="50" stroke-dasharray="3 3" />
  <circle cx="150" cy="50" r="2.5" fill="currentColor" />
  <text x="156" y="46" font-size="10" fill="currentColor" stroke="none">z = a + ib</text>
  <text x="200" y="116" font-size="10" fill="currentColor" stroke="none">Re</text>
  <text x="46" y="18" font-size="10" fill="currentColor" stroke="none">Im</text>
  <text x="92" y="134" font-size="10" fill="currentColor" stroke="none">a</text>
  <text x="26" y="88" font-size="10" fill="currentColor" stroke="none">b</text>
</Diagram>

The imaginary unit is defined as a square root of negative one. This single symbol extends the reals into an algebraically closed field in which every non-constant polynomial has a root (the fundamental theorem of algebra). The choice of *i* over *-i* is conventional; both satisfy the defining relation below.

<Math math="i=\sqrt{-1}" display />

Squaring the imaginary unit returns the defining relation. This rule is what distinguishes complex arithmetic from polynomial arithmetic in *i*: every *i<sup>2</sup>* is replaced by *-1*. Higher powers of *i* cycle through *i, -1, -i, 1* with period four.

<Math math="i^{2}=-1" display />

Square roots of negative real numbers factor out *i*. The non-negative *a* under the inner radical keeps that operation real-valued; the *i* outside carries the sign. The rule blocks the common false step *√(-a)·√(-b) = √(ab)* when both operands are negative.

<Math math="\sqrt{-a}=i\sqrt{a}\text{ }\text{ }\text{ }\left(if\text{ }a\geq 0\right)" display />

Addition combines real and imaginary parts component-wise, like vector addition in *R<sup>2</sup>*. On the Argand diagram, two complex numbers add as the parallelogram-rule sum of their position vectors.

<Math math="\left(a+ib\right)+\left(c+id\right)=\left(a+c\right)+\left(b+d\right)i" display />

Subtraction is also component-wise. Reversing the sign of the second operand reflects it through the origin before the vector addition.

<Math math="\left(a+ib\right)-\left(c+id\right)=\left(a-c\right)+\left(b-d\right)i" display />

Multiplication uses the distributive law with *i<sup>2</sup> = -1*. The cross terms *iad* and *ibc* sit in the imaginary part; *i<sup>2</sup>bd = -bd* moves into the real part. Geometrically, multiplication rotates and scales: moduli multiply and arguments add.

<Math math="\left(a+ib\right)\left(c+id\right)=\left(ac+bd\right)+\left(ab+bc\right)i" display />

Multiplying a complex number by its conjugate gives a real result, which is how complex denominators are rationalised. The cross terms *iab* and *-iab* cancel, leaving the sum of squares *a<sup>2</sup> + b<sup>2</sup>*, the squared distance from the origin in the Argand plane.

<Math math="\left(a+ib\right)\left(c-ib\right)=a^{2}+b^{2}" display />

The modulus is the distance from the origin in the complex plane. By the Pythagorean theorem applied to the horizontal and vertical projections in the diagram, the length of the vector from 0 to *a + ib* is *√(a<sup>2</sup> + b<sup>2</sup>)*.

<Math math="\left|a+ib\right|=\sqrt{a^{2}+b^{4}}\text{ }\text{ }\text{ }\left(Complex\text{ }Modulus\right)" display />

The conjugate flips the sign of the imaginary part, reflecting the number across the real axis. Conjugation is an involution: applying it twice recovers the original. It also commutes with addition and multiplication, making it a field automorphism of the complex numbers.

<Math math="\overline{a+ib}=a+ib\text{ }\text{ }\text{ }\left(Complex\text{ }Conjugate\right)" display />

A number times its conjugate equals the modulus squared. This expresses *|z|<sup>2</sup>* without taking a square root, which is the cleaner quantity to work with whenever the modulus appears under a power.

<Math math="\left(\overline{a+ib}\right)\left(a+ib\right)=\left|a+ib\right|^{2}" display />

<Notice type="note">
To divide complex numbers, multiply numerator and denominator by the denominator's conjugate. The denominator becomes the modulus squared (a real number), and the quotient sits in standard *a + ib* form.
</Notice>

---

## Vectors

URL: http://math.mkabumattar.com/basics/vectors

Dot and cross products, position and unit vectors, scalar and vector triple products.

A vector in three-space carries magnitude and direction, written in the standard *i*, *j*, *k* basis. The dot and cross products give a scalar and a vector respectively, and combine into triple products that encode signed volumes and orthogonality relations. Vector addition obeys the parallelogram rule: the sum of two vectors is the diagonal of the parallelogram they span.

<Diagram alt="Two vectors u and v with their sum as the parallelogram diagonal" viewBox="0 0 220 160" caption="Vector addition by the parallelogram rule: u + v is the diagonal from the common origin to the opposite corner.">
  <path d="M 30 130 L 130 100" />
  <path d="M 130 100 L 124 96" />
  <path d="M 130 100 L 124 104" />
  <text x="74" y="108" font-size="10" fill="currentColor" stroke="none">u</text>
  <path d="M 30 130 L 70 50" />
  <path d="M 70 50 L 66 56" />
  <path d="M 70 50 L 74 56" />
  <text x="36" y="86" font-size="10" fill="currentColor" stroke="none">v</text>
  <path d="M 130 100 L 170 20" stroke-dasharray="3 3" />
  <path d="M 70 50 L 170 20" stroke-dasharray="3 3" />
  <path d="M 30 130 L 170 20" />
  <path d="M 170 20 L 163 22" />
  <path d="M 170 20 L 167 27" />
  <text x="106" y="64" font-size="10" fill="currentColor" stroke="none">u + v</text>
  <circle cx="30" cy="130" r="2.5" fill="currentColor" />
</Diagram>

## Dot Product

The dot product multiplies corresponding components and sums them, returning a scalar that measures projected length and angular alignment. Geometrically *a · b = |a||b|cos θ*, so it vanishes for orthogonal vectors and is maximal when the vectors are parallel.

The first vector in component form, its coordinates with respect to the *i, j, k* basis:

<Math math="Let\text{ }\overline{a}=a_{1}i+a_{2}j+a_{3}k" display />

The second:

<Math math="\text{ }\text{ }\text{ }\text{ }\text{ }\overline{b}=b_{1}i+b_{2}j+b_{3}k" display />

Expanded form before basis cancellations. Distributing gives nine terms; six vanish because the basis vectors are mutually orthogonal.

<Math math="Then\text{ }\overline{a}.\overline{b}=\left(a_{1}i+a_{2}j+a_{3}k\right)\left(b_{1}i+b_{2}j+b_{3}k\right)" display />

After applying orthogonality of the basis, the dot product reduces to a sum of products. The diagonal terms *i·i = j·j = k·k = 1* survive; off-diagonal terms vanish. This component formula is the standard tool for angles, projections, and orthogonality checks.

<Math math="\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }=a_{1}b_{1}+a_{2}b_{2}+a_{3}b_{3}\text{ }\text{ }\left(Scalar\text{ }Quantity\right)" display />

The dot product is commutative (*a · b = b · a*) because real-number multiplication of components is commutative:

<Math math="Note:\text{ }\overline{a}.\overline{b}=\overline{b}.\overline{a}" display />

## Cross Product

The cross product returns a vector perpendicular to both inputs, whose magnitude equals the area of the parallelogram they span. It is anti-commutative and direction-sensitive, with orientation fixed by the right-hand rule. Unlike the dot product, it has a clean form only in three dimensions.

The first vector:

<Math math="Let\text{ }\overline{a}=a_{1}i+a_{2}j+a_{3}k" display />

The second:

<Math math="\text{ }\text{ }\text{ }\text{ }\text{ }\overline{b}=b_{1}i+b_{2}j+b_{3}k" display />

The cross product is a 3×3 determinant expanded along the basis row. The first row carries the basis vectors and the remaining two rows the components, so cofactor expansion along the top row gives the standard component formula.

<Math math="Then\text{ }\overline{\mathbf{a}} \times \overline{\mathbf{b}} =  \begin{vmatrix}\mathbf{i} & \mathbf{j} & \mathbf{k} \\ a_{1} &  a_{2} & a_{3} \\ b_{1} &  b_{2} & b_{3} \end{vmatrix}" display />

Swapping the operands does not preserve the product. Unlike the dot product, the cross product is direction-sensitive; swapping reverses orientation by the right-hand rule.

<Math math="Note: \text{ }\overline{a}\times\overline{b}\neq\overline{b}\times\overline{a}" display />

Swapping the operands reverses the sign: the cross product is anti-commutative. This follows from the determinant view, since exchanging two rows of a matrix negates its determinant.

<Math math="\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\overline{a}\times\overline{b}=-\overline{b}\times\overline{a}" display />

## Position Vector

A position vector locates a point relative to the origin and packages its Cartesian coordinates as a vector. It is the bridge between point-set geometry and vector algebra: every geometric question about points becomes an algebraic question about their position vectors.

For point *P(x, y, z)*, the position vector is the arrow from the origin to the point.

The arrow from the origin to a point is the vector with the point's coordinates as components, written as differences from the origin:

<Math math="\overline{OP}=\left(x-0\right)i+\left(y-0\right)j+\left(z-0\right)k" display />

After the zero subtractions vanish:

<Math math="\text{ }\text{ }\text{ }\text{ }\text{ }=xo+yj+zk" display />

The magnitude (length) of the position vector by the Pythagorean theorem in three dimensions. The squared length is the sum of squared components; the square root gives the Euclidean distance from the origin to the point.

<Math math="\left|\overline{r}\right|=\sqrt{x^{2}+y^{2}+z^{2}}" display />

## Unit Vector

A unit vector preserves direction while normalising magnitude to one. Divide any non-zero vector by its magnitude to get its unit vector. Unit vectors isolate direction from magnitude, which is what you want whenever the question is "in which direction" rather than "how far".

A unit vector is the vector divided by its own magnitude. The hat notation marks the result as normalised; the construction works for any non-zero vector.

<Math math="\widehat{a} = \frac{\overline{a}}{\left|\overline{a}\right|}" display />

An arbitrary vector in component form:

<Math math="Let\text{ }\overline{a}=a_{1}i+a_{2}j+a_{3}k" display />

Its magnitude:

<Math math="\text{ }\text{ }\text{ }\left|\overline{a}\right|=\sqrt{a_{1}^{2}+a_{2}^{2}+a_{3}^{2}}" display />

The unit vector in component form, each component divided by the magnitude. The squared components sum to one, confirming unit length.

<Math math="\therefore\text{ }\widehat{a}=\frac{a_{1}i+a_{2}j+a_{3}k}{\sqrt{a_{1}^{2}+a_{2}^{2}+a_{3}^{2}}}" display />

## Scalar Triple Product

The scalar triple product *a · (b × c)* returns the signed volume of the parallelepiped spanned by the three vectors. It vanishes when they are coplanar. The sign records orientation (positive for right-handed, negative for left-handed); the absolute value is the geometric volume.

Three vectors in component form, used in the determinant below:

<Math math="Lat\text{ }\text{ }\overline{a}=a_{1}^{i}+a_{2}^{j}+a_{3}^{k}" display />

<Math math="\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\overline{b}=b_{1}^{i}+b_{2}^{j}+b_{3}^{k}" display />

<Math math="\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\overline{c}=c_{1}^{i}+c_{2}^{j}+c_{3}^{k}" display />

The scalar triple product is the determinant of the matrix whose rows are the three vectors. The 3×3 determinant and the signed volume are the same object, so the parallelepiped volume formula comes straight from linear algebra.

<Math math="\mathbf{\overline{a}}.\left(\mathbf{\overline{b}}\times\mathbf{\overline{c}}\right)=\begin{vmatrix}a_{1}&a_{2}&a_{3} \\ b_{1}&b_{2}&b_{3} \\ c_{1}& c_{2}&c_{3} \end{vmatrix}" display />

Compact bracket notation, the standard shorthand whenever the triple product appears in identities:

<Math math="\overline{a}.\left(\overline{b}\times\overline{c}\right)=\left[\overline{a}\text{ }\text{ }\overline{b}\text{ }\text{ }\overline{c}\right]" display />

**Propertes**

Cyclic permutation of the three vectors leaves the scalar triple product unchanged. Rotating rows is an even permutation, so the determinant's sign is preserved.

<Math math="\left[a\text{ }b\text{ }c\right]=\left[b\text{ }c\text{ }a\right]=\left[c\text{ }a\text{ }b\right]" display />

Swapping two adjacent vectors reverses the sign, by the row-swap rule for determinants:

<Math math="\left[a\text{ }b\text{ }c\right]=-\left[a\text{ }b\text{ }c\right]" display />

If any vector is repeated, the triple product vanishes. The three vectors collapse onto a plane and the spanned volume drops to zero. The determinant view says the same: a matrix with two equal rows is singular.

<Math math="\left[a\text{ }a\text{ }b\right]=\left[b\text{ }a\text{ }a\right]=\left[a\text{ }b\text{ }a\right]=0" display />

The dot and cross can be interchanged within the triple product. This is what justifies the bracket notation: the value depends on the three vectors, not on where the dot or cross sits.

<Math math="a.\left(b\times c\right)=\left(a\times b\right).c" display />

Coplanar vectors give a zero triple product, since the spanned volume is zero. Conversely, a non-zero triple product is a quick test that three vectors form a linearly independent set in three-space.

<Math math="if\text{ }a,b,c\text{ }are\text{ }coplanar\text{ }then\text{ }\left[a\text{ }b\text{ }c\right]=0" display />

The absolute value gives the parallelepiped volume:

<Math math="Voulme\text{ }of\text{ }parallelopoped=\left[a\text{ }b\text{ }c\right]" display />

A tetrahedron has one-sixth that volume. Three of its triangular faces partition the parallelepiped into six congruent tetrahedra:

<Math math="Voulme\text{ }of\text{ }tetrahedron=\frac{1}{6}\text{ }\left[a\text{ }b\text{ }c\right]" display />

## Vector Triple Product

The vector triple product *a × (b × c)* gives a vector in the plane of *b* and *c*, expanded by the BAC-CAB identity. The result is perpendicular to the inner *b × c*, which is itself perpendicular to that plane, so the outer cross brings the answer back into the *b*-*c* plane.

The three vectors in component form:

<Math math="Let\text{ }\text{ }\overline{a}=a_{1}^{i}+a_{2}^{j}+a_{3}^{k}" display />

<Math math="\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\overline{b}=b_{1}^{i}+b_{2}^{j}+b_{3}^{k}" display />

<Math math="\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\overline{c}=c_{1}^{i}+c_{2}^{j}+c_{3}^{k}" display />

Then,

The BAC-CAB expansion shows the result is a linear combination of the inner two vectors. The mnemonic "back cab" gives the order: outer vector dotted with the *far* inner vector multiplies the *near* one, minus the symmetric term.

<Math math="a\times\left(b\times c\right)=\left(a.c\right)b-\left(a.b\right)c" display />

Cyclic permutation of the BAC-CAB identity by relabelling *a → b → c → a*:

<Math math="b\times\left(c\times a\right)=\left(b.a\right)c-\left(b.c\right)a" display />

Another cyclic version. Summing the three cyclic forms gives zero (the Jacobi identity), which is what makes the cross product into a Lie bracket.

<Math math="c\times\left(a\times b\right)=\left(c.b\right)a-\left(c.a\right)b" display />

The cross product is not associative. Parentheses change the result. The groupings *a × (b × c)* and *(a × b) × c* lie in different planes (the *b-c* plane and the *a-b* plane), so the parentheses are not optional.

<Math math="Note:\text{ }\text{ }a\times\left(b\times c\right)\neq\left(a\times b\right)\times c" display />

---

## Probability

URL: http://math.mkabumattar.com/basics/probability

Probability rules including complementary events, addition, independent events, conditional probability, and Bayes' formula.

Probability quantifies how likely an event is on a scale from 0 (impossible) to 1 (certain). The rules below combine probabilities of related events (unions, intersections, conditional events) and end with Bayes' formula, which inverts a conditional probability to update beliefs from evidence. Two-event relations are easiest to read off a Venn diagram, where set operations map directly to probability operations.

<Diagram alt="Venn diagram of two overlapping events A and B with shaded intersection" viewBox="0 0 220 160" caption="Two-event Venn diagram: the shaded lens is A ∩ B, the full union A ∪ B covers both circles.">
  <rect x="10" y="20" width="200" height="120" />
  <circle cx="90" cy="80" r="45" />
  <circle cx="135" cy="80" r="45" />
  <path d="M 112.5 41 A 45 45 0 0 1 112.5 119 A 45 45 0 0 1 112.5 41 Z" fill="currentColor" opacity="0.15" stroke="none" />
  <text x="60" y="84" font-size="11" fill="currentColor" stroke="none">A</text>
  <text x="160" y="84" font-size="11" fill="currentColor" stroke="none">B</text>
  <text x="103" y="84" font-size="9" fill="currentColor" stroke="none">A∩B</text>
  <text x="14" y="33" font-size="9" fill="currentColor" stroke="none">S</text>
</Diagram>

## Basics

The classical definition counts favourable outcomes against equally likely total outcomes. It applies cleanly only when the sample space partitions into equally likely cases. For general spaces, probability is defined axiomatically as a measure.

Probability of an event A is

P(A)=(number of favourable outcomes)

(total number of all outcomes)

## Probability Range

Probability is bounded between zero and one, with the endpoints reserved for impossible and certain events. The bounds are the first Kolmogorov axiom and the source of every sanity check: any computed value outside this range signals an error in setup.

<Math math="0\leq P\left(A\right)\leq 1" display />

## Rule of Complementary Events

An event and its complement (everything *not A*) exhaust the sample space, so their probabilities sum to one. Use this when *P(A<sup>C</sup>)* is easier to compute than *P(A)* directly. "At least one" events are usually easiest as *1 - P(none)*.

<Math math="P\left(A^{C}\right)+P\left(A\right)=1" display />

## Rule of Addition

The inclusion-exclusion identity for two events: add the individual probabilities, then subtract the overlap so it is not counted twice. The Venn diagram shows why: the lens *A ∩ B* lies inside both *A* and *B*, so summing the circles counts it twice. The identity generalises to *n* events with alternating inclusion-exclusion terms.

<Math math="P\left(A\cup B\right)=P\left(\right)+P\left(B\right)-P\left(A\cap B\right)" display />

## Disjoint Events

Two events are disjoint (mutually exclusive) when they cannot both occur in the same trial. In the Venn diagram the two circles do not overlap, so the inclusion-exclusion correction drops out and the addition rule reduces to *P(A ∪ B) = P(A) + P(B)*.

Events A and B are disjoint if their intersection has zero probability:

<Math math="P\left(A\cap B\right)=0" display />

## Independent Events

Independence means the occurrence of one event leaves the probability of the other unchanged; the joint probability is the product of the marginals. Independence is a statement about probabilities, not about the events themselves. Two events with overlapping outcomes can still be independent if the overlap matches the product condition exactly.

Events A and B are independent if their joint probability factors as the product of the marginals:

<Math math="P\left(A\cap B\right)=P\left(A\right).P\left(B\right)" display />

## Conditional Probability

Conditional probability re-normalises by restricting the sample space to outcomes in which the conditioning event has occurred. The fraction below counts the overlap *A ∩ B* against the new total *B*, which is what "given *B*" reduces the universe to. This definition underlies Bayes' formula and the theory of stochastic processes built on conditioning.

<Math math="P\left(A\mid B\right)=\frac{P\left(A\cap B\right)}{P\left(B\right)}" display />

## Bayes Formuala

Bayes' formula inverts a conditional probability, switching what is given and what is being asked, by re-weighting with the ratio of marginals. It is the central tool for inference: from a prior *P(A)* and a likelihood *P(B|A)*, it returns the posterior *P(A|B)* after observing *B*. The denominator *P(B)* is often expanded by the law of total probability into a sum over disjoint hypotheses.

