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Trigonometry

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.

Trigonometry Basics
θa (opp)b (adj)c (hyp)
sin θ = opp/hyp, cos θ = adj/hyp, tan θ = opp/adj.

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.

θ(cos θ, sin θ)cos θsin θ
On the unit circle, coordinates of the terminal point are (cos θ, sin θ).

Sine.

sin⁡ θ=Oppositehypotenuse=ac\sin\text{ }\theta=\frac{