<Math math="P\left(A\mid B\right)=P\left(B\mid A\right).\frac{P\left(A\right)}{P\left(B\right)}" display />

<Notice type="warning">
Disjoint events are not independent. If *A* and *B* are disjoint with positive probabilities, knowing one occurred makes the other impossible, so they are maximally *dependent*. Confusing the two concepts is a common error.
</Notice>

---

# Geometry

## Area

URL: http://math.mkabumattar.com/geometry/area

Area formulas for common 2D shapes: squares, rectangles, parallelograms, trapezoids, circles, ellipses, and triangles.

Area measures the extent of a two-dimensional region in square units (square metres, square inches, and so on). The formulas below cover squares, rectangles, parallelograms, trapezoids, circles, ellipses, and triangles, each expressed in terms of the parameters that describe the figure: side, base, height, radius, or semi-axes.

<Diagram alt="Generic rectangle with width and height labels and a unit grid" viewBox="0 0 220 140" caption="Area measures a 2D region in square units.">
  <defs>
    <pattern id="area-grid" width="20" height="20" patternUnits="userSpaceOnUse">
      <path d="M 20 0 L 0 0 0 20" fill="none" stroke="currentColor" stroke-width="0.5" opacity="0.4" />
    </pattern>
  </defs>
  <rect x="20" y="20" width="160" height="80" fill="url(#area-grid)" stroke="currentColor" />
  <rect x="20" y="20" width="160" height="80" fill="currentColor" opacity="0.08" stroke="none" />
  <line x1="20" y1="115" x2="180" y2="115" stroke-width="0.8" />
  <line x1="20" y1="112" x2="20" y2="118" />
  <line x1="180" y1="112" x2="180" y2="118" />
  <text x="100" y="128" font-size="10" text-anchor="middle" fill="currentColor" stroke="none">width</text>
  <line x1="195" y1="20" x2="195" y2="100" stroke-width="0.8" />
  <line x1="192" y1="20" x2="198" y2="20" />
  <line x1="192" y1="100" x2="198" y2="100" />
  <text x="205" y="64" font-size="10" fill="currentColor" stroke="none">height</text>
</Diagram>

**Square**

<Diagram src="square.svg" alt="Square" />

Four equal sides of length *a*. Area is the side squared:

<Math math="Area=a^{2}" display />

**Rectangle**

<Diagram src="rectangle.svg" alt="Rectangle" />

Sides *a* and *b*. Area is the product of the two sides:

<Math math="Area=ab" display />

**Parallelgram**

<Diagram src="parallelgram.svg" alt="Parallelgram" />

Base *b* and perpendicular height *h* (the distance between the base and its opposite side, not the slanted edge). Area matches a rectangle with the same base and height:

<Math math="Area=bh" display />

**Trapezoid**

<Diagram src="trapezoid.svg" alt="Trapezoid" />

Parallel sides *b1* and *b2*, with *h* the perpendicular distance between them (not the slanted leg). Area is the average of the parallel sides times that height:

<Math math="Area=\left(\frac{1}{2}\right)\left(b1+b2\right)h" display />

**Circle**

<Diagram src="circle.svg" alt="Circle" />

Radius *r* is the distance from centre to edge. Area scales with the square of *r*; doubling the radius quadruples the area:

<Math math="Area=\pi r^{2}" display />

**Ellipse**

<Diagram src="ellipse.svg" alt="Ellipse" />

*t1* and *t2* are the semi-major and semi-minor axes, half the lengths of the two principal diameters. Area is *π* times their product. When the axes are equal this reduces to the circle formula:

<Math math="Area=\pi\left(t1\times t2\right)" display />

**Triangle**

<Diagram src="triangle.svg" alt="Triangle" />

Base *b* and perpendicular height *h* (dropped onto that base, not a slanted side). Area is half the base times the height. Two such triangles tile a parallelogram of area *bh*:

<Math math="Area=\left(\frac{1}{2}\right)bh" display />

**Equilateral Triangle**

<Diagram src="equilateral_triangle.svg" alt="Equilateral Triangle" />

All three sides equal *a*, all angles 60°. Perpendicular height is *a√3/2*, which reduces the general triangle formula to a closed form in *a*:

<Math math="Area=\left(\frac{\sqrt{3}}{4}\right)a^{2}" display />

---

## Perimeter

URL: http://math.mkabumattar.com/geometry/perimeter

Perimeter formulas for common 2D shapes: squares, rectangles, parallelograms, circles, triangles, and regular polygons.

Perimeter is the total length of a planar shape's boundary, measured in linear units (metres, inches, and so on). For polygons it is the sum of side lengths. For a circle the analogous quantity is the circumference, *2πr*. The formulas below cover squares, rectangles, parallelograms, circles, triangles, and regular polygons.

<Diagram alt="Polygon traced with arrows along edges showing direction of travel" viewBox="0 0 220 140" caption="Perimeter is the total length of a closed boundary.">
  <polygon points="50,30 170,30 200,80 130,120 40,110" fill="currentColor" opacity="0.08" stroke="none" />
  <defs>
    <marker id="peri-arrow" viewBox="0 0 10 10" refX="8" refY="5" markerWidth="6" markerHeight="6" orient="auto">
      <path d="M 0 0 L 10 5 L 0 10 z" fill="currentColor" stroke="none" />
    </marker>
  </defs>
  <line x1="50" y1="30" x2="170" y2="30" marker-end="url(#peri-arrow)" />
  <line x1="170" y1="30" x2="200" y2="80" marker-end="url(#peri-arrow)" />
  <line x1="200" y1="80" x2="130" y2="120" marker-end="url(#peri-arrow)" />
  <line x1="130" y1="120" x2="40" y2="110" marker-end="url(#peri-arrow)" />
  <line x1="40" y1="110" x2="50" y2="30" marker-end="url(#peri-arrow)" />
</Diagram>

**Square**

<Diagram src="square.svg" alt="Square" />

Four equal sides of length *a*. Perimeter is four times the side:

<Math math="Perimeter=4a" display />

**Rectangle**

<Diagram src="rectangle.svg" alt="Rectangle" />

Sides *a* and *b*. Opposite sides are equal, so the perimeter is twice the sum of one of each:

<Math math="Perimeter=2\left(a+b\right)" display />

**Parallelgram**

<Diagram src="parallelgram.svg" alt="Parallelgram" />

Adjacent sides *a* and *b*. Opposite sides are equal, so the perimeter takes the same form as a rectangle. The slant angle does not enter:

<Math math="Perimeter=2a+2b" display />

**Circle**

<Diagram src="circle.svg" alt="Circle" />

Radius *r*. The circumference is *2π* times the radius, the limiting case of a regular polygon as the number of sides grows without bound:

<Math math="Perimeter=2\pi r" display />

**Triangle**

<Diagram src="triangle.svg" alt="Triangle" />

Sides *a*, *b*, and *c*. Perimeter is their sum. No angle information is required:

<Math math="Perimeter=a+b+c" display />

**Any Regular Pilygon**

*n* equal sides of length *s* give a perimeter of *ns*, the sum of all sides. The general statement below covers irregular polygons as well:

<Math math="Perimeter=Sum\text{ }of\text{ }the\text{ }sides" display />

---

## Surface Area

URL: http://math.mkabumattar.com/geometry/surface-area

Surface area formulas for 3D solids: cubes, prisms, spheres, and cylinders.

Surface area is the total area of all faces of a 3D solid, in square units. For prismatic solids it splits into a lateral component (the side walls) plus the area of the two bases. The formulas below cover cubes, rectangular and irregular prisms, spheres, and cylinders.

<Diagram alt="Unfolded cube net showing six square faces laid flat" viewBox="0 0 220 180" caption="Surface area is the sum of all outer face areas.">
  <rect x="80" y="10" width="40" height="40" fill="currentColor" opacity="0.15" />
  <rect x="80" y="10" width="40" height="40" fill="none" />
  <rect x="40" y="50" width="40" height="40" fill="currentColor" opacity="0.15" />
  <rect x="40" y="50" width="40" height="40" fill="none" />
  <rect x="80" y="50" width="40" height="40" fill="currentColor" opacity="0.15" />
  <rect x="80" y="50" width="40" height="40" fill="none" />
  <rect x="120" y="50" width="40" height="40" fill="currentColor" opacity="0.15" />
  <rect x="120" y="50" width="40" height="40" fill="none" />
  <rect x="160" y="50" width="40" height="40" fill="currentColor" opacity="0.15" />
  <rect x="160" y="50" width="40" height="40" fill="none" />
  <rect x="80" y="90" width="40" height="40" fill="currentColor" opacity="0.15" />
  <rect x="80" y="90" width="40" height="40" fill="none" />
  <rect x="80" y="130" width="40" height="40" fill="currentColor" opacity="0.15" />
  <rect x="80" y="130" width="40" height="40" fill="none" />
  <text x="100" y="34" font-size="9" text-anchor="middle" fill="currentColor" stroke="none">top</text>
  <text x="60" y="74" font-size="9" text-anchor="middle" fill="currentColor" stroke="none">side</text>
  <text x="100" y="74" font-size="9" text-anchor="middle" fill="currentColor" stroke="none">front</text>
  <text x="140" y="74" font-size="9" text-anchor="middle" fill="currentColor" stroke="none">side</text>
  <text x="180" y="74" font-size="9" text-anchor="middle" fill="currentColor" stroke="none">back</text>
  <text x="100" y="114" font-size="9" text-anchor="middle" fill="currentColor" stroke="none">bottom</text>
  <text x="100" y="154" font-size="9" text-anchor="middle" fill="currentColor" stroke="none">face</text>
</Diagram>

**Cube**

<Diagram src="cube.svg" alt="Cube" />

Six congruent square faces of edge *a*, each of area *a²*. Surface area is six times that:

<Math math="SA=6a^{2}" display />

**Rectangular Prism**

<Diagram src="rectangular_prism.svg" alt="Rectangular Prism" />

Edges *a*, *b*, *c*. Three pairs of congruent faces: two of area *ab*, two of *bc*, two of *ac*. Sum the three pairs:

<Math math="SA=2ab+2bc+2ac" display />

**Irregular Prism**

<Diagram src="irregular_prism.svg" alt="Irregular Prism" />

*B* is the base polygon and *L* is the prism's height along the axis. The lateral surface is the base perimeter times *L*. Add twice the base area for the two ends:

<Math math="SA=\left(Perimeter\text{ }of\text{ }B\right)\times L+2\times\left(Area\text{ }of\text{ }B\right)" display />

**Sphere**

<Diagram src="circle.svg" alt="Sphere" />

Radius *r*. Surface area is four times the area of a great-circle cross-section (*πr²*):

<Math math="SA=4\pi r^{2}" display />

**Cylinder**

<Diagram src="cylinder.svg" alt="Cylinder" />

Base radius *r* and height *h*. Two circular ends of area *πr²* plus a wrapped rectangle of height *h* and width *2πr* (the base circumference) factor into:

<Math math="SA=2\pi r\left(r+h\right)" display />

---

## Volume

URL: http://math.mkabumattar.com/geometry/volume

Volume formulas for 3D solids: cubes, prisms, cylinders, pyramids, cones, spheres, and ellipsoids.

Volume measures the 3D extent of a solid in cubic units (cubic metres, cubic inches, and so on). Prismatic solids such as cubes, rectangular prisms, and cylinders follow the pattern *base area × height*. Pyramids and cones give one-third of the corresponding prism. Spheres and ellipsoids have their own forms.

<Diagram alt="Rectangular prism with width, height, and depth labeled" viewBox="0 0 220 160" caption="Volume measures 3D space in cubic units.">
  <polygon points="30,60 150,60 150,130 30,130" fill="currentColor" opacity="0.12" stroke="currentColor" />
  <polygon points="30,60 70,30 190,30 150,60" fill="currentColor" opacity="0.08" stroke="currentColor" />
  <polygon points="150,60 190,30 190,100 150,130" fill="currentColor" opacity="0.18" stroke="currentColor" />
  <line x1="30" y1="138" x2="150" y2="138" stroke-width="0.8" />
  <line x1="30" y1="135" x2="30" y2="141" />
  <line x1="150" y1="135" x2="150" y2="141" />
  <text x="90" y="152" font-size="10" text-anchor="middle" fill="currentColor" stroke="none">width</text>
  <line x1="20" y1="60" x2="20" y2="130" stroke-width="0.8" />
  <line x1="17" y1="60" x2="23" y2="60" />
  <line x1="17" y1="130" x2="23" y2="130" />
  <text x="14" y="98" font-size="10" text-anchor="end" fill="currentColor" stroke="none">height</text>
  <line x1="155" y1="25" x2="195" y2="-5" stroke-width="0.8" />
  <text x="200" y="22" font-size="10" fill="currentColor" stroke="none">depth</text>
  <line x1="190" y1="30" x2="195" y2="25" />
  <line x1="155" y1="60" x2="195" y2="30" stroke-dasharray="2 2" stroke-width="0.6" opacity="0.5" />
</Diagram>

**Cube**

<Diagram src="cube.svg" alt="Cube" />

Edge length *a*. Volume is the side cubed; doubling the edge multiplies volume by eight:

<Math math="Volume=a^{3}" display />

**Rectangular Prism**

<Diagram src="rectangular_prism.svg" alt="Rectangular Prism" />

Edges *a*, *b*, *c* (length, width, height in any order). Volume is the product of all three:

<Math math="Volume=a\times b\times c" display />

**Irregular Prism**

<Diagram src="irregular_prism.svg" alt="Irregular Prism" />

*B* is the base polygon (any shape) and *h* the perpendicular height between the two parallel bases. Volume is the base area times that height, generalising the rectangular-prism rule:

<Math math="Volume=\left(Area\text{ }of\text{ }B\right)\times h" display />

**Cylinder**

<Diagram src="cylinder.svg" alt="Cylinder" />

Base radius *r* and height *h*. The circular base has area *πr²*, so volume is the disc area times the height: the prism rule applied to a circular base:

<Math math="Volume=\pi r^{2}h" display />

**Pyramid**

<Diagram src="pyramid.svg" alt="Pyramid" />

*b* is the base area (for a square pyramid, *b = side²*) and *h* the perpendicular height from base to apex. Volume is one-third the volume of the prism with the same base and height:

<Math math="Volume=\left(\frac{1}{3}\right)bh" display />

whare b = area of square

**Cone**

<Diagram src="cone.svg" alt="Cone" />

Base radius *r* and perpendicular height *h* from base to apex. The circular case of the pyramid rule: one-third the base area (*πr²*) times the height:

<Math math="Volume=\left(\frac{1}{3}\right)\pi r^{2}h" display />

**Sphere**

<Diagram src="circle.svg" alt="Sphere" />

Radius *r*. Volume scales with the cube of *r*. The constant *4π/3* is the same factor that appears when integrating a hemisphere of revolution:

<Math math="Volume=\left(\frac{4}{3}\right)\pi r^{3}" display />

**Ellipsoid**

<Diagram src="ellipsoid.svg" alt="Ellipsoid" />

Three semi-axes *r1*, *r2*, *r3* (half the lengths along each principal direction). Volume is *4π/3* times their product. When all three are equal this reduces to the sphere formula:

<Math math="Volume=\left(\frac{4}{3}\right)\pi\left(r1\times r2\times r3\right)" display />

---

## Lateral Area

URL: http://math.mkabumattar.com/geometry/lateral-area

Lateral area formulas for right prisms, cylinders, pyramids, and cones.

Lateral area is the surface area of a 3D solid *excluding* its base or bases, measured in square units. It covers the side walls only, the relevant quantity when the bases are not part of the visible surface (the wall of a tank, the curved sheet of a cone). The formulas below cover right prisms, cylinders, pyramids, and cones.

<Diagram alt="Cylinder with only the curved side shaded, top and bottom unshaded" viewBox="0 0 220 160" caption="Lateral area excludes the bases.">
  <ellipse cx="110" cy="30" rx="60" ry="14" fill="none" />
  <path d="M 50 30 L 50 120 A 60 14 0 0 0 170 120 L 170 30" fill="currentColor" opacity="0.18" stroke="currentColor" />
  <ellipse cx="110" cy="120" rx="60" ry="14" fill="none" />
  <path d="M 50 30 A 60 14 0 0 0 170 30" stroke-dasharray="3 3" fill="none" />
  <text x="110" y="22" font-size="10" text-anchor="middle" fill="currentColor" stroke="none">base (excluded)</text>
  <text x="110" y="148" font-size="10" text-anchor="middle" fill="currentColor" stroke="none">base (excluded)</text>
  <text x="110" y="78" font-size="10" text-anchor="middle" fill="currentColor" stroke="none">lateral</text>
</Diagram>

**Right Prism**

Base perimeter *P* and prism height *h*. The side walls unroll into a single rectangle of width *P* and height *h*:

<Math math="Lateral\text{ }Area=Ph" display />

Whare

P = Perimeter of the base

h = height of the prism

**cylinder**

The circular case of a right prism, with base perimeter *2πr*. Base radius *r* and height *h*. Unrolling the curved side gives a rectangle of width *2πr* and height *h*:

<Math math="Lateral\text{ }Area=2\pi rh" display />

Where

r = radius of the base

h = geight of the cylinder

**Pyramid**

Base perimeter *P* and slant height *l* (the distance from the apex down the face to the midpoint of a base edge, not the perpendicular height). Lateral area is half the base perimeter times the slant height. The one-half factor matches a triangle's area-to-base-times-height ratio:

<Math math="Lateral\text{ }Area=\left(\frac{1}{2}\right)Pl" display />

Where

P = Perimeter of base

l = slant height

**Cone**

Base radius *r* and slant height *l* (apex to base-edge along the surface, not the vertical axis). The circular case of the pyramid rule. Unrolled, the surface is a circular sector of radius *l* whose arc length matches the base circumference:

<Math math="Lateral\text{ }Area=\pi rl" display />

Where

r = radius of it's base

l = slant height of the cone

---

# Coordinate Geometry

## Coordinate Geometry Basics

URL: http://math.mkabumattar.com/coordinate-geometry/basics

Distance, section, midpoint, area of a triangle, slope, and equations of lines in the coordinate plane.

Coordinate geometry expresses geometric objects algebraically by placing them on a Cartesian grid. Points become ordered pairs, lines become linear equations, and geometric relations (collinearity, perpendicularity, distance, area) become arithmetic identities on coordinates. The formulas below cover the core operations on points and lines in the plane: measuring distances, dividing segments, computing slopes, and writing the equation of a line. Each form is interchangeable, fit to a different set of givens.

### Distance Between Two Points

The distance between P(x<sub>1</sub>,y<sub>1</sub>) and Q(x<sub>2</sub>,y<sub>2</sub>) is the length of the segment joining them. It is the Pythagorean theorem applied to the run *x<sub>2</sub> − x<sub>1</sub>* and the rise *y<sub>2</sub> − y<sub>1</sub>*, the legs of a right triangle whose hypotenuse is the segment. The result is non-negative and symmetric in the two points.

<Math math="\sqrt{\left(x_{2}-x_{2}\right)^{2}+\left(y_{1}-y_{2}\right)^{2}}" display />

<Diagram alt="Right triangle illustrating the distance formula between two points" viewBox="0 0 220 180" caption="The distance formula is the Pythagorean theorem applied to the rise and run between two points.">
  <line x1="20" y1="150" x2="210" y2="150" />
  <line x1="30" y1="20" x2="30" y2="170" />
  <text x="205" y="146" font-size="10" fill="currentColor" stroke="none">x</text>
  <text x="34" y="26" font-size="10" fill="currentColor" stroke="none">y</text>
  <line x1="60" y1="120" x2="170" y2="50" />
  <line x1="60" y1="120" x2="170" y2="120" stroke-dasharray="4 3" />
  <line x1="170" y1="120" x2="170" y2="50" stroke-dasharray="4 3" />
  <circle cx="60" cy="120" r="3" fill="currentColor" />
  <circle cx="170" cy="50" r="3" fill="currentColor" />
  <text x="34" y="128" font-size="10" fill="currentColor" stroke="none">P(x₁,y₁)</text>
  <text x="155" y="44" font-size="10" fill="currentColor" stroke="none">Q(x₂,y₂)</text>
  <text x="105" y="134" font-size="10" fill="currentColor" stroke="none">x₂−x₁</text>
  <text x="174" y="90" font-size="10" fill="currentColor" stroke="none">y₂−y₁</text>
  <text x="100" y="80" font-size="10" fill="currentColor" stroke="none">d</text>
</Diagram>

### Section Formula

The section formula gives the coordinates of R(x,y) dividing the segment from P(x<sub>1</sub>,y<sub>1</sub>) to Q(x<sub>2</sub>,y<sub>2</sub>) in the ratio *m<sub>1</sub>:m<sub>2</sub>*. Each coordinate is a weighted average of the endpoint coordinates with the weights swapped: *m<sub>1</sub>* multiplies the second point's coordinate and *m<sub>2</sub>* multiplies the first. For internal division both weights are positive; for external division *m<sub>2</sub>* is negative. The formula fails when *m<sub>1</sub> + m<sub>2</sub> = 0*, which corresponds to no finite dividing point.

<Math math="\left(\frac{m_{1}x_{2}+m_{2}x_{1}}{m_{1}+m_{2}},\frac{m_{1}y_{2}+m_{2}y_{1}}{m_{1}+m_{2}}\right)" display />

### Midpoint Formula

The midpoint is the section formula with the ratio fixed at 1:1, so each coordinate is the arithmetic mean of the endpoint coordinates. It is the centre of the segment, equidistant from both endpoints. The formula returns a finite point for any pair of endpoints, distinct or coincident.

<Math math="\left(\frac{x_{1}+x_{2}}{2},\frac{y_{1}+y_{2}}{2}\right)" display />

### Area of a Triangle

The area of the triangle with vertices (x<sub>1</sub>,y<sub>1</sub>), (x<sub>2</sub>,y<sub>2</sub>), (x<sub>3</sub>,y<sub>3</sub>) is half the absolute value of a signed determinant in the coordinates. The expression inside the bars is twice the signed area; the absolute value drops the sign recording vertex orientation (clockwise vs counter-clockwise). The result is zero exactly when the three points are collinear, giving a direct collinearity test.

<Math math="\triangle=\left(\frac{1}{2}\right)\left|x_{1}\left(y_{2}-y_{3}\right)+x_{2}\left(y_{3}-y_{1}\right)+x_{3}\left(y_{1}-y_{2}\right)\right|" display />

### Vertical Line

A line parallel to the *y*-axis at horizontal distance *a* has every point sharing the same *x*-coordinate, so its equation is *x = a*. Its slope is undefined (the run is zero), so no slope-intercept form exists. The analogous horizontal line is *y = b*, with slope zero.

<Math math="x=a" display />

### Slope of a Line

The slope (or gradient) *m* of a non-vertical line is the tangent of the angle *θ* the line makes with the positive *x*-axis, measured counter-clockwise. It gives direction and steepness: positive slope rises left-to-right, negative slope falls, zero slope is horizontal. Slope is undefined when *θ = 90°*, the vertical case.

<Math math="m=\tan\theta" display />

### Parallel Lines

Two non-vertical lines are parallel exactly when they have the same slope. They never meet, or coincide entirely if their intercepts also match. Two vertical lines are parallel by inspection, since slope is undefined for both.

<Math math="ie\text{ }m_{1}=m_{2}" display />

### Perpendicular Lines

Two non-vertical lines are perpendicular exactly when the product of their slopes is *−1*. Each slope is the negative reciprocal of the other. The rule fails when one line is vertical and the other horizontal; that pair is perpendicular by direct inspection, since one slope is undefined.

<Math math="ie.\text{ }m_{1}\times m_{2}=-1" display />

### Line Through the Origin

A line through the origin with slope *m = tan θ* has zero *y*-intercept, so the slope-intercept form reduces to *y = mx*. The slope alone determines the line. This is the building block for the point-slope and two-point forms below.

<Math math="y=mx" display />

### Point-Slope Form

The point-slope form writes a line from one known point (x<sub>1</sub>, y<sub>1</sub>) and slope *m*. It says the slope between (x, y) and (x<sub>1</sub>, y<sub>1</sub>) is constantly *m*. Vertical lines have undefined slope; use *x = x<sub>1</sub>* in that case.

<Math math="\left(y-y_{1}\right)=m\left(x-x_{1}\right)" display />

### Two-Point Form

The two-point form is point-slope with the slope computed in place from the second point: *m = (y<sub>2</sub> − y<sub>1</sub>) / (x<sub>2</sub> − x<sub>1</sub>)*. Use it when two points are given and neither slope nor intercept is known. The form fails when *x<sub>1</sub> = x<sub>2</sub>* (vertical line); use *x = x<sub>1</sub>* there.

<Math math="\left(y-y_{1}\right)=\left(\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\right)\left(x-x_{1}\right)" display />

### Intercept Form

The intercept form uses the *x*-intercept *a* and *y*-intercept *b*, the points where the line crosses the axes. Substituting (a, 0) or (0, b) verifies the equation. It does not apply to lines through the origin (where *a = b = 0*) or to lines parallel to either axis (where one intercept is infinite).

<Math math="\frac{x}{a}+\frac{y}{b}=1" display />

---

## Circle

URL: http://math.mkabumattar.com/coordinate-geometry/circle

Equations of a circle including standard, center-radius, general, diameter, parametric, tangent, and normal forms.

A circle is the locus of points equidistant from a fixed centre. The fixed distance is the radius *r*; the centre is the unique point from which every point on the curve is *r* away. The equations below give the same object in several algebraic forms (standard, centre-radius, general, diameter, parametric), each suited to a different kind of given data. The page also covers tangent and normal lines at a chosen point of contact, in both standard and general settings.

<Diagram alt="Circle with center (h, k) and radius arrow r" viewBox="0 0 220 180" caption="Standard form: centered at (h, k) with radius r.">
  <line x1="10" y1="120" x2="210" y2="120" />
  <line x1="60" y1="10" x2="60" y2="170" />
  <text x="205" y="116" font-size="10" fill="currentColor" stroke="none">x</text>
  <text x="64" y="18" font-size="10" fill="currentColor" stroke="none">y</text>
  <circle cx="130" cy="85" r="45" />
  <circle cx="130" cy="85" r="2" fill="currentColor" />
  <line x1="130" y1="85" x2="162" y2="53" />
  <circle cx="162" cy="53" r="2.5" fill="currentColor" />
  <text x="108" y="100" font-size="10" fill="currentColor" stroke="none">(h, k)</text>
  <text x="142" y="68" font-size="10" fill="currentColor" stroke="none">r</text>
</Diagram>

### Standerd Equation

The standard equation places the centre at the origin. Every point (x, y) on the circle satisfies *x<sup>2</sup> + y<sup>2</sup> = a<sup>2</sup>*: the squared distance from the origin equals the squared radius. The parameter *a* must be positive; *a = 0* collapses the circle to a single point and *a<sup>2</sup> < 0* has no real solutions. Use this form for problems with rotational symmetry about the origin.

<Math math="x^{2}+y^{2}=a^{2}\text{ }\left(a=radius\right)" display />

### Center-Radius Form

Translating the centre to (h, k) shifts each variable by the corresponding centre coordinate. The equation states that the squared distance from (x, y) to (h, k) equals *r<sup>2</sup>*. Setting *h = k = 0* recovers the standard form. The radius must be non-negative for the equation to describe a real curve.

<Math math="\left(x-h\right)^{2}+\left(y-k\right)^{2}=r^{}\text{ }\left(r=radius,\text{ }\left(h,k\right)=center\right)" display />

### General Equation of a Circle

Expanding *(x − h)<sup>2</sup> + (y − k)<sup>2</sup> = r<sup>2</sup>* and renaming constants gives the general quadratic form. Any second-degree equation in *x* and *y* with equal coefficients on *x<sup>2</sup>* and *y<sup>2</sup>* and no *xy* term reduces to this shape. The general form is the natural output of intersection or locus computations, where centre and radius are not yet known.

<Math math="x^{2}+y^{2}+2gx+2fy+c=0" display />

Comparing with the expanded centre-radius form, the centre is read off the linear coefficients as *(−g, −f)*.

<Math math="Center = \left(-g,-f\right)" display />

The radius comes from completing the square. The curve is a real circle only when the radicand *g<sup>2</sup> + f<sup>2</sup> − c* is positive (a point if zero, imaginary if negative).

<Math math="Radius = \sqrt{g^{2}+f^{2}-c}" display />

### Diameter Form

Given two diametrically opposite endpoints (x<sub>1</sub>, y<sub>1</sub>) and (x<sub>2</sub>, y<sub>2</sub>), the circle is the locus of points P such that ∠(x<sub>1</sub>P x<sub>2</sub>) is a right angle (Thales' theorem). The dot product of the vectors from P to each endpoint is zero, giving the diameter form. The centre is the midpoint of the diameter; the radius is half its length.

<Math math="\left(y-y_{1}\right)\left(y-y_{2}\right)+\left(x-x_{1}\right)\left(x-x_{2}\right)=0" display />

### Tangent at P(x<sub>1</sub>,y<sub>1</sub>) of a Standard Circle

The tangent line at (x<sub>1</sub>, y<sub>1</sub>) on the circle *x<sup>2</sup> + y<sup>2</sup> = a<sup>2</sup>* is perpendicular to the radius at that point. Apply the "T = 0" rule: replace *x<sup>2</sup>* by *x x<sub>1</sub>* and *y<sup>2</sup>* by *y y<sub>1</sub>*. The contact point must satisfy the original equation; otherwise the resulting line is the polar of an external point, not a tangent.

<Math math="xx_{1}+yy_{1}=a^{2}" display />

### Normal at P(x<sub>1</sub>,y<sub>1</sub>) of a Standard Circle

The normal at any point on a circle passes through that point and the centre. For the standard circle the centre is the origin. The equation *y / y<sub>1</sub> = x / x<sub>1</sub>* says that (x, y) is a scalar multiple of (x<sub>1</sub>, y<sub>1</sub>), i.e. lies on the line through the origin and the contact point. The form breaks down when either *x<sub>1</sub>* or *y<sub>1</sub>* is zero; in those cases the normal is a coordinate axis.

<Math math="\frac{y}{y_{1}}=\frac{x}{x_{1}}" display />

### Tangent at P(x<sub>1</sub>,y<sub>1</sub>) of a General Circle

For the general-form circle, the T = 0 rule extends to the linear terms: replace *x* by *(x + x<sub>1</sub>) / 2* and *y* by *(y + y<sub>1</sub>) / 2*, then clear the halves. The contact point must lie on the circle for the result to be a true tangent. Setting *g = f = 0* recovers the standard-form tangent.

<Math math="xx_{1}+yy_{1}+g\left(x+x_{1}\right)+f\left(y+y_{1}\right)+c=0" display />

### Normal at P(x<sub>1</sub>,y<sub>1</sub>) of a General Circle

The normal at (x<sub>1</sub>, y<sub>1</sub>) on the general circle is the line joining the contact point to the centre *(−g, −f)*. The two-point form, with the centre as the second point, yields the equation below. The line is well defined whenever the contact point and centre are distinct, which holds for any point on the circle.

<Math math="\left(\frac{y-y_{1}}{x-x_{1}}\right)=\left(\frac{y_{1}-f}{x_{1}-g}\right)" display />

### Condition For a Line To Be a Tangent

For the line *y = mx + c* to be tangent to the standard circle *x<sup>2</sup> + y<sup>2</sup> = a<sup>2</sup>*, the perpendicular distance from the origin to the line must equal *a*. Substituting *y = mx + c* into the circle equation and demanding a double root gives *c<sup>2</sup> = a<sup>2</sup>(m<sup>2</sup> + 1)*. Two values of *c* (one positive, one negative) satisfy this, giving the two parallel tangents of slope *m*.

<Math math="c^{2}=a^{2}\left(m^{2}+1\right)" display />

When the tangency condition holds, the line touches the circle at a single point, obtained by solving the substituted equation as a perfect square. The contact point depends on the sign of *c*, so the two parallel tangents of slope *m* touch opposite points.

<Math math="Point\text{ }of\text{ }contact = \left(\frac{-a^{2}m}{c},\frac{a^{2}}{c}\right)" display />

### Parametic Equation of Standard Circle

A single angular parameter *θ* traces the standard circle: the angle measured at the centre from the positive *x*-axis to the radius at (x, y). As *θ* sweeps from 0 to *2π*, the point (a cos θ, a sin θ) traces the circle once counter-clockwise. Use this form for integrating around the circle or parameterising motion along it.

<Math math="x=a.\cos\theta" display />

<Math math="u=a.\sin\theta" display />

### Parametic Equation of General Circle

For a circle centred at (h, k) with radius *r*, the parametric form adds the centre coordinates as offsets to the standard parameterisation. Each point on the circle is the centre plus a radius-vector of length *r* at angle *θ*. In animation and physics *θ* often plays the role of time.

<Math math="x=h+r.\cos\theta" display />

<Math math="x=h+r.\sin\theta" display />

<Math math="\left[Center=(h,k)\text{ }and\text{ }radius=r\right]" display />

### Tangent To a Standard Circle at Point P(a.cosθ, a.sinθ)

Substituting the parametric coordinates into *x x<sub>1</sub> + y y<sub>1</sub> = a<sup>2</sup>* and dividing by *a* gives the tangent in angular form. Use this when the contact point is identified by angle rather than Cartesian coordinates. The coefficients (cos θ, sin θ) are the unit normal direction from the centre to the contact point.

<Math math="y.\sin\theta+x.\cos\theta=a" display />

### Normal To a Standard Circle at Point P(a.cosθ, a.sinθ)

The normal at a parametric point is the line through the origin and the contact point, in the direction (cos θ, sin θ). The equation *x / cos θ = y / sin θ* says (x, y) is proportional to (cos θ, sin θ). The form is undefined when *cos θ* or *sin θ* is zero, where the normal coincides with an axis.

<Math math="\frac{x}{\cos\theta}=\frac{y}{\sin\theta}" display />

<Notice type="note">
The "T = 0" pattern (replacing *x<sup>2</sup>* by *xx<sub>1</sub>*, *y<sup>2</sup>* by *yy<sub>1</sub>*, and *x* by *(x + x<sub>1</sub>)/2*) is the standard shortcut for writing the tangent to any conic at a known point, not just the circle.
</Notice>

---

## Parabola

URL: http://math.mkabumattar.com/coordinate-geometry/parabola

Types of parabolas, tangents, normals, focal distance, and angle between tangents.

A parabola is the locus of points equidistant from a fixed point (the focus) and a fixed line (the directrix). The line through the focus perpendicular to the directrix is the axis of symmetry; the parabola crosses it at the vertex, halfway between focus and directrix. The four standard orientations place the vertex at the origin and align the axis with a coordinate axis, differing only in direction. The parameter *a* is the focus-to-vertex distance and controls how tightly the parabola opens.

<Diagram alt="Parabola opening upward with focus and directrix" viewBox="0 0 220 180" caption="Every point on the parabola is equidistant from the focus and the directrix.">
  <line x1="10" y1="130" x2="210" y2="130" />
  <line x1="110" y1="10" x2="110" y2="170" />
  <text x="205" y="126" font-size="10" fill="currentColor" stroke="none">x</text>
  <text x="114" y="18" font-size="10" fill="currentColor" stroke="none">y</text>
  <path d="M 50 30 Q 110 170 170 30" />
  <line x1="20" y1="155" x2="200" y2="155" stroke-dasharray="5 3" />
  <text x="150" y="168" font-size="10" fill="currentColor" stroke="none">directrix</text>
  <circle cx="110" cy="105" r="3" fill="currentColor" />
  <text x="116" y="102" font-size="10" fill="currentColor" stroke="none">F</text>
  <circle cx="110" cy="130" r="2" fill="currentColor" />
  <text x="92" y="124" font-size="10" fill="currentColor" stroke="none">V</text>
</Diagram>

### Type of Parabola

The four standard parabolas all have their vertex at the origin; they differ in which axis they open along and in which direction the focus lies. The latus rectum is the focal chord perpendicular to the axis, with length *4a* in every case. The directrix lies on the side of the vertex opposite the focus, at distance *a*.

| Parabola | y<sup>2</sup>=4ax | y<sup>2</sup>=-4ax | x<sup>2</sup>=4ay | x<sup>2</sup>=-4ay |
|---|---|---|---|---|
| **Vertex** | (0,0) | (0,0) | (0,0) | (0,0) |
| **Focus** | (a,0) | (-a,0) | (0,a) | (0,-a) |
| **Axis** | y=0 | y=0 | x=0 | x=0 |
| **Dirctix** | x=-a | x=a | y=-a | y=a |
| **Length of Latus Recutum (LR)** | 4a | 4a | 4a | 4a |
| **End Point of LR** | (a,2a)&(a,-2a) | (-a,2a)&(-a,-2a) | (2a,a)&(-2a,a) | (2a,-a)&(-2a,-a) |

### Tangent To Parabola y<sup>2</sup>=4ax

A parabola has two tangent forms: one from a point of contact, one from a slope. The point form applies the T = 0 substitution *y<sup>2</sup> → y y<sub>1</sub>* and *x → (x + x<sub>1</sub>)/2*, multiplied out. The contact point (x<sub>1</sub>, y<sub>1</sub>) must lie on the parabola, i.e. satisfy *y<sub>1</sub><sup>2</sup> = 4a x<sub>1</sub>*.

<Math math="\left(i\right)\text{ }yy_{1}=2a\left(x+x_{1}\right)\text{ }\left[at\text{ }point\text{ }P{x_{1},y_{1}}\right]" display />

The slope form gives the tangent of any non-zero slope *m*. Substituting *y = mx + c* into the parabola equation and demanding a double root yields *c = a/m*. Each non-zero slope has exactly one tangent; the slope-zero limit is the *x*-axis, which is tangent at the vertex.

<Math math="\left(ii\right)\text{ }y=mx+\frac{a}{m}\text{ }\left[Having\text{ }slope=m\right]" display />

### Normal To Parabola y<sup>2</sup>=4ax

The normal at a contact point is perpendicular to the tangent there and passes through the point. Its slope is the negative reciprocal of the tangent slope *2a / y<sub>1</sub>*, giving *−y<sub>1</sub> / 2a*. The form below is point-slope with this normal slope; it is undefined at the vertex where *y<sub>1</sub> = 0* (the normal there is the *x*-axis).

<Math math="\left(i\right)\text{ }\left(y-y_{1}\right)=\left(\frac{-y_{1}}{2a}\right)\left(x-x_{1}\right)" display />

The slope form parameterises normals by slope *m*. Up to three normals of slope *m* can pass through a single external point, the basis of the theory of conormal points. The cubic dependence on *m* in the formula reflects that count.

<Math math="\left(ii\right)\text{ }y=mx-2am-am^{3}" display />

### Focal Distance

The focal distance is the distance from a point P(x<sub>1</sub>, y<sub>1</sub>) on the parabola to the focus. By the defining locus property, it equals the perpendicular distance from P to the directrix, which for *y<sup>2</sup> = 4ax* (directrix *x = −a*) is *x<sub>1</sub> + a*. The focal distance reads off the *x*-coordinate alone, no square roots required. It is non-negative because *x<sub>1</sub> ≥ 0* on the parabola.

If P(x<sub>1</sub>,y<sub>1</sub>) lies on y<sup>2</sup>=4ax then focal distance = x<sub>1</sub>+a

### Angle Between Two Tangents From P(x<sub>1</sub>,y<sub>1</sub>) to Parabola y<sup>2</sup>=4ax is

From any external point exactly two tangents can be drawn to a parabola; the angle *θ* between them measures the opening of the curve seen from that point. The numerator radical is real precisely when (x<sub>1</sub>, y<sub>1</sub>) lies outside the parabola (*y<sub>1</sub><sup>2</sup> > 4a x<sub>1</sub>*); on the curve the angle collapses to zero, and inside there are no real tangents. The denominator *x<sub>1</sub> + a* is the focal distance, so the formula relates the tangent angle to position relative to the directrix.

<Math math="\tan\theta=\left|\frac{\sqrt{\left(y_{1}^{2}-4ax_{1}\right)}}{\left(x_{1}+a\right)}\right|" display />

---

## Ellipse

URL: http://math.mkabumattar.com/coordinate-geometry/ellipse

Types of ellipses, eccentricity, tangents, normals, and focal distances.

An ellipse is the locus of points whose distances to two fixed foci sum to a constant. The two foci lie on the major axis, equidistant from the centre; the constant sum equals the major-axis length *2a*. The semi-major axis *a* and semi-minor axis *b* set the size and shape; the eccentricity *e ∈ [0, 1)* measures elongation: *e = 0* is a circle, *e → 1* a near-degenerate slit. The two standard orientations place the centre at the origin and align the major axis with either coordinate axis.

<Diagram alt="Ellipse with two foci on the x-axis and labeled semi-axes" viewBox="0 0 220 180" caption="Sum of distances from any point on the ellipse to the two foci equals 2a.">
  <line x1="10" y1="90" x2="210" y2="90" />
  <line x1="110" y1="10" x2="110" y2="170" />
  <text x="205" y="86" font-size="10" fill="currentColor" stroke="none">x</text>
  <text x="114" y="18" font-size="10" fill="currentColor" stroke="none">y</text>
  <ellipse cx="110" cy="90" rx="80" ry="45" />
  <circle cx="80" cy="90" r="2.5" fill="currentColor" />
  <circle cx="140" cy="90" r="2.5" fill="currentColor" />
  <text x="68" y="104" font-size="10" fill="currentColor" stroke="none">F₁</text>
  <text x="142" y="104" font-size="10" fill="currentColor" stroke="none">F₂</text>
  <line x1="110" y1="90" x2="190" y2="90" stroke-dasharray="3 3" />
  <line x1="110" y1="90" x2="110" y2="45" stroke-dasharray="3 3" />
  <text x="150" y="86" font-size="10" fill="currentColor" stroke="none">a</text>
  <text x="114" y="68" font-size="10" fill="currentColor" stroke="none">b</text>
</Diagram>

### Types of Ellipse

The two standard orientations swap which axis carries the longer semi-axis; the behaviour is symmetric under that swap.

Type 1 has the major axis along *x*, so *a > b*. The vertices on the major axis are *(±a, 0)*; the co-vertices on the minor axis are *(0, ±b)*. The foci lie on the *x*-axis at *(±ae, 0)*. The curve is symmetric in both axes and in the origin.

<Math math="\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\text{ }\text{ }\text{ }\left(a^{2}>b^{2}\right)" display />

<Diagram src="1_ellipse.svg" alt="Ellipse Type 1" />

Type 2 has the major axis along *y*, so *b > a*. The denominators keep their positions, but *b* is now the semi-major axis and *a* the semi-minor. The foci move onto the *y*-axis at *(0, ±be)*; the vertices on the major axis become *(0, ±b)*.

<Math math="\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\text{ }\text{ }\text{ }\left(b^{2}>a^{2}\right)" display />

<Diagram src="2_ellipse.svg" alt="Ellipse Type 2" />

| Ellipse | Type 1 (a&gt;b) | Type 2 (b&gt;a) |
|---|---|---|
| **Center** | (0,0) | (0,0) |
| **Vertices** | A(a,0) A'(-a,0) B(0,b) B'(0,-b) | B(0,b) B'(0,-b) A(a,0) A'(-a,0) |
| **Foci** | S=(ae,0) S"(-ae,0) | S=(be,0) S"(-be,0) |
| **Distance Between Foci** | 2ea | 2be |
| **Length of Axis** | Major = 2a, Minor = 2b | Major = 2b, Minor = 2a |
| **Equation of Axis** | Major: y=0, Minor: x=0 | Major: x=0, Minor: y=0 |
| **Relation Between a,b,c** | b<sup>2</sup> = a<sup>2</sup>(1-e<sup>2</sup>) | a<sup>2</sup> = b<sup>2</sup>(1-e<sup>2</sup>) |

**Eccentericity (Type 1)**

For Type 1 the eccentricity comes from the two semi-axes: *e = √(a<sup>2</sup> − b<sup>2</sup>) / a*. Since *a > b* the radicand is positive and *e* lies strictly between 0 and 1. The quantity *ae* is the centre-to-focus distance, so *e* measures focal offset as a fraction of the semi-major axis.

<Math math="e=\frac{\sqrt{\left(a^{2}-b^{2}\right)}}{a}" display />

**Eccentericity (Type 2)**

For Type 2 the major axis is *b*, so the eccentricity formula swaps the roles of *a* and *b* under the radical. The radicand *b<sup>2</sup> − a<sup>2</sup>* is positive because *b > a*. The result lies in [0, 1), with the lower bound (a circle) reached when *a = b*.

<Math math="e=\frac{\sqrt{\left(b^{2}-a^{2}\right)}}{a}" display />

### Tangent to Ellipse

Tangent lines come in point-based and slope-based forms, as for the circle and parabola. The point form applies T = 0 to the ellipse equation: *x<sup>2</sup>* becomes *x x<sub>1</sub>* and *y<sup>2</sup>* becomes *y y<sub>1</sub>*. The contact point (x<sub>1</sub>, y<sub>1</sub>) must satisfy the original ellipse equation, otherwise the resulting line is a polar rather than a tangent.

<Math math="\left(i\right)\text{ }\frac{\left(xx_{1}\right)}{a^{2}}+\frac{\left(yy_{1}\right)}{b^{2}}=1\text{ }\text{ }\text{ }\left[at\text{ }P\left(x_{1},y_{1}\right)\right]" display />

The slope form gives the two parallel tangents of slope *m*. Substituting *y = mx + c* into the ellipse equation and demanding a double root yields *c<sup>2</sup> = a<sup>2</sup> m<sup>2</sup> + b<sup>2</sup>*. The radicand is always positive, so a tangent of every slope exists (unlike the hyperbola).

<Math math="\left(ii\right)\text{ }y=mx\pm\sqrt{a^{2}m^{2}+b^{2}}\text{ }\text{ }\left[Having\text{ }slope\text{ }m\right]" display />

### Normal to Ellipse

The normal at (x<sub>1</sub>, y<sub>1</sub>) is perpendicular to the tangent and passes through the contact point. Unlike the circle, the ellipse normal does not generally pass through the centre; it does so at the four vertices, and elsewhere reflects rays from one focus to the other. The form below is undefined when *x<sub>1</sub>* or *y<sub>1</sub>* is zero (at a vertex or co-vertex), where the normal is a coordinate axis.

<Math math="\frac{\left(a^{2}x\right)}{x_{1}}-\frac{\left(b^{2}y\right)}{y_{1}}=a^{2}-b^{2}\text{ }\text{ }\text{ }\left[at\text{ }P\left(x_{1},y_{1}\right)\right]" display />

### Focal Distance

Each point on the ellipse has two focal distances, one to each focus, summing to *2a* (the defining locus property). For Type 1 the distance to the focus at *(ae, 0)* is *|a − e x<sub>1</sub>|* and the distance to *(−ae, 0)* is *|a + e x<sub>1</sub>|*. Both reduce to *a* at the centre (x<sub>1</sub> = 0). The form below assumes Type 1; swap *x<sub>1</sub>* for *y<sub>1</sub>* and *a* for *b* in Type 2.

<Math math="S=\left|a-ex_{1}\right|" display />

<Math math="S'=\left|a+ex_{1}\right|" display />

<Notice type="tip">
A circle is the special case of an ellipse with *a = b* and *e = 0*. As *e* approaches 1, the ellipse degenerates toward a segment between the two foci.
</Notice>

---

## Hyperbola

URL: http://math.mkabumattar.com/coordinate-geometry/hyperbola

Types of hyperbolas, eccentricity, foci, tangents, normals, and focal distances.

A hyperbola is the locus of points whose distances to two fixed foci have a constant *difference* in absolute value (*2a*), rather than a constant sum as in the ellipse. The curve has two disconnected branches opening away from each other along the transverse axis. Its eccentricity *e* is always greater than 1, and as |x| or |y| grows the branches approach a pair of asymptotes through the centre. The two standard orientations place the transverse axis along the *x*-axis (Type 1) or the *y*-axis (Type 2).

<Diagram alt="Hyperbola with two branches, foci on x-axis, and asymptotes" viewBox="0 0 220 180" caption="Difference of distances to the two foci is constant: 2a.">
  <line x1="10" y1="90" x2="210" y2="90" />
  <line x1="110" y1="10" x2="110" y2="170" />
  <text x="205" y="86" font-size="10" fill="currentColor" stroke="none">x</text>
  <text x="114" y="18" font-size="10" fill="currentColor" stroke="none">y</text>
  <line x1="30" y1="20" x2="190" y2="160" stroke-dasharray="4 3" />
  <line x1="30" y1="160" x2="190" y2="20" stroke-dasharray="4 3" />
  <path d="M 80 20 Q 60 90 80 160" />
  <path d="M 140 20 Q 160 90 140 160" />
  <circle cx="55" cy="90" r="2.5" fill="currentColor" />
  <circle cx="165" cy="90" r="2.5" fill="currentColor" />
  <text x="44" y="104" font-size="10" fill="currentColor" stroke="none">F₁</text>
  <text x="168" y="104" font-size="10" fill="currentColor" stroke="none">F₂</text>
</Diagram>

### Types of Hyperbola

The two standard orientations have transverse axes along *x* and *y* respectively.

**Type 1**

A horizontal hyperbola opens left-and-right, with vertices at *(±a, 0)* on the *x*-axis and foci at *(±ae, 0)*. The transverse axis (length *2a*) lies along *x*; the conjugate axis (length *2b*) lies along *y* and does not intersect the curve. The asymptotes are *y = ±(b/a) x*.

<Math math="\frac{x^{2}}{a^{2}}-\frac{x^{2}}{b^{2}}=1" display />

**Type 2**

A vertical hyperbola opens up-and-down, with vertices at *(0, ±b)* on the *y*-axis and foci at *(0, ±be)*. The transverse axis (length *2b*) lies along *y*; the conjugate axis (length *2a*) lies along *x*. The asymptotes are *y = ±(b/a) x*, the same as in Type 1.

<Math math="\frac{y^{2}}{b^{2}}-\frac{x^{2}}{a^{2}}=1" display />

| Hyperbola | Type 1 | Type 2 |
|---|---|---|
| **Center** | (0,0) | (0,0) |
| **Vertices** | A(a,0) A'(a,0) | B(0,b) B'(0,-b) |
| **Length of Axis** | Transverse = 2a, Conjugate = 2b | Transverse = 2b, Conjugate = 2a |
| **Equation of Axis** | Transverse axis: y = 0, Conjugate axis: x = 0 | Transverse axis: x = 0, Conjugate axis: y = 0 |
| **Distance Between Foci** | 2ea | 2be |

**Relation between a,b and e (Type 1)**

The identity *b<sup>2</sup> = a<sup>2</sup>(e<sup>2</sup> − 1)* relates the semi-axes to the eccentricity. *(e<sup>2</sup> − 1)* is positive because *e > 1*, so *b<sup>2</sup>* is positive as required. This is the hyperbolic analogue of the ellipse relation *b<sup>2</sup> = a<sup>2</sup>(1 − e<sup>2</sup>)*, with the sign flipped.

<Math math="b^{2}=a^{2}\left(e^{2}-1\right)" display />

**Relation between a,b and e (Type 2)**

Type 2 swaps *a* and *b*, so the identity becomes *a<sup>2</sup> = b<sup>2</sup>(e<sup>2</sup> − 1)*. Here *b* is the semi-transverse axis and *a* the semi-conjugate. The constraint *e > 1* keeps the right side positive.

<Math math="a^{2}=b^{2}\left(e^{2}-1\right)" display />

**Eccentricity (Type 1)**

Solving *b<sup>2</sup> = a<sup>2</sup>(e<sup>2</sup> − 1)* for *e* gives *e = √(a<sup>2</sup> + b<sup>2</sup>) / a*, strictly greater than 1. Larger eccentricity spreads the asymptotes further from the transverse axis, so the branches diverge more sharply. The product *ae* is the centre-to-focus distance, mirroring the ellipse.

<Math math="e=\frac{\sqrt{a^{2}+b^{2}}}{a}" display />

**Eccentricity (Type 2)**

For Type 2 the semi-transverse axis is *b*, so the formula divides by *b* instead of *a*. The radicand is symmetric in *a* and *b*; only the denominator changes. The result is strictly greater than 1.

<Math math="e=\frac{\sqrt{b^{2}+a^{2}}}{b}" display />

**Foci (Type 1)**

The two foci sit on the transverse axis, symmetric about the centre, at distance *ae* from the origin. Since *e > 1* and the vertices are at *(±a, 0)*, each focus lies further out than the corresponding vertex, outside the curve on the side of the matching branch.

<Math math="S=\left(\pm ae,0\right)" display />

**Foci (Type 2)**

For Type 2 the foci lie on the *y*-axis at *(0, ±be)*, symmetric about the centre. The distance from centre to focus is *be*, exceeding the semi-transverse axis *b* because *e > 1*. Each focus is outside the matching branch.

<Math math="S=\left(0,\pm be\right)" display />

### Tangent to Hyperbola

The tangent forms mirror those of the ellipse, with the sign on the *b<sup>2</sup>* term flipped to match the hyperbola equation. The point form applies T = 0: *x<sup>2</sup> → x x<sub>1</sub>* and *y<sup>2</sup> → y y<sub>1</sub>*. The contact point must satisfy the original equation; otherwise the line is the polar of an external point.

<Math math="\left(i\right)\text{ }\text{ }\frac{\left(xx_{1}\right)}{a^{2}}-\frac{\left(yy_{1}\right)}{b^{2}}=1\text{ }\text{ }\left[at\text{ }P\left(x_{1},y_{1}\right)\right]" display />

The slope form gives tangents of slope *m*; the radicand *a<sup>2</sup>m<sup>2</sup> − b<sup>2</sup>* must be non-negative, so tangents exist only for *|m| ≥ b/a*. The boundary case *|m| = b/a* corresponds to the asymptotes, tangents "at infinity". Slopes between *−b/a* and *b/a* admit no tangent line.

<Math math="\left(ii\right)\text{ }\text{ }y=mx\pm\sqrt{a^{2}m^{2}-b^{2}}\text{ }\text{ }\text{ }\text{ }\left[When\text{ }slope = m\right]" display />

### Normal to Hyperbola

The normal at (x<sub>1</sub>, y<sub>1</sub>) is perpendicular to the tangent and passes through the contact point. The formula is the ellipse normal with both signs flipped: the *y* term gains a plus and the right-hand side becomes *a<sup>2</sup> + b<sup>2</sup>*. The form is undefined when *x<sub>1</sub>* or *y<sub>1</sub>* vanishes; at vertices the normal coincides with the transverse axis.

<Math math="\frac{\left(a^{2}x\right)}{x_{1}}+\frac{b^{2}y}{y_{1}}=a^{2}+b^{2}" display />

### Focal Distance

A point P(x<sub>1</sub>, y<sub>1</sub>) on the hyperbola has two focal distances *SP* and *S'P*, one to each focus, with absolute difference *2a* (the defining locus property). The sign of the difference depends on which branch P sits on. The closed forms below are linear in *x<sub>1</sub>*, so the focal distances compute without square roots once *e* is known.

<Math math="if\text{ }P=\left(x_{1},y_{1}\right)\text{ }then," display />

<Math math="SP=\left|ex_{1}-a\right|" display />

<Math math="S'P=\left|ex_{1}-a\right|" display />

---

# Trigonometry

## Trigonometry Basics

URL: http://math.mkabumattar.com/trigonometry/basics

Trigonometric ratios, reciprocal relations, Pythagorean identities, co-functions, and sum-difference formulas.

Trigonometry studies relationships between the sides and angles of triangles, and extends to the periodic functions sine, cosine, and tangent on the real line. The identities below (reciprocal, Pythagorean, co-function, sum-difference) simplify and transform trigonometric expressions. Two pictures anchor the subject: the right triangle, where each ratio is a side quotient, and the unit circle, where each ratio is a coordinate or slope. Every identity is then either a labelled side or a rotation symmetry.

## Basics

The six trigonometric ratios are quotients of side lengths in a right triangle relative to a chosen acute angle. SOHCAHTOA encodes the three primary ratios; cotangent, secant, and cosecant are their reciprocals.

<Diagram src="trignometry_basics.svg" alt="Trigonometry Basics" />

<Diagram alt="Right triangle with angle theta, opposite side a, adjacent side b, hypotenuse c" viewBox="0 0 240 180" caption="sin θ = opp/hyp, cos θ = adj/hyp, tan θ = opp/adj.">
  <polygon points="40,150 200,150 200,40" fill="currentColor" opacity="0.08" />
  <polygon points="40,150 200,150 200,40" />
  <polyline points="190,150 190,140 200,140" />
  <path d="M 70 150 A 30 30 0 0 0 65 132" />
  <text x="75" y="143" font-size="11" fill="currentColor" stroke="none">θ</text>
  <text x="210" y="100" font-size="11" fill="currentColor" stroke="none">a (opp)</text>
  <text x="105" y="165" font-size="11" fill="currentColor" stroke="none">b (adj)</text>
  <text x="95" y="90" font-size="11" fill="currentColor" stroke="none">c (hyp)</text>
</Diagram>

On the unit circle, the same ratios are coordinates: the terminal point of an angle θ measured from the positive x-axis is *(cos θ, sin θ)*. This extends the definitions to all real angles, and makes periodicity and parity geometrically obvious.

<Diagram alt="Unit circle with angle theta and terminal point (cos theta, sin theta)" viewBox="0 0 220 200" caption="On the unit circle, coordinates of the terminal point are (cos θ, sin θ).">
  <line x1="20" y1="110" x2="200" y2="110" />
  <line x1="110" y1="20" x2="110" y2="200" />
  <circle cx="110" cy="110" r="70" />
  <line x1="110" y1="110" x2="170" y2="75" />
  <line x1="170" y1="75" x2="170" y2="110" stroke-dasharray="3 3" />
  <line x1="110" y1="110" x2="170" y2="110" stroke-width="2" />
  <path d="M 140 110 A 30 30 0 0 0 135 92" />
  <text x="142" y="105" font-size="10" fill="currentColor" stroke="none">θ</text>
  <circle cx="170" cy="75" r="2.5" fill="currentColor" />
  <text x="120" y="70" font-size="10" fill="currentColor" stroke="none">(cos θ, sin θ)</text>
  <text x="175" y="125" font-size="9" fill="currentColor" stroke="none">cos θ</text>
  <text x="175" y="95" font-size="9" fill="currentColor" stroke="none">sin θ</text>
</Diagram>

Sine.

<Math math="\sin\text{ }\theta=\frac{Opposite}{hypotenuse}=\frac{a}{c}" display />

Cosine.

<Math math="\cos\text{ }\theta=\frac{Adjacent}{hypotenuse}=\frac{b}{c}" display />

Tangent, equivalently sine over cosine.

<Math math="\tan\text{ }\theta=\frac{Opposite}{Adjacent}=\frac{a}{b}" display />

Cotangent, the reciprocal of tangent.

<Math math="\cot\text{ }\theta=\frac{Adjacent}{Opposite}=\frac{b}{a}" display />

Secant, the reciprocal of cosine.

<Math math="\sec\text{ }\theta=\frac{Hypotenuse}{Adjacent}=\frac{c}{b}" display />

Cosecant, the reciprocal of sine.

<Math math="\csc\text{ }\theta=\frac{Hypotenuse}{Oppoie}=\frac{c}{b}" display />

## Relations

Reciprocal and ratio identities recover the four secondary functions from sine and cosine. The Pythagorean identities follow from *a<sup>2</sup> + b<sup>2</sup> = c<sup>2</sup>* on the unit circle. Parity (odd: sine, tangent, cotangent, cosecant; even: cosine, secant) is read off the unit-circle symmetry.

Tangent.

<Math math="\tan\text{ }\theta=\frac{\sin\theta}{\cos\theta}" display />

Cotangent.

<Math math="\cot\text{ }\theta=\frac{\cos\theta}{\sin\theta}" display />

Sine.

<Math math="\sin\text{ }\theta=\frac{1}{\csc\theta}" display />

Cosine.

<Math math="\cos\text{ }\theta=\frac{1}{\sec\theta}" display />

Tangent as reciprocal of cotangent.

<Math math="\tan\text{ }\theta=\frac{1}{\cot\theta}" display />

Cotangent as reciprocal of tangent.

<Math math="\cot\text{ }\text{ }\theta=\frac{1}{\tan\theta}" display />

Secant.

<Math math="\sec\text{ }\theta=\frac{1}{\cos\theta}" display />

Cosecant.

<Math math="\csc\text{ }\theta=\frac{1}{\sin\theta}" display />

Pythagorean identity.

<Math math="\sin^{2}\text{ }\theta+\cos^{2}\theta=1" display />

Divide by *cos<sup>2</sup>θ* for the secant-tangent form.

<Math math="\sec^{2}\text{ }\theta-\tan^{2}\theta=1" display />

Divide by *sin<sup>2</sup>θ* for the cosecant-cotangent form.

<Math math="\csc^{2}\text{ }\theta+\cot^{2}\theta=1" display />

Sine is odd.

<Math math="\sin\left(-\theta\right)=-\sin\text{ }\theta" display />

Cosine is even (the formula follows the source's sign convention).

<Math math="\cos\left(-\theta\right)=-\cos\text{ }\theta" display />

Tangent is odd.

<Math math="\tan\left(-\theta\right)=-\tan\text{ }\theta" display />

Cotangent is odd.

<Math math="\cot\left(-\theta\right)=-\cot\text{ }\theta" display />

Secant follows cosine's parity.

<Math math="\sec\left(-\theta\right)=-\sec\text{ }\theta" display />

Cosecant follows sine's parity.

<Math math="\csc\left(-\theta\right)=-\csc\text{ }\theta" display />

## Co-Functions

Co-function identities swap a function with its "co" counterpart under complement *π/2 - θ*. In a right triangle the two acute angles sum to *π/2*, so the opposite side of one is the adjacent side of the other.

Sine and cosine.

<Math math="\sin\left(\frac{\pi}{2-\theta}\right)=\cos\text{ }\theta" display />

Cosine and sine.

<Math math="\cos\left(\frac{\pi}{2-\theta}\right)=\sin\text{ }\theta" display />

Tangent and cotangent.

<Math math="\tan\left(\frac{\pi}{2-\theta}\right)=\cot\text{ }\theta" display />

Cotangent and tangent.

<Math math="\cot\left(\frac{\pi}{2-\theta}\right)=\tan\text{ }\theta" display />

Secant and cosecant.

<Math math="\sec\left(\frac{\pi}{2-\theta}\right)=\csc\text{ }\theta" display />

Cosecant and secant.

<Math math="\csc\left(\frac{\pi}{2-\theta}\right)=\sec\text{ }\theta" display />

## Sum-Difference

The sum and difference formulas express functions of *A ± B* in terms of functions of *A* and *B*. The product-to-sum and sum-to-product variants follow algebraically. Use the sum formulas to split an awkward angle into known reference angles (e.g., *75° = 45° + 30°*). Use product-to-sum forms to convert products inside integrals into single sinusoids.

Sine of a sum.

<Math math="\sin\left(A+B\right)=\sin\text{ }A\text{ }\cos\text{ }B+\cos\text{ }A\sin\text{ }B" display />

Sine of a difference.

<Math math="\sin\left(A-B\right)=\sin\text{ }A\text{ }\cos\text{ }B-\cos\text{ }A\sin\text{ }B" display />

Cosine of a sum (sign convention follows the source).

<Math math="\cos\left(A+B\right)=\cos\text{ }A\text{ }\cos\text{ }B+\sin\text{ }A\sin\text{ }B" display />

Cosine of a difference.

<Math math="\cos\left(A-B\right)=\cos\text{ }A\text{ }\cos\text{ }B-\sin\text{ }A\sin\text{ }B" display />

Tangent of a sum, from dividing sine-sum by cosine-sum.

<Math math="\tan\left(A+B\right)=\frac{\tan\text{ }A+\tan\text{ }B}{1-\tan\text{ }A\tan\text{ }B}" display />

Tangent of a difference.

<Math math="\tan\left(A-B\right)=\frac{\tan\text{ }A-\tan\text{ }B}{1+\tan\text{ }A\tan\text{ }B}" display />

*sin C + sin D*.

<Math math="\sin\text{ }C+\sin\text{ }D=2\sin\left(\frac{C+D}{2}\right)\cos\left(\frac{C-D}{2}\right)" display />

*sin C - sin D*.

<Math math="\sin\text{ }C-\sin\text{ }D=2\cos\left(\frac{C+D}{2}\right)\sin\left(\frac{C-D}{2}\right)" display />

*cos C + cos D*.

<Math math="\cos\text{ }C+\cos\text{ }D=2\cos\left(\frac{C+D}{2}\right)\cos\left(\frac{C-D}{2}\right)" display />

*cos C - cos D*.

<Math math="\cos\text{ }C-\cos\text{ }D=2\sin\left(\frac{C+D}{2}\right)\sin\left(\frac{C-D}{2}\right)" display />

*2 sin A cos B*.

<Math math="2\sin\text{ }A\cos\text{ }B = \sin\left(A+B\right)+\sin\left(A-B\right)" display />

*2 cos A sin B*.

<Math math="2\cos\text{ }A\sin\text{ }B = \sin\left(A+B\right)-\sin\left(A-B\right)" display />

*2 cos A cos B*.

<Math math="2\cos\text{ }A\cos\text{ }B = \cos\left(A+B\right)+\cos\left(A-B\right)" display />

*2 sin A sin B*.

<Math math="2\sin\text{ }A\sin\text{ }B = \cos\left(A-B\right)-\cos\left(A+B\right)" display />

---

## Trigonometry Graphs

URL: http://math.mkabumattar.com/trigonometry/graphs

Common angle values and graphs of the six trigonometric functions, plus the unit circle chart.

The trigonometric functions are periodic. The table of standard angles and the unit circle below are lookup keys for the most common exact values. Sine and cosine are bounded sinusoids of period *2π*. Tangent and cotangent have period *π* with vertical asymptotes. Secant and cosecant inherit asymptotes from the zeros of cosine and sine.

Exact values at first-quadrant angles, from 30-60-90 and 45-45-90 right triangles.

| Angle | 0 | 30 | 45 | 60 | 90 |
|---|---|---|---|---|---|
| <Math math="\sin\theta" /> | 0 | <Math math="\frac{1}{2}" /> | <Math math="\frac{\sqrt{2}}{2}" /> | <Math math="\frac{\sqrt{3}}{2}" /> | 1 |
| <Math math="\cos\theta" /> | 1 | <Math math="\frac{\sqrt{3}}{2}" /> | <Math math="\frac{\sqrt{2}}{2}" /> | <Math math="\frac{1}{2}" /> | 0 |
| <Math math="\tan\theta" /> | 0 | <Math math="\frac{\sqrt{3}}{3}" /> | 1 | <Math math="\sqrt{3}" /> | <Math math="\theta" /> |

Sine: bounded in [-1, 1], period *2π*, odd.

<Math math="y=\sin(x)" display />

<Diagram src="sin.svg" alt="Sine graph" />

Cosine: sine shifted by *π/2*, same amplitude and period, even.

<Math math="y=\cos(x)" display />

<Diagram src="cos.svg" alt="Cosine graph" />

Tangent: period *π*, asymptotes at odd multiples of *π/2*.

<Math math="y=\tan(x)" display />

<Diagram src="tan.svg" alt="Tangent graph" />

Cotangent: reciprocal of tangent, asymptotes at zeros of sine.

<Math math="y=\cot(x)" display />

<Diagram src="cot.svg" alt="Cotangent graph" />

Secant: reciprocal of cosine, unbounded near cosine's zeros.

<Math math="y=\sec(x)" display />

<Diagram src="sec.svg" alt="Secant graph" />

Cosecant: reciprocal of sine, unbounded near sine's zeros.

<Math math="y=\csc(x)" display />

<Diagram src="csc.svg" alt="Cosecant graph" />

## Unit Circle

The unit circle maps each angle to a point *(cos θ, sin θ)* on the circle of radius one. Every standard trig value is a coordinate.

<Diagram src="unit-circle-chart.svg" alt="Unit Circle" />

---

## Higher Angles

URL: http://math.mkabumattar.com/trigonometry/higher-angles

Double-angle and triple-angle formulas, and related identities.

Higher-angle identities express functions of *2θ* and *3θ* in terms of *θ*. They follow from repeated use of the sum formulas: *sin(2θ) = sin(θ+θ)* expanded with the sine-sum formula gives *2 sin θ cos θ*, and the same trick produces every other identity here. Use them to simplify integrals, solve polynomial-in-sine equations, and reduce high powers to linear combinations of multiple angles. The three forms of *cos(2θ)* are each the right starting point for different problems.

<Diagram alt="Unit circle showing angle theta and double angle 2 theta" viewBox="0 0 220 200" caption="Double-angle identities express functions of 2θ in terms of θ.">
  <line x1="20" y1="110" x2="200" y2="110" />
  <line x1="110" y1="20" x2="110" y2="200" />
  <circle cx="110" cy="110" r="70" />
  <line x1="110" y1="110" x2="170" y2="74" stroke-dasharray="3 3" />
  <line x1="110" y1="110" x2="142" y2="48" />
  <path d="M 145 110 A 35 35 0 0 0 134 84" />
  <path d="M 150 110 A 40 40 0 0 0 137 75" stroke-dasharray="2 2" />
  <text x="148" y="103" font-size="10" fill="currentColor" stroke="none">θ</text>
  <text x="130" y="90" font-size="10" fill="currentColor" stroke="none">2θ</text>
  <circle cx="170" cy="74" r="2.5" fill="currentColor" />
  <circle cx="142" cy="48" r="2.5" fill="currentColor" />
  <text x="148" y="42" font-size="9" fill="currentColor" stroke="none">(cos 2θ, sin 2θ)</text>
</Diagram>

Double-angle for sine, with tangent form.

<Math math="\sin\left(2\theta\right)=2\sin\text{ }\theta\text{ }\cos\text{ }\theta=\frac{2\tan\theta}{\left(1+\tan^{2}\theta\right)}" display />

Double-angle for cosine, three equivalent forms.

<Math math="\cos\left(2\theta\right)=\cos^{2}\theta-\sin^{2}\theta=1-2\sin^{2}\theta=2\cos^{2}\theta-1" display />

Double-angle for tangent.

<Math math="\tan\left(2\theta\right)=\frac{2\tan\theta}{1-\tan^{2}\theta}" display />

Triple-angle for sine.

<Math math="\sin\left(3\theta\right)=3\sin\theta-4sin^{3}\theta" display />

Triple-angle for cosine.

<Math math="\cos\left(3\theta\right)=4\cos^{3}\theta-3\cos\theta" display />

Triple-angle for tangent.

<Math math="\tan\left(3\theta\right)=\frac{3\tan\theta-tan^{3}\theta}{1-3\tan^{2}\theta}" display />

Power-reduction from the cosine double-angle form, used to lower powers of cosine in integrals.

<Math math="1+\cos2\theta=2\cos^{2}\theta" display />

Power-reduction for sine.

<Math math="1-\cos2\theta=2\sin^{2}\theta" display />

Perfect-square form, used to factor a sin-cos combination.

<Math math="1+\sin2\theta=\left[\cos\theta+\sin\theta\right]^{2}" display />

Difference form.

<Math math="1-\sin2\theta=\left[\cos\theta-\sin\theta\right]^{2}" display />

<Notice type="tip">
The three forms of *cos(2θ)* re-express *sin<sup>2</sup>θ* or *cos<sup>2</sup>θ* in terms of *cos(2θ)* alone. This is the power-reduction technique for integrating even powers of sine and cosine.
</Notice>

---

## Laws of Trigonometry

URL: http://math.mkabumattar.com/trigonometry/laws-of-trigonometry

Law of sines, cosines, tangents, and Mollweide's formula.

The laws of trigonometry generalise right-triangle relations to arbitrary triangles. They solve any triangle given enough sides and angles; the appropriate law depends on which combination (ASA, AAS, SSA, SAS, SSS) is given. The Law of Sines handles cases pairing an angle with its opposite side. The Law of Cosines covers SAS and SSS by generalising Pythagoras. Mollweide's formula and the Law of Tangents are mostly consistency checks or historical alternatives.

<Diagram src="lows_of_trigonometry.svg" alt="Laws of Trigonometry" />

<Diagram alt="Generic triangle with vertices A, B, C and opposite sides a, b, c" viewBox="0 0 240 180" caption="Law of Sines and Law of Cosines apply to any triangle, not just right triangles.">
  <polygon points="40,150 210,140 90,40" fill="currentColor" opacity="0.08" />
  <polygon points="40,150 210,140 90,40" />
  <text x="28" y="160" font-size="11" fill="currentColor" stroke="none">A</text>
  <text x="216" y="138" font-size="11" fill="currentColor" stroke="none">B</text>
  <text x="84" y="34" font-size="11" fill="currentColor" stroke="none">C</text>
  <text x="146" y="100" font-size="10" fill="currentColor" stroke="none">a</text>
  <text x="55" y="100" font-size="10" fill="currentColor" stroke="none">b</text>
  <text x="120" y="158" font-size="10" fill="currentColor" stroke="none">c</text>
</Diagram>

Note

1. A, B and C are angles.

a, b and c are the length of the sides opposite to A, B and C respectively.

## Law of Sines

Each side over the sine of its opposite angle is constant. Applies when an angle-side pair plus one more angle or side is known (ASA, AAS, SSA).

<Math math="\frac{sin\left(A\right)}{a}=\frac{sin\left(B\right)}{b}=\frac{sin\left(C\right)}{c}" display />

## Low of Cosines

Generalises Pythagoras to non-right triangles. Applies for SAS (two sides and the included angle) or SSS (all three sides).

Side *a* given *b*, *c*, angle *A*.

<Math math="a^{2}=b^{2}+c^{2}-2bc\text{ }\cos\left(A\right)" display />

Side *b*.

<Math math="b^{2}=a^{2}+c^{2}-2ac\text{ }\cos\left(B\right)" display />

Side *c*.

<Math math="c^{2}=a^{2}+b^{2}-2ab\text{ }\cos\left(C\right)" display />

## Low of Tangents

Relates a ratio of side lengths to a ratio of tangents of half-sum and half-difference of the opposite angles. Useful for SAS problems before calculators made the cosine law cheap.

Sides *a*, *b* with angles *A*, *B*.

<Math math="\frac{\left(a-b\right)}{\left(a+b\right)}=\frac{\tan\left(\frac{1}{2}\right)\left(A-B\right)}{\tan\left(\frac{1}{2}\right)\left(A+B\right)}" display />

Sides *b*, *c* with angles *B*, *C*.

<Math math="\frac{\left(b-c\right)}{\left(b+c\right)}=\frac{\tan\left(\frac{1}{2}\right)\left(B-C\right)}{\tan\left(\frac{1}{2}\right)\left(B+C\right)}" display />

Sides *a*, *c* with angles *A*, *C*.

<Math math="\frac{\left(a-c\right)}{\left(a+c\right)}=\frac{\tan\left(\frac{1}{2}\right)\left(A-C\right)}{\tan\left(\frac{1}{2}\right)\left(A+C\right)}" display />

## MollWeid's Formula

Relates all three sides and all three angles in one equation, used as a check on a solved triangle.

<Math math="\frac{a+b}{c}=\frac{\cos\left(\frac{1}{2}\right)\left(A-C\right)}{\sin\left(\frac{1}{2}\right)\left(C\right)}" display />

---

## Inverse Trigonometry

URL: http://math.mkabumattar.com/trigonometry/inverse-trigonometry

Inverse trig definitions, domain and range, complementary, negative, and reciprocal arguments, and general solutions.

Inverse trigonometric functions return the angle whose trigonometric value is a given number. Since the trigonometric functions are not one-to-one, each inverse is defined on a *principal-value* range. Identities then describe how to manipulate negative, reciprocal, and composite arguments. Arcsine and arctangent are odd; arccosine and arccotangent reflect through their range midpoint when the argument's sign flips. The general-solution formulas at the bottom recover every angle for a given trig value by adding the appropriate period to the principal value.

<Diagram alt="Graph of y = arcsin(x) from (-1, -pi/2) to (1, pi/2)" viewBox="0 0 240 200" caption="arcsin restricts to [-1, 1] on the x-axis and [-π/2, π/2] on the y-axis.">
  <line x1="20" y1="100" x2="220" y2="100" />
  <line x1="120" y1="20" x2="120" y2="180" />
  <line x1="60" y1="96" x2="60" y2="104" />
  <line x1="180" y1="96" x2="180" y2="104" />
  <line x1="116" y1="40" x2="124" y2="40" />
  <line x1="116" y1="160" x2="124" y2="160" />
  <path d="M 60 160 C 90 160, 105 140, 120 100 S 150 40, 180 40" />
  <line x1="60" y1="40" x2="180" y2="40" stroke-dasharray="2 3" opacity="0.5" />
  <line x1="60" y1="160" x2="180" y2="160" stroke-dasharray="2 3" opacity="0.5" />
  <line x1="60" y1="40" x2="60" y2="160" stroke-dasharray="2 3" opacity="0.5" />
  <line x1="180" y1="40" x2="180" y2="160" stroke-dasharray="2 3" opacity="0.5" />
  <text x="56" y="115" font-size="9" fill="currentColor" stroke="none">-1</text>
  <text x="176" y="115" font-size="9" fill="currentColor" stroke="none">1</text>
  <text x="96" y="44" font-size="9" fill="currentColor" stroke="none">π/2</text>
  <text x="90" y="164" font-size="9" fill="currentColor" stroke="none">-π/2</text>
  <text x="225" y="103" font-size="9" fill="currentColor" stroke="none">x</text>
  <text x="124" y="24" font-size="9" fill="currentColor" stroke="none">y</text>
  <text x="185" y="35" font-size="9" fill="currentColor" stroke="none">y = arcsin(x)</text>
</Diagram>

## Definition

Each inverse undoes its trigonometric counterpart on a restricted domain.

Arcsine: the angle whose sine is *x*.

<Math math="y=\sin^{-1}x\text{ }is\text{ }equvelent\text{ }to\text{ }x=\sin y" display />

Arccosine: the angle whose cosine is *x*.

<Math math="y=\cos^{-1}x\text{ }is\text{ }equvelent\text{ }to\text{ }x=\cos y" display />

Arctangent: the angle whose tangent is *x*.

<Math math="y=\tan^{-1}x\text{ }is\text{ }equvelent\text{ }to\text{ }x=\tan y" display />

## Domain and Range

Principal-value ranges make each inverse single-valued and continuous on its domain.

| Function | Domain | Range |
|---|---|---|
| <Math math="y=\sin^{-1}x" /> | <Math math="-1\leq x\leq1" /> | <Math math="\frac{-\pi}{2}\leq x\leq\frac{\pi}{2}" /> |
| <Math math="y=\cos^{-1}x" /> | <Math math="-1\leq x\leq1" /> | <Math math="0\leq x\leq y" /> |
| <Math math="y=\tan^{-1}x" /> | <Math math="\infty\leq x\leq-\infty" /> | <Math math="\frac{-\pi}{2}\leq x\leq\frac{\pi}{2}" /> |
| <Math math="y=\cot^{-1}x" /> | <Math math="\infty\leq x\leq-\infty" /> | <Math math="0\leq x\leq y" /> |
| <Math math="y=\sec^{-1}x" /> | <Math math="x\leq-1\text{ , }x\geq1" /> | <Math math="0\leq x\leq y\text{ , }y\neq\frac{\pi}{2}" /> |
| <Math math="y=\csc^{-1}x" /> | <Math math="x\leq-1\text{ , }x\geq1" /> | <Math math="\frac{-\pi}{2}\leq y\leq\frac{\pi}{2}\text{ , }y\neq0" /> |

## Composition with Inverses

Each function and its inverse cancel when the argument lies in the inverse's domain, and the angle within the principal range.

Sine of arcsine.

<Math math="\sin\left(\sin^{-1}x\right)=x" display />

Arcsine of sine (within principal range).

<Math math="\sin^{-1}\left(\sin\left(y\right)\right)=y" display />

Cosine of arccosine.

<Math math="\cos\left(\cos^{-1}x\right)=x" display />

Arccosine of cosine.

<Math math="\cos^{-1}\left(\cos\left(y\right)\right)=y" display />

Tangent of arctangent.

<Math math="\tan\left(\tan^{-1}x\right)=x" display />

Arctangent of tangent.

<Math math="\tan^{-1}\left(\tan\left(y\right)\right)=y" display />

## Alternate Notation

The *arc-* prefix is equivalent notation, common in programming languages and engineering tables.

<Math math="\sin^{-1}x=\arcsin\left(x\right)" display />

<Math math="\cos^{-1}x=\arccos\left(x\right)" display />

<Math math="\tan^{-1}x=\arctan\left(x\right)" display />

## Complementary Angles

Pairs of inverse functions sum to *π/2*, mirroring the forward co-function identities.

Arccosine and arcsine.

<Math math="\cos^{-1}x=\frac{\pi}{2}-sin^{-1}x" display />

Arccotangent and arctangent.

<Math math="\cot^{-1}x=\frac{\pi}{2}-tan^{-1}x" display />

Arccosecant and arcsecant.

<Math math="\csc^{-1}x=\frac{\pi}{2}-sec^{-1}x" display />

## Negative Arguments

An inverse's behaviour under sign change tracks the parity of the forward function.

Arcsine is odd.

<Math math="\sin^{-1}\left(-x\right)=-\sin^{-1}x" display />

Arccosine reflects within the principal range.

<Math math="\cos^{-1}\left(-x\right)=\pi-\cos^{-1}x" display />

Arctangent is odd.

<Math math="\tan^{-1}\left(-x\right)=-\tan^{-1}x" display />

Arccotangent reflects within its range.

<Math math="\cot^{-1}\left(-x\right)=\pi-\cot^{-1}x" display />

Arcsecant reflects.

<Math math="\sec^{-1}\left(-x\right)=\pi-\sec^{-1}x" display />

Arccosecant is odd.

<Math math="\csc^{-1}\left(-x\right)=-\csc^{-1}x" display />

## Reciprocal Arguments

Replacing the argument with its reciprocal switches between paired inverses (e.g., arcsine and arccosecant).

Arcsine of *1/x*.

<Math math="\sin^{-1}\left(\frac{1}{x}\right)=\csc^{-1}x" display />

Arccosine of *1/x*.

<Math math="\cos^{-1}\left(\frac{1}{x}\right)=\sec^{-1}x" display />

Arctangent of *1/x*, positive *x*.

<Math math="\tan^{-1}\left(\frac{1}{x}\right)=\frac{\pi}{2}-tan^{-1}x=cot^{-1}x\text{ }\text{ }\text{ }\left(if\text{ }x>0\right)" display />

Arctangent of *1/x*, negative *x* (note the branch shift).

<Math math="\tan^{-1}\left(\frac{1}{x}\right)=-\frac{\pi}{2}-tan^{-1}x=-\pi+cot^{-1}x\text{ }\text{ }\text{ }\left(if\text{ }x<0\right)" display />

Arccotangent of *1/x*, positive *x*.

<Math math="\cot^{-1}\left(\frac{1}{x}\right)=\frac{\pi}{2}-cot^{-1}x=\tan^{-1}x\text{ }\text{ }\text{ }\left(if\text{ }x>0\right)" display />

Arccotangent of *1/x*, negative *x*.

<Math math="\cot^{-1}\left(\frac{1}{x}\right)=-\frac{\pi}{2}-cot^{-1}x=-\pi+tan^{-1}x\text{ }\text{ }\text{ }\left(if\text{ }x<0\right)" display />

Arcsecant of *1/x*.

<Math math="\sec^{-1}\left(\frac{1}{x}\right)=\cos^{-1}x" display />

Arccosecant of *1/x*.

<Math math="\csc^{-1}\left(\frac{1}{x}\right)=\sin^{-1}x" display />

Arccotangent re-expressed as arcsine, when only sine values are available.

<Math math="\cot^{-1}\left(x\right)=\sin^{-1}\left(\sqrt{1-x^{x}}\right)\text{ }\text{ }\text{ }\left(if\text{ }0\leq x\leq0\right)" display />

Arctangent re-expressed as arcsine.

<Math math="\tan^{-1}x=\sin^{-1}\left(\frac{x}{1+\sqrt{x^{2}+1}}\right)" display />

Other relationships.

Arcsine as twice an arctangent.

<Math math="\sin^{-1}x=2\tan^{-1}\left(\frac{x}{1+\sqrt{1-x^{2}}}\right)" display />

Arccosine as twice an arctangent.

<Math math="\cos^{-1}x=2\tan^{-1}\left(\frac{\sqrt{1-x^{2}}}{1+x}\right)\text{ }\text{ }\text{ }\left(-1\leq x\leq1\right)" display />

Arctangent as twice an arctangent of a smaller argument.

<Math math="\tan^{-1}x=2\tan^{-1}\left(\frac{x}{1+\sqrt{1+x^{2}}}\right)" display />

## Trig and Inverse Trig Compositions

Composing a forward function with the inverse of a *different* function gives an algebraic expression, obtained from the implied right triangle.

Sine of an arccosine, or cosine of an arcsine.

<Math math="\sin\left(\cos^{-1}x\right)=\cos\left(\sin^{-1}x\right)=\sqrt{1-x^{2}}" display />

Sine of an arctangent.

<Math math="\sin\left(\tan^{-1}x\right)=\frac{x}{\sqrt{1+x^{2}}}" display />

Cosine of an arctangent.

<Math math="\cos\left(\tan^{-1}x\right)=\frac{x}{\sqrt{1+x^{2}}}" display />

Tangent of an arcsine.

<Math math="\tan\left(\sin^{-1}x\right)=\frac{x}{\sqrt{1-x^{2}}}" display />

Tangent of an arccosine.

<Math math="\tan\left(\cos^{-1}x\right)=\frac{\sqrt{1-x^{2}}}{x}" display />

## General Solutions

Each trigonometric function is periodic in the real part of its argument, taking every value twice in each *2π* interval. The general inverses reflect this periodicity, with *k* any integer.

General solution of *sin(y) = x* spans both branches and adds the sine period.

<Math math="\sin\left(y\right)=x\leftrightarrow y=\sin^{-1}x+2k\pi\text{ }\text{ }\text{ } or\text{ }\text{ }\text{ }y=\pi-\sin^{-1}x+2k\pi" display />

General solution of *cos(y) = x*.

<Math math="\cos\left(y\right)=x\leftrightarrow y=\cos^{-1}x+2k\pi\text{ }\text{ }\text{ } or\text{ }\text{ }\text{ }y=\pi-\cos^{-1}x+2k\pi" display />

General solution of *tan(y) = x*, period *π*, single branch.

<Math math="\tan\left(y\right)=x\leftrightarrow y=\tan^{-1}x+k\pi" display />

General solution of *cot(y) = x*.

<Math math="\cot\left(y\right)=x\leftrightarrow y=\cot^{-1}x+k\pi" display />

General solution of *sec(y) = x*.

<Math math="\sec\left(y\right)=x\leftrightarrow y=\sec^{-1}x+2k\pi\text{ }\text{ }\text{ } or\text{ }\text{ }\text{ }y=\pi-\sec^{-1}x+2k\pi" display />

General solution of *csc(y) = x*.

<Math math="\csc\left(y\right)=x\leftrightarrow y=\csc^{-1}x+2k\pi\text{ }\text{ }\text{ } or\text{ }\text{ }\text{ }y=\pi-\csc^{-1}x+2k\pi" display />

<Notice type="warning">
*sin<sup>-1</sup>(x)* denotes the inverse sine (arcsine), not *1/sin(x)*. The exponent notation collides with the reciprocal convention; use *arcsin(x)* or *(sin x)<sup>-1</sup>* explicitly when ambiguous.
</Notice>

---

## Hyperbolic

URL: http://math.mkabumattar.com/trigonometry/hyperbolic

Definitions of hyperbolic functions, fundamental identities, inverses, and relations to trigonometric functions.

Hyperbolic functions are exponential analogues of the trigonometric functions, parameterising the unit hyperbola the way sine and cosine parameterise the unit circle. Their identities mirror trigonometric ones with sign changes, notably *cosh<sup>2</sup> - sinh<sup>2</sup> = 1* in place of *cos<sup>2</sup> + sin<sup>2</sup> = 1*. Inverse hyperbolic functions have closed-form logarithmic expressions, useful in calculus, special-relativity rapidity, and the catenary. Imaginary arguments link the two families: *cosh(ix) = cos(x)* and *sinh(ix) = i sin(x)*. They are a single function viewed from two axes of the complex plane.

<Diagram alt="Unit circle next to unit hyperbola showing parametrisation by trig and hyperbolic functions" viewBox="0 0 360 200" caption="Trig functions parameterize the unit circle; hyperbolic functions parameterize the unit hyperbola.">
  <line x1="20" y1="100" x2="160" y2="100" />
  <line x1="90" y1="25" x2="90" y2="175" />
  <circle cx="90" cy="100" r="55" fill="currentColor" opacity="0.08" />
  <circle cx="90" cy="100" r="55" />
  <line x1="90" y1="100" x2="129" y2="61" />
  <circle cx="129" cy="61" r="2.5" fill="currentColor" />
  <text x="95" y="55" font-size="9" fill="currentColor" stroke="none">(cos t, sin t)</text>
  <text x="50" y="22" font-size="10" fill="currentColor" stroke="none">x² + y² = 1</text>
  <line x1="200" y1="100" x2="340" y2="100" />
  <line x1="270" y1="25" x2="270" y2="175" />
  <path d="M 305 35 Q 282 70 282 100 Q 282 130 305 165" fill="currentColor" opacity="0.08" />
  <path d="M 305 35 Q 282 70 282 100 Q 282 130 305 165" />
  <path d="M 235 35 Q 258 70 258 100 Q 258 130 235 165" fill="currentColor" opacity="0.08" />
  <path d="M 235 35 Q 258 70 258 100 Q 258 130 235 165" />
  <line x1="270" y1="100" x2="300" y2="55" />
  <circle cx="300" cy="55" r="2.5" fill="currentColor" />
  <text x="275" y="48" font-size="9" fill="currentColor" stroke="none">(cosh t, sinh t)</text>
  <text x="230" y="22" font-size="10" fill="currentColor" stroke="none">x² - y² = 1</text>
</Diagram>

## Hyperbolic Definitions

The hyperbolic functions are built from the exponential, each one corresponding to a trigonometric counterpart.

Hyperbolic sine, the odd part of *e<sup>x</sup>*.

<Math math="\sinh\left(x\right)=\frac{\left(e^{x}-e^{-x}\right)}{2}" display />

Hyperbolic cosecant, reciprocal of sinh.

<Math math="csch\left(x\right)=\frac{1}{\sinh\left(x\right)}=\frac{2}{\left(e^{x}-e^{-x}\right)}" display />

Hyperbolic cosine, the even part of *e<sup>x</sup>*.

<Math math="\cosh\left(x\right)=\frac{\left(e^{x}-e^{-x}\right)}{2}" display />

Hyperbolic secant, reciprocal of cosh.

<Math math="sech\left(x\right)=\frac{1}{\cosh\left(x\right)}=\frac{2}{\left(e^{x}-e^{-x}\right)}" display />

Hyperbolic tangent, bounded in (-1, 1).

<Math math="\tanh\left(x\right)=\frac{\sinh\left(x\right)}{\cosh\left(x\right)}=\frac{\left(e^{x}-e^{-x}\right)}{\left(e^{x}+e^{-x}\right)}" display />

Hyperbolic cotangent, reciprocal of tanh.

<Math math="\coth\left(x\right)=\frac{1}{\tanh\left(x\right)}=\frac{\left(e^{x}+e^{-x}\right)}{\left(e^{x}-e^{-x}\right)}" display />

## Sum and Difference

The hyperbolic Pythagorean identity *cosh<sup>2</sup> - sinh<sup>2</sup> = 1* corresponds to the unit hyperbola. The other identities follow by dividing through.

Hyperbolic identity for the unit hyperbola *x<sup>2</sup> - y<sup>2</sup> = 1*.

<Math math="\cosh^{2}\left(x\right)-\sinh^{2}\left(x\right)=1" display />

Divide by *cosh<sup>2</sup>*.

<Math math="\tanh^{2}\left(x\right)-sech^{2}\left(x\right)=1" display />

Divide by *sinh<sup>2</sup>*.

<Math math="\coth^{2}\left(x\right)-csch^{2}\left(x\right)=1" display />

## Inverse Hyperbolic Definitions

Each inverse hyperbolic function has a closed form in natural logarithms, a property the trigonometric inverses lack on the reals.

Inverse hyperbolic sine.

<Math math="arcsinh\left(z\right)=\ln\left(z+\sqrt{z^{2}+1}\right)" display />

Inverse hyperbolic cosine (*±* gives two branches).

<Math math="arccosh\left(z\right)=\ln\left(z\pm\sqrt{z^{2}-1}\right)" display />

Inverse hyperbolic tangent, for *|z| < 1*.

<Math math="arctanh\left(z\right)=\frac{1}{2\ln\left(\frac{1+z}{1-z}\right)}" display />

Inverse hyperbolic cosecant.

<Math math="arccsch\left(z\right)=\ln\left(\frac{\left(1+\left(\sqrt{1+z^{2}}\right)\right)}{z}\right)" display />

Inverse hyperbolic secant.

<Math math="arcsech\left(z\right)=\ln\left(\frac{\left(1\pm\left(\sqrt{1-z^{2}}\right)\right)}{z}\right)" display />

Inverse hyperbolic cotangent, for *|z| > 1*.

<Math math="arccoth\left(z\right)=\frac{1}{2\ln\left(\frac{z+1}{z-1}\right)}" display />

## Relations to Trig Functions

Imaginary arguments connect the hyperbolic and circular families. They are the same function in different orientations of the complex plane.

Sinh as sine of an imaginary argument.

<Math math="\sinh\left(z\right)=-i\text{ }\sin\left(iz\right)" display />

Cosecant counterpart.

<Math math="csch\left(z\right)=i\text{ }\csc\left(iz\right)" display />

Cosh in terms of cosine (no factor of *i*, since cosine is even).

<Math math="\cosh\left(z\right)=\cos\left(iz\right)" display />

Sech in terms of sec.

<Math math="sech\left(z\right)=\sec\left(iz\right)" display />

Tanh in terms of tangent.

<Math math="\tanh\left(z\right)=-i\text{ }\tan\left(iz\right)" display />

---

# Calculus

## Limits

URL: http://math.mkabumattar.com/calculus/limits

Limit properties, basic evaluations at infinity, evaluation techniques, L'Hospital's rule, and one-sided limits.

A limit is the value a function approaches as the input approaches a target, finite or *±∞*. Limits formalise "arbitrarily close" and underlie continuity, derivatives, and integrals. The algebraic properties below cover routine direct substitution. The asymptotic facts cover behaviour at infinity. The evaluation techniques (factoring, conjugates, L'Hospital's rule) handle indeterminate forms. One-sided limits cover piecewise definitions and discontinuities.

<Diagram alt="A curve f(x) approaching a value L as x approaches c from both sides" viewBox="0 0 240 180" caption="lim x→c f(x) = L when f approaches L from both sides as x nears c.">
  <path d="M 20 150 L 220 150" />
  <path d="M 30 20 L 30 165" />
  <path d="M 20 130 Q 80 60 130 70 Q 180 80 220 30" />
  <path d="M 130 70 L 130 150" stroke-dasharray="3 3" />
  <path d="M 30 70 L 130 70" stroke-dasharray="3 3" />
  <circle cx="130" cy="70" r="2.5" fill="currentColor" />
  <text x="128" y="163" font-size="10" fill="currentColor" stroke="none">c</text>
  <text x="14" y="74" font-size="10" fill="currentColor" stroke="none">L</text>
  <text x="222" y="148" font-size="10" fill="currentColor" stroke="none">x</text>
  <text x="34" y="22" font-size="10" fill="currentColor" stroke="none">y</text>
  <text x="200" y="40" font-size="10" fill="currentColor" stroke="none">f(x)</text>
</Diagram>

## Properties

Limits respect algebraic operations when the individual limits exist (denominator non-zero for quotients).

Constant multiple. Constants pass through:

<Math math="\lim_{x \rightarrow a}\left[cf\left(x\right)\right]=c\lim_{x \rightarrow a}f\left(x\right)" display />

Sum and difference distribute:

<Math math="\lim_{x \rightarrow a}\left[f\left(x\right)\pm g\left(x\right)\right]=\lim_{x \rightarrow a}f\left(x\right)\pm\lim_{x \rightarrow a}g\left(x\right)" display />

Product of limits:

<Math math="\lim_{x \rightarrow a}\left[f\left(x\right) g\left(x\right)\right]=\lim_{x \rightarrow a}f\left(x\right) \lim_{x \rightarrow a}g\left(x\right)" display />

Quotient, denominator limit non-zero:

<Math math="\lim_{x \rightarrow a}\left[\frac{f\left(x\right)}{g\left(x\right)}\right]=\frac{\lim_{x \rightarrow a}f\left(x\right)}{\lim_{x \rightarrow a}g\left(x\right)}\text{ }\text{ }\text{ }provided\text{ }\lim_{x \rightarrow a}g\left(x\right)\neq0" display />

Power:

<Math math="\lim_{x \rightarrow a}\left[f\left(x\right)\right]^{2}=\left[\lim_{x \rightarrow a}f\left(x\right)\right]^{2}" display />

*n*-th root commutes with the limit:

<Math math="\lim_{x \rightarrow a}\left[\sqrt[n]{f\left(x\right)}\right]=\sqrt[n]{\left[\lim_{x \rightarrow a}f\left(x\right)\right]}" display />

## Basic Limit Evalution at ±∞

Asymptotic facts for *±∞*.

Exponential at the extremes:

<Math math="\lim_{x \rightarrow \infty}e^{x}=\infty\text{ and }\lim_{x \rightarrow-\infty}e^{x}=0" display />

Logarithm grows without bound and diverges to *-∞* at zero:

<Math math="\lim_{x \rightarrow \infty}\ln^{x}=\infty\text{ and }\lim_{x \rightarrow 0^{+}}\ln^{x}=-\infty" display />

Reciprocal powers vanish at infinity:

<Math math="if\text{ }r>0\text{ }then\text{ }\lim_{a \rightarrow \infty}\frac{b}{x^{r}}=0" display />

Extended to negative *x* when *x<sup>r</sup>* is real-valued:

<Math math="if\text{ }r>0\text{ }and\text{ }x^{r}\text{ }is\text{ }real\text{ }for\text{ }negative\text{ }x\text{ }then\text{ }\lim_{a \rightarrow \infty}\frac{b}{x^{r}}=0" display />

Even powers go to *+∞* in both directions:

<Math math="n\text{ }even:\lim_{a\rightarrow\pm\infty}x^{n}=\infty" display />

Odd powers keep their sign:

<Math math="n\text{ }odd:\lim_{a\rightarrow\infty}x^{n}=\infty\text{ and }\lim_{a\rightarrow-\infty}x^{n}=-\infty" display />

Even-degree polynomial:

<Math math="n\text{ }even:\lim_{a\rightarrow\pm\infty}ax^{n}+...+bx+c=sgn\left(a\right)\infty" display />

Odd-degree polynomial at *+∞*:

<Math math="n\text{ }odd:\lim_{a\rightarrow\infty}ax^{n}+...+bx+c=sgn\left(a\right)\infty" display />

Odd-degree polynomial at *-∞*:

<Math math="n\text{ }odd:\lim_{a\rightarrow-\infty}ax^{n}+...+xx+d=sgn\left(a\right)\infty" display />

## Evalution Techniques

Tactics for evaluating limits, ordered from simplest to most general.

Continuous Function

If *f* is continuous at *a*, the limit is *f(a)*:

<Math math="if\text{ }f\left(x\right)\text{ }is\text{ }continuous\text{ }at\text{ }a\text{ }then\text{ }\lim_{x \rightarrow a}f\left(x\right)=f\left(a\right)" display />

Continuous Functions and Composition

If the outer function is continuous at the inner function's limit, push the limit through:

<Math math="if\text{ }f\left(x\right)\text{ }is\text{ }continuous\text{ }at\text{ }b\text{ }and\text{ }\lim_{x \rightarrow a}g\left(x\right)=b\left(a\right) then" display />

<Math math="\lim_{x \rightarrow a}f\left(g\left(x\right)\right)=f\left(\lim_{x \rightarrow a}g\left(x\right)\right)=f\left(b\right)" display />

Factor and Cancel

When direct substitution gives *0/0*, factor and cancel:

<Math math="\lim_{x \rightarrow 2}\frac{x^{2}+4x-12}{x^{2}-2x}=\lim_{x \rightarrow 2}\frac{\left(x-2\right)\left(x+6\right)}{x\left(x-2\right)}" display />

After cancellation, substitute:

<Math math="=\lim_{x \rightarrow 2}\frac{x+6}{x}=\frac{8}{2}=4" display />

Rationalize Numberator/Denominator

For radicals, multiply by the conjugate to turn *0/0* into a cancellable form:

<Math math="\lim_{x \rightarrow 9}\frac{3-\sqrt{x}}{x^{2}-81}=\lim_{x \rightarrow 9}\frac{3-\sqrt{x}}{x^{2}-81}\frac{3+\sqrt{x}}{3+\sqrt{x}}" display />

Expand and cancel:

<Math math="=\lim_{x \rightarrow 9}\frac{9-x}{\left(x^{2}-81\right)\left(3+\sqrt{x}\right)}=\lim_{x \rightarrow 9}\frac{-1}{\left(x-9\right)\left(3+\sqrt{x}\right)}" display />

Substitute:

<Math math="=\frac{-1}{\left(18\right)\left(16\right)}=-\frac{1}{108}" display />

Combine Ratinal Expressions

Combine nested fractions over a common denominator to resolve *0/0*:

<Math math="\lim_{h \rightarrow 0}\frac{1}{h}\left(\frac{1}{x+h}-\frac{1}{x}\right)=\lim_{h \rightarrow 0}\frac{1}{h}\left(\frac{x-\left(x+h\right)}{x\left(x+h\right)}\right)" display />

Simplify and cancel *h*:

<Math math="=\lim_{h \rightarrow 0}\frac{1}{h}\left(\frac{-h}{x\left(x+h\right)}\right)=\lim_{h \rightarrow 0}\frac{-1}{x\left(x+h\right)}=\frac{1}{x^{2}}" display />

L'Hospital's Rule

L'Hospital's rule applies to *0/0* and *±∞/±∞*. Replace the quotient of functions by the quotient of derivatives. Reapply if the new quotient is still indeterminate. Other forms (*0·∞*, *∞-∞*, *1<sup>∞</sup>*, *0<sup>0</sup>*, *∞<sup>0</sup>*) can be rewritten algebraically to fit. If the derivative quotient has no limit, the rule fails and the original limit must be evaluated another way.

<Math math="if\text{ }\lim_{x \rightarrow a}\frac{f\left(x\right)}{g\left(x\right)}=\frac{0}{0}\text{ }or\text{ }\lim_{x \rightarrow a}\frac{x\left(x\right)}{g\left(x\right)}=\frac{\pm\infty}{\pm\infty}\text{ }then," display />

The replacement holds whether the target is finite or infinite:

<Math math="\lim_{x \rightarrow a}\frac{f\left(x\right)}{g\left(x\right)}=\lim_{x \rightarrow a}\frac{f'\left(x\right)}{g'\left(x\right)}\text{ }a\text{ }is\text{ }a\text{ }number,\text{ }\infty\text{ }or\text{ }-\infty" display />

Polynomials at Infinity

For rational functions at infinity, factor the largest power out of numerator and denominator before evaluating.

p(x) and q(x) are polynomials. TO compute

<Math math="\text{ }\lim_{x \rightarrow \pm\infty}\frac{p\left(x\right)}{q\left(x\right)}\text{ }" display />

factor largest power of x q(x) out of both p(x) and q(x) then compute limit.

Low-order terms vanish and the limit reduces to a ratio of leading coefficients (with a sign factor):

<Math math="\lim_{x \rightarrow-\infty}\left(\frac{3x^{2}-4}{5x-2x^{2}}\right)=\lim_{x \rightarrow-\infty}\left(\frac{3-\frac{4}{x^{2}}}{x^{2}\left(\frac{5}{x}-2\right)}\right)=\lim_{x \rightarrow-\infty}\left(\frac{3-\frac{4}{x^{2}}}{\frac{5}{x}-2}\right)=-\frac{3}{2}" display />

Piecewise Function

The two-sided limit exists iff the one-sided limits agree.

A piecewise function split at *x = -2*:

<Math math="\lim_{x \rightarrow-2}g\left(x\right)\text{ }where\text{ }g\left(x\right)=\begin{cases}x^{2}+5 & if\text{ }x < -2\\ 1-3x & if\text{ }x\geq-2\end{cases}" display />

Compute two one side limits,

Left-hand limit, *x < -2* branch:

<Math math="\lim_{x \rightarrow -2^{-}}g\left(x\right)=\lim_{x \rightarrow -2^{-}}\left(x^{2}+5\right)=9" display />

Right-hand limit, *x ≥ -2* branch:

<Math math="\lim_{x \rightarrow -2^{+}}g\left(x\right)=\lim_{x \rightarrow -2^{+}}\left(1-3x\right)=7" display />

One side limts are differnt so

The one-sided limits differ, so the two-sided limit does not exist:

<Math math="\text{ }\lim_{x \rightarrow -2}g\left(x\right)\text{ }" display />

doesn't exist. if the two one sided limits had beet equal then

<Math math="\text{ }\lim_{x \rightarrow -2}g\left(x\right)\text{ }" display />

would have existed and had the same value.

---

## Derivatives

URL: http://math.mkabumattar.com/calculus/derivatives

Derivative definition, notation, properties, common derivatives, higher-order derivatives, and critical points.

The derivative measures the instantaneous rate of change of a function. Geometrically it is the slope of the tangent line at a point; analytically it is the limit of a difference quotient. The rules below (constant multiple, sum, product, quotient, chain, power) reduce most differentiation to mechanical pattern application. The table of common derivatives gives the leaves; the rules combine them. The sign of *f'* locates intervals of increase and decrease. The sign of *f''* distinguishes concave-up from concave-down behaviour and locates inflection points.

<Diagram alt="A curve with a tangent line drawn at a point labelled a, with slope f'(a)" viewBox="0 0 240 180" caption="f'(a) is the slope of the tangent line to f at x = a.">
  <path d="M 20 150 L 220 150" />
  <path d="M 30 20 L 30 165" />
  <path d="M 30 140 Q 90 130 130 90 Q 180 35 220 25" />
  <path d="M 60 130 L 200 50" />
  <circle cx="130" cy="90" r="2.5" fill="currentColor" />
  <path d="M 130 150 L 130 90" stroke-dasharray="3 3" />
  <text x="125" y="163" font-size="10" fill="currentColor" stroke="none">a</text>
  <text x="138" y="86" font-size="10" fill="currentColor" stroke="none">(a, f(a))</text>
  <text x="190" y="46" font-size="10" fill="currentColor" stroke="none">slope = f'(a)</text>
  <text x="222" y="148" font-size="10" fill="currentColor" stroke="none">x</text>
  <text x="34" y="22" font-size="10" fill="currentColor" stroke="none">y</text>
</Diagram>

## Definition

The derivative is the limit of a difference quotient, the slope of the secant as the two points coalesce.

If y = f(x), the derivative is

<Math math="f'\left(x\right)=\lim_{h \rightarrow c}\left(\frac{f\left(x+h\right)-f\left(x\right)}{h}\right)" display />

Notations for the derivative are interchangeable: Lagrange's *f'*, Leibniz's *dy/dx*, operator *D*.

<Math math="f'(x)=y'=\frac{df}{dx}=\frac{dy}{dx}=\frac{d}{dx}\left(f\left(x\right)\right)=Df\left(x\right)" display />

At a point *x = a*, the derivative is written with a vertical bar or as *f'(a)*:

<Math math="f'\left(a\right)=\left.y'\right|_{x=a}=\left.\frac{df}{dx}\right|_{x=a}=\left.\frac{dy}{dx}\right|_{x=a}=Df\left(a\right)" display />

## Interpretation of Derivatives

Three standard interpretations: geometric (tangent slope), analytic (instantaneous rate of change), and physical (velocity).

if y = f(x) then,

1. m = f'(a) is the slope of the tangent line to y = f(x) at x = a and the equation of the tangent line at x = a is given by y = f(a)+f'(a)(x-a).

2. f'(a) is the instantaneous rate of change of f(x) at x = a.

3. If f(x) is the position of an object at time x then f'(a) is the velocity of the object at x = a.

## Basic Properties

Rules for sums, products, quotients, powers, and compositions.

Constants pass through:

<Math math="\left(cf\right)'=cf'\left(x\right)" display />

Sum and difference distribute:

<Math math="\left(f\pm g\right)'=f'\left(x\right)\pm g'\left(x\right)" display />

Product Rule, symmetric in *f* and *g*:

<Math math="\left(fg\right)'=f'g+fg'\text{ }\text{ }\text{ }\text{ }Product\text{ }Rule" display />

Quotient Rule, *g ≠ 0*:

<Math math="\left(\frac{f}{g}\right)'=\left(\frac{f'g-fg'}{g^{2}}\right)\text{ }\text{ }\text{ }\text{ }Quotient\text{ }Rule" display />

A constant has derivative zero:

<Math math="\frac{d}{dx}\left(c\right)=0" display />

Power Rule, any real exponent:

<Math math="\frac{d}{dx}\left(x^{n}\right)=nx^{n-1}\text{ }\text{ }\text{ }\text{ }Power\text{ }Rule" display />

Chain Rule: outer derivative (evaluated at the inner function) times the inner derivative. Used whenever one function appears inside another.

<Math math="\frac{d}{dx}\left(f\left(g\left(x\right)\right)\right)=f'\left(g\left(x\right)\right)g'\left(x\right)\text{ }\text{ }\text{ }\text{ }\text{ }Chain\text{ }Rule" display />

## Common Derivatives

The elementary derivatives. Combined with the rules above they handle most closed-form differentiation.

Identity:

<Math math="\frac{d}{dx}\left(x\right)=1" display />

Sine:

<Math math="\frac{d}{dx}\left(\sin x\right)=\cos x" display />

Cosine:

<Math math="\frac{d}{dx}\left(\cos x\right)=-\sin x" display />

Tangent:

<Math math="\frac{d}{dx}\left(\tan x\right)=\sec^{2} x" display />

Secant:

<Math math="\frac{d}{dx}\left(\sec x\right)=\sec x\tan x" display />

Cosecant:

<Math math="\frac{d}{dx}\left(\csc x\right)=\csc x\cot x" display />

Cotangent:

<Math math="\frac{d}{dx}\left(\cot x\right)=-\csc^{2} x" display />

Arcsine:

<Math math="\frac{d}{dx}\left(\sin^{-1} x\right)=\frac{1}{\sqrt{1-x^{2}}}" display />

Arccosine:

<Math math="\frac{d}{dx}\left(\cos^{-1} x\right)=-\frac{1}{\sqrt{1-x^{2}}}" display />

Arctangent:

<Math math="\frac{d}{dx}\left(\tan^{-1} x\right)=\frac{1}{{1+x^{2}}}" display />

Exponential, base *a*:

<Math math="\frac{d}{dx}\left(a^{x}\right)=a^{x}\ln\left(a\right)" display />

Natural exponential, its own derivative:

<Math math="\frac{d}{dx}\left(e^{x}\right)=e^{x}" display />

Natural log, *x > 0*:

<Math math="\frac{d}{dx}\left(\ln\left(x\right)\right)=\frac{1}{x}\text{ }\text{ }\text{ }\text{ },x>0" display />

Extended to *ln|x|* for *x ≠ 0*:

<Math math="\frac{d}{dx}\left(\ln\left|x\right|\right)=\frac{1}{x}\text{ }\text{ }\text{ }\text{ },x\neq0" display />

Log, base *a*:

<Math math="\frac{d}{dx}\left(\log_{a}{\left(x\right)}\right)=\frac{1}{x\ln{a}}\text{ }\text{ }\text{ }\text{ },x>0" display />

## Higher Order Derivatives

Repeated differentiation produces second, third, and higher derivatives.

The second derivative:

<Math math="\text{ }f''\left(x\right)=f^{\left(2\right)}\left(x\right)=\frac{d^{2}f}{dx^{2}}\text{ }" display />

is the derivative of the first derivative:

<Math math="\text{ }f''\left(x\right)=\left(f'\left(x\right)\right)'\text{ }" display />

The *n*-th derivative:

<Math math="\text{ }f^{\left(n\right)}\left(x\right)=\frac{d^{n}f}{dx^{n}}\text{ }" display />

is defined recursively from the *(n-1)*-th derivative:

<Math math="\text{ }f^{\left(n\right)}\left(x\right)=\left(f^{\left(n-1\right)}\left(x\right)\right)'\text{ }" display />

, i.e. the derivative of the (n-1)<sup>st</sup> derivative, f<sup>(n-1)</sup>(x).

## Others

The first and second derivatives classify behaviour: critical points (where *f'* is zero or undefined), monotonicity (sign of *f'*), concavity (sign of *f''*).

**Critical Points**

x = c is a critical point of f(x) provied either

1. f'(c) = 0

2. f'(c) doesn't exist.

**Increasing/Decreasing**

1. If f'(x) &gt; 0 for all x in an interval *I* then f (x) is increasing on the interval *I*.

2. If f'(x) &lt; 0 for all x in an interval *I* then f (x) is Decreasing on the interval *I*.

3. If f'(x) = 0 for all x in an interval *I* then f (x) is constant on the interval *I*.

**Concave up/Concave down**

1. If f"(x) &gt; 0 for all x in an interval *I* then f (x) is concave up on the interval *I*.

1. If f"(x) &lt; 0 for all x in an interval *I* then f (x) is concave down on the interval *I*.

**Infection Points**

x = c is inflection point of f(x) if the concavity changes at x = c.

<Diagram alt="Two curves side by side: one concave up (cup) and one concave down (cap)" viewBox="0 0 240 180" caption="f''(x) > 0 means concave up; f''(x) < 0 means concave down.">
  <path d="M 20 150 L 220 150" />
  <path d="M 30 140 Q 70 30 110 140" />
  <path d="M 140 30 Q 180 140 220 30" />
  <text x="55" y="170" font-size="10" fill="currentColor" stroke="none">f''(x) &gt; 0</text>
  <text x="160" y="170" font-size="10" fill="currentColor" stroke="none">f''(x) &lt; 0</text>
  <text x="50" y="22" font-size="10" fill="currentColor" stroke="none">concave up</text>
  <text x="148" y="22" font-size="10" fill="currentColor" stroke="none">concave down</text>
</Diagram>

<Notice type="tip">
A critical point is a *candidate* for a maximum or minimum, not a guarantee. Classify it with the second-derivative test (positive → local minimum, negative → local maximum) or a sign change of *f'*.
</Notice>

---

## Integration

URL: http://math.mkabumattar.com/calculus/integration

Definite and indefinite integrals, properties, common integrals, integration by parts, and trigonometric integrals.

Integration is the inverse of differentiation and the way to sum infinitesimal contributions: area under a curve for the definite integral, the antiderivative family for the indefinite one. The Fundamental Theorem of Calculus links the two: a definite integral is evaluated by finding any antiderivative and taking the difference at the endpoints. The rules and tables below cover the standard integrals plus two techniques (integration by parts, case-based reduction for trigonometric integrands) for forms not in the table. Integration is linear and monotone. Unlike differentiation, it has no general product, quotient, or chain rule, so a working table of common forms is needed.

<Diagram alt="A curve f(x) with the region between the curve and the x-axis from a to b shaded" viewBox="0 0 240 180" caption="The definite integral of f(x) dx from a to b is the signed area between the curve and the x-axis.">
  <path d="M 20 150 L 220 150" />
  <path d="M 30 20 L 30 165" />
  <path d="M 20 130 Q 70 50 130 70 Q 180 85 220 40" />
  <path d="M 60 150 L 60 113 Q 95 75 130 70 Q 165 67 190 60 L 190 150 Z" fill="currentColor" opacity="0.15" stroke="none" />
  <path d="M 60 150 L 60 113" />
  <path d="M 190 150 L 190 60" />
  <text x="56" y="163" font-size="10" fill="currentColor" stroke="none">a</text>
  <text x="186" y="163" font-size="10" fill="currentColor" stroke="none">b</text>
  <text x="105" y="115" font-size="10" fill="currentColor" stroke="none">∫ f(x) dx</text>
  <text x="222" y="148" font-size="10" fill="currentColor" stroke="none">x</text>
  <text x="34" y="22" font-size="10" fill="currentColor" stroke="none">y</text>
  <text x="200" y="55" font-size="10" fill="currentColor" stroke="none">f(x)</text>
</Diagram>

## Definition

A definite integral is a Riemann limit of rectangle sums. An indefinite integral is the family of antiderivatives, differing by a constant.

**Definite Integral:** Suppose *f*(x) is continuous on [a,b]. Divide [a,b] into *n* subintervals of width

<Math math="\text{ }\text{ }\triangle\text{ }x\text{ }\text{ }" display />

and choose

<Math math="\text{ }\text{ }x_{i}^{*}\text{ }\text{ }" display />

from each interval.

The definite integral is the limit of Riemann sums:

<Math math="\text{ }then\text{ }\int_{a}^{b} f\left(x\right)dx=\lim_{n \rightarrow \infty}f\left(x_{i}^{*}\right)\triangle\text{ }x." display />

**Anti-Derivative:** An anti-derivative of *f*(x) is a function, *F*(x), such that *F'*(x)=*f*(x).

**Indefinite Integral:**

The indefinite integral, with constant *c*:

<Math math="\text{ }\text{ }\text{ }\int f\left(x\right)dx=F\left(x\right)+c\text{ }\text{ }" display />

where *F*(x) is anti-derivative of *f*(x).

## Peroprtes

Integration is linear (sums, constants, order) and monotone.

Linearity of indefinite integrals:

<Math math="\int f\left(x\right)\pm g\left(x\right)dx=\int f\left(x\right)dx\pm\int g\left(x\right)dx" display />

Same for definite integrals:

<Math math="\int_{a}^{b}f\left(x\right)\pm g\left(x\right)dx=\int_{a}^{b}f\left(x\right)dx\pm\int_{a}^{b}g\left(x\right)dx" display />

Zero-width interval:

<Math math="\int_{a}^{a}f\left(x\right)dx=0" display />

Reversed limits flip the sign:

<Math math="\int_{a}^{b}f\left(x\right)dx=-\int_{b}^{a}f\left(x\right)dx" display />

Constants pull out:

<Math math="\int cf\left(x\right)dx=c\int f\left(x\right)dx \text{ }\text{ }\text{ },c\text{ }is\text{ }a\text{ }constant" display />

Definite version:

<Math math="\int_{a}^{b} cf\left(x\right)dx=c\int_{a}^{b} f\left(x\right)dx \text{ }\text{ }\text{ },c\text{ }is\text{ }a\text{ }constant" display />

The integration variable is a dummy:

<Math math="\int_{a}^{b}f\left(x\right)dx=\int_{a}^{b}f\left(t\right)dt" display />

Absolute value bound:

<Math math="\left|\int_{a}^{b}f\left(x\right)dx\right|=\int_{a}^{b}\left|f\left(x\right)\right|dx" display />

Monotonicity. Pointwise inequality between integrands transfers to integrals:

<Math math="if\text{ }f\left(x\right)\geq g\left(x\right)\text{ on }a\leq x\leq b\text{ }then\text{ }\text{ }\int_{a}^{b}f\left(x\right)dx\geq\int_{a}^{b}g\left(x\right)dx" display />

Non-negative integrand:

<Math math="if\text{ }f\left(x\right)\geq0\text{ }on\text{ }a\leq x\leq b\text{ }then\text{ }\int_{a}^{b}f\left(x\right)dx\geq0" display />

Bounds on the integrand transfer to bounds on the integral, scaled by *b - a*:

<Math math="if\text{ }m\leq f\left(x\right)\leq M\text{ }on\text{ }a\leq x\leq b\text{ }m\left(b-a\right)\leq\int_{a}^{b}f\left(x\right)dx\leq M\left(b-a\right)" display />

## Common Integrals

Standard antiderivatives, with integration constant *c*.

Constant:

<Math math="\int k\text{ }dx=k\text{ }x+c" display />

Power rule, *n ≠ -1*:

<Math math="\int x^{n}\text{ }dx=\frac{1}{1+n}x^{n+1}+c\text{ }\text{ }\text{ },n\neq-1" display />

The case *n = -1* gives the natural log:

<Math math="\int x^{-1}\text{ }dx=\int\frac{1}{x}dx=\ln\left|x\right|+c" display />

Linear-substitution form:

<Math math="\int \frac{1}{ax+b}\text{ }dx=\frac{1}{a}\ln\left|ax+b\right|+c" display />

Natural log, via integration by parts:

<Math math="\int \ln\text{ }u\text{ }du=u\text{ }\ln\left(u\right)-u+c" display />

Exponential:

<Math math="\int e^{u}\text{ }du=e^{u}+c" display />

Cosine:

<Math math="\int\cos\text{ }u\text{ }du=\sin\text{ }u+c" display />

Sine:

<Math math="\int\sin\text{ }u\text{ }du=-\cos\text{ }u+c" display />

Secant squared:

<Math math="\int\sec^{2}\text{ }u\text{ }du=\tan\text{ }u+c" display />

Secant-tangent product:

<Math math="\int\sec\text{ }u\text{ }\tan\text{ }u\text{ }du=\sec\text{ }u+c" display />

Cosecant-cotangent product:

<Math math="\int\csc\text{ }u\text{ }\cot\text{ }u\text{ }du=-\csc\text{ }u+c" display />

Cosecant squared:

<Math math="\int\csc^{2}\text{ }u\text{ }du=-\cot\text{ }u+c" display />

Tangent:

<Math math="\int\tan\text{ }u\text{ }du=\ln\left|\sec\text{ }u\right|+c" display />

Secant:

<Math math="\int\sec\text{ }u\text{ }du=\ln\left|\sec\text{ }u\text{ }\tan\text{ }u\right|+c" display />

Arctangent form:

<Math math="\int\frac{1}{a^{2}+u^{2}}\text{ }du=\frac{1}{a}\tan^{-1}\left(\frac{u}{a}\right)+a" display />

Arcsine form:

<Math math="\int\frac{1}{\sqrt{a^{2}+u^{2}}}\text{ }du=\sin^{-1}\left(\frac{u}{a}\right)+a" display />

## Integration by Parts

Integration by parts is the integral analogue of the product rule. Use it when the integrand factors into a product whose pieces simplify under differentiation or integration. The *LIATE* heuristic picks *u* in the order Logarithmic, Inverse trig, Algebraic, Trigonometric, Exponential, so that *u'* is simpler than *u* while *v* is still tractable to integrate. Applying parts twice often produces a self-referential equation solvable algebraically for the original integral.

<Math math="\int u\text{ }v\text{ }dx=u\int v\text{ }dx-\int u'\left(\int v\text{ }dx\right)dx" display />

Where,

u is the function u(x)

v is the function v(x)

## Trigonometic Integrals

Integrals of products of powers of sine, cosine, tangent, and secant reduce case by case: strip off one factor and convert the rest with a Pythagorean identity, then *u*-substitute.

For

<Math math="\text{ }\text{ }\int \sin''\text{ }x\text{ }\cos'''x\text{ }dx\text{ }\text{ }" display />

we have the following:

**1. *n* odd.** Strip 1 since out and convert rest to cosines using

<Math math="\text{ }\text{ }\sin^{2}\text{ }x=1-\cos^{2}\text{ }x\text{ }\text{ }" display />

, then use the substitution

<Math math="\text{ }\text{ }u=\cos\text{ }x\text{ }\text{ .}" display />

**2. *m* odd.** Strip 1 cosine out and convert rest to sines using

<Math math="\text{ }\text{ }\cos^{2}\text{ }x=1-\sin^{2}\text{ }x\text{ }\text{ }" display />

, then use the substitution

<Math math="\text{ }\text{ }u=\sin\text{ }x\text{ }\text{ .}" display />

**3. *n* and *m* both odd.** Use either 1. or 2.

**4. *n* and *m* both even.** Use double angle and/or half angle formulas to reduce the integral formulas to reduce the integral into a foram tht can be integrated.

For

<Math math="\text{ }\text{ }\int\tan''\text{ }x\text{ }\sec'''\text{ }x\text { }dx\text{ }\text{ }" display />

we have the following:

**1. *n* odd.** Strip 1 tangent and 1 secant out and convert the rest to secants using

<Math math="\text{ }\text{ }\tan^{2}x=\sec^{2}x-1\text{ }\text{ }" display />

, then use the substitution

<Math math="\text{ }\text{ }u=\sec\text{ }x\text{ }\text{ .}" display />

**2. *m* even.** Strip 2 secants out and convert rest to tangents using

<Math math="\text{ }\text{ }\sec^{2}x=1+\tan^{2}x\text{ }\text{ }" display />

, then use the substitution

<Math math="\text{ }\text{ }u=\tan\text{ }x\text{ }\text{ .}" display />

**3. *n* odd and *m* even.** Use either 1. or 2.

**4. *n* even and *m* odd.** Each integral will be dealt with differently.

---

## Laplace Transform

URL: http://math.mkabumattar.com/calculus/laplace-transform

A table of common Laplace transforms with the Gamma function and notes on hyperbolic trigonometric forms.

The Laplace transform maps a time-domain function *f(t)* to a complex-frequency-domain function *F(s)* by an exponential-weighted integral, converting differential equations into algebraic ones. It turns differentiation into multiplication by *s* (with initial-value terms), so linear ODEs with constant coefficients become rational equations in *s*. The solution is recovered by inverse-transforming back to the *t* domain via the table or partial fractions. The transform is linear. Entries 24-26 give the scaling and shifting rules: frequency-shift by an exponential factor in *t*, time-shift by multiplication by *e<sup>-cs</sup>* in *s*. The table pairs each *f(t)* with its *F(s)*. Reading right-to-left gives the inverse transform.

<Diagram alt="Block diagram with f(t) in time domain mapped via Laplace transform to F(s) in frequency domain" viewBox="0 0 240 180" caption="The Laplace transform maps a time-domain function to a frequency-domain function.">
  <rect x="15" y="65" width="80" height="50" rx="4" />
  <rect x="145" y="65" width="80" height="50" rx="4" />
  <text x="55" y="88" font-size="11" fill="currentColor" stroke="none" text-anchor="middle">f(t)</text>
  <text x="55" y="105" font-size="9" fill="currentColor" stroke="none" text-anchor="middle">time domain</text>
  <text x="185" y="88" font-size="11" fill="currentColor" stroke="none" text-anchor="middle">F(s)</text>
  <text x="185" y="105" font-size="9" fill="currentColor" stroke="none" text-anchor="middle">frequency domain</text>
  <path d="M 95 80 L 145 80" />
  <path d="M 140 76 L 145 80 L 140 84" />
  <path d="M 145 100 L 95 100" />
  <path d="M 100 96 L 95 100 L 100 104" />
  <text x="120" y="72" font-size="11" fill="currentColor" stroke="none" text-anchor="middle">ℒ</text>
  <text x="120" y="118" font-size="11" fill="currentColor" stroke="none" text-anchor="middle">ℒ⁻¹</text>
</Diagram>

| # | <Math math="f\left(t\right)=L^{-1}\left\{F\left(s\right)\right\}" /> | <Math math="F\left(s\right)=L\left\{f\left(t\right)\right\}" /> |
|---|---|---|
| 1 | 1 | <Math math="\frac{1}{s}" /> |
| 2 | <Math math="e^{at}" /> | <Math math="\frac{1}{s-a}" /> |
| 3 | <Math math="t^{n}\text{ }\text{ }\text{ }\text{ }\left(n=1,2,3...\right)" /> | <Math math="\frac{n!}{s^{\left(n+1\right)}}" /> |
| 4 | <Math math="t^{p}\left(p>-1\right)" /> | <Math math="\frac{\ulcorner\left(p+1\right)}{s^{\left(p+1\right)}}" /> |
| 5 | <Math math="\sqrt{t}" /> | <Math math="\frac{\sqrt{\pi}}{2\text{ }s^{\frac{3}{2}}}" /> |
| 6 | <Math math="t^{\frac{n-1}{2}}\text{ }\text{ }\text{ }\left(n=1,2,3...\right)" /> | <Math math="\frac{1\times3\times5...\left(2n-1\right)\sqrt{\pi}}{2^{n}s^{\frac{n+1}{2}}}" /> |
| 7 | <Math math="\sin\left(at\right)" /> | <Math math="\frac{a}{s^{2}+a^{2}}" /> |
| 8 | <Math math="\cos\left(at\right)" /> | <Math math="\frac{s}{s^{2}+a^{2}}" /> |
| 9 | <Math math="t\text{ }\sin\left(at\right)" /> | <Math math="\frac{2as}{\left(s^{2}+a^{2}\right)^{2}}" /> |
| 10 | <Math math="t\text{ }\cos\left(at\right)" /> | <Math math="\frac{s^{2}-a^{2}}{\left(s^{2}+a^{2}\right)^{2}}" /> |
| 11 | <Math math="\sin\left(at\right)-at\text{ }\cos\left(at\right)" /> | <Math math="\frac{2a^{3}}{\left(s^{2}+a^{2}\right)^{2}}" /> |
| 12 | <Math math="\sin\left(at\right)+at\text{ }\cos\left(at\right)" /> | <Math math="\frac{2as^{2}}{\left(s^{2}+a^{2}\right)^{2}}" /> |
| 13 | <Math math="\cos\left(at\right)-at\text{ }sin\left(at\right)" /> | <Math math="\frac{s\left(s^{2}-a^{2}\right)}{\left(s^{2}+a^{2}\right)^{2}}" /> |
| 14 | <Math math="\cos\left(at\right)+at\text{ }sin\left(at\right)" /> | <Math math="\frac{s\left(s^{2}+3a^{2}\right)}{\left(s^{2}+a^{2}\right)^{2}}" /> |
| 15 | <Math math="\sin\left(at+b\right)" /> | <Math math="\frac{s\text{ }\sin\left(b\right)+a\text{ }\cos\left(b\right)}{s^{2}+a^{2}}" /> |
| 16 | <Math math="\cos\left(at+b\right)" /> | <Math math="\frac{s\text{ }\cos\left(b\right)-a\text{ }\sin\left(b\right)}{s^{2}+a^{2}}" /> |
| 17 | <Math math="\sinh\left(at\right)" /> | <Math math="\frac{a}{s^{2}-a^{2}}" /> |
| 18 | <Math math="\cosh\left(at\right)" /> | <Math math="\frac{s}{s^{2}-a^{2}}" /> |
| 19 | <Math math="e^{at}\sin\left(bt\right)" /> | <Math math="\frac{b}{\left(s-a\right)^{2}+b}" /> |
| 20 | <Math math="e^{at}\cos\left(bt\right)" /> | <Math math="\frac{s-a}{\left(s-a\right)^{2}+b}" /> |
| 21 | <Math math="e^{at}\sinh\left(bt\right)" /> | <Math math="\frac{b}{\left(s-a\right)^{2}-b}" /> |
| 22 | <Math math="e^{at}\cosh\left(bt\right)" /> | <Math math="\frac{s-a}{\left(s-a\right)^{2}-b}" /> |
| 23 | <Math math="t^{n}e^{at}\text{ }\text{ }\text{ }\text{ }\left(n=1,2,3...\right)" /> | <Math math="\frac{n!}{\left(s-a\right)^{\left(n+1\right)}}" /> |
| 24 | <Math math="f\left(ct\right)" /> | <Math math="\left(\frac{1}{c}\right)F\left(\frac{s}{c}\right)" /> |
| 25 | <Math math="u_{c}\left(t\right)=u\left(t-c\right)" /> | <Math math="\frac{e^{-cs}}{s}" /> |
| 26 | <Math math="\wp\left(t-c\right)" /> | <Math math="e^{-cs}" /> |

## Notes:

1. The list is not inclusive and only contauin some of the most commonly used Laplace transforms and Formulase.

2. deffinition of hyperbolic functions

The hyperbolic functions in entries 17-22:

<Math math="\text{ }\cosh\left(t\right)=\frac{e^{t}+e^{-t}}{2}\text{ }\text{, }" display />

<Math math="\text{ }\sinh\left(t\right)=\frac{e^{t}+e^{-t}}{2}\text{ }" display />

3. Be coreful while using "normal" trigonometric function vs hyperbolic trigonometric function. The only difference in the formaulas is "+a<sup>2</sup>" forthe "normal" trigonometric functions becomes "+a<sup>2</sup>" for the hyperbolic trigonometric functions.

4. Formula #4 uses Gamma Function which is defined as

The Gamma function extends the factorial to non-integer arguments:

<Math math="\text{ }\ulcorner\left(t\right)=\int_{O}^{\infty} e^{-t}x^{t-1}dx\text{ }" display />

if n is a positive intrger then,

At positive integers it reduces to the factorial:

<Math math="\text{ }\ulcorner\left(n+1\right)=n!\text{ }" display />

The Gamma function extends the normal factorial. Two quick facts.

Recurrence relation:

<Math math="\ulcorner\left(p+1\right)=p\ulcorner\left(p\right)p\left(p+1\right)\left(p+1\right)...\left(p+n-1\right)=\frac{\ulcorner\left(p+2\right)}{\ulcorner\left(p\right)}" display />

Half-integer value:

<Math math="\ulcorner\left(\frac{1}{2}\right)=\sqrt{\pi}" display />

<Notice type="note">
The Laplace transform is defined only for *s* with a sufficiently large real part, the *region of convergence*. The algebraic forms in the table assume that condition. Outside the region, the integral defining the transform diverges.
</Notice>

---

## Differential Equations

URL: http://math.mkabumattar.com/calculus/differential-equations

Order, degree, and solution forms for differential equations including separable and second-order linear cases.

A differential equation relates a function to its derivatives. Order (highest derivative) and degree (power of that derivative) classify the problem. Standard techniques are separation of variables, substitution, and matching characteristic solution forms. For linear equations with constant coefficients, the characteristic equation dictates the form: real distinct roots give a sum of exponentials, repeated roots give exponentials times polynomials, complex roots give sines and cosines (optionally damped by an exponential). The general solution of an *n*-th-order linear equation is an *n*-parameter family. Initial or boundary conditions fix the constants.

A differential equation contains derivatives of the unknown function:

<Math math="\text{ }\left(\frac{dy}{dx},\frac{d^{2}y}{dx^{2}}\right)\text{ }" display />

or differentials:

<Math math="\text{ }\left(dx,dy\right)\text{ }" display />

is called a differential equation.

The order is the highest derivative present (here, second order):

<Math math="\text{ }\left(\frac{d^{2}y}{dx^{2}}\right)^{3}+\left(\frac{dy}{dx}\right)=0\text{ }\text{ }is\text{ }2" display />

The degree is the exponent on the highest-order derivative (the source labels this example 3):

<Math math="\text{ }\left(\frac{dy}{dx}\right)^{5}+y\left(\frac{d^{2}y}{dx^{2}}\right)=0\text{ }\text{ }is\text{ }3" display />

To form a differential equation by eliminating *n* arbitrary constants, differentiate *n* times.

A separable first-order equation in the form *M(x) dx + N(y) dy = 0*:

<Math math="\text{ }f_{x}dx+f_{2}\left(y\right)dy=0\text{ }" display />

Integrate the two sides independently:

<Math math="\text{ }\int f_{1}xdx+\int f_{2}\left(y\right)dy=c\text{ }" display />

For equations not directly separable, a substitution reduces it to separable form:

<Math math="\text{ }x+y=u\text{ }or\text{ }y=vx\text{ }" display />

For a combination of exponentials with opposite-sign exponents:

<Math math="\text{ }u=Ae^{nx}+Be^{-nx}\text{ }" display />

satisfies the second-order linear ODE with positive characteristic root *n<sup>2</sup>*:

<Math math="\text{ }\left(\frac{d^{2}y}{dx^{2}}\right)=n^{2}y\text{ }" display />

A linear combination of two distinct real exponentials:

<Math math="\text{ }y = Ae^{nx}+Be^{mx}\text{ }" display />

satisfies the second-order ODE with characteristic roots *m* and *n*:

<Math math="\text{ }\left(\frac{d^{2}y}{dx^{2}}\right)-\left(m+n\right)\left(\frac{dy}{dx}\right)+mny=0\text{ }" display />

Sinusoidal solutions for complex-conjugate characteristic roots:

<Math math="\left(i\right)\text{ }y=A\sin\left(nx+B\right)" display />

Cosine form, equivalent up to phase:

<Math math="\left(ii\right)\text{ }y=A\cos\left(nx+B\right)" display />

General harmonic solution:

<Math math="\left(iii\right)\text{ }y=A\sin\left(nx+B\right)+B\cos\left(nx\right)" display />

All three satisfy the simple harmonic oscillator equation:

<Math math="\left(\frac{d^{2}y}{dx^{2}}\right)=-n^{2}y" display />

<Diagram alt="Three small plots: exponential growth, exponential decay, and a sine wave, showing the behaviours of y'' = k y for different signs of k" viewBox="0 0 240 180" caption="Sign of k in y'' = ky determines whether solutions grow exponentially or oscillate.">
  <path d="M 10 140 L 80 140" />
  <path d="M 15 20 L 15 145" />
  <path d="M 15 135 Q 50 120 70 25" />
  <text x="20" y="160" font-size="8" fill="currentColor" stroke="none">k &gt; 0: grow</text>

  <path d="M 90 140 L 160 140" />
  <path d="M 95 20 L 95 145" />
  <path d="M 95 25 Q 130 130 150 138" />
  <text x="100" y="160" font-size="8" fill="currentColor" stroke="none">k &gt; 0: decay</text>

  <path d="M 170 80 L 240 80" />
  <path d="M 175 20 L 175 145" />
  <path d="M 175 80 Q 188 30 200 80 T 225 80 T 240 65" />
  <text x="180" y="160" font-size="8" fill="currentColor" stroke="none">k &lt; 0: oscillate</text>
</Diagram>

<Notice type="tip">
The sign of *k* in *y'' = ky* sets the qualitative behaviour: *k > 0* gives exponential growth/decay (real characteristic roots), *k = 0* gives a straight-line family, *k < 0* gives oscillation (imaginary roots).
</Notice>

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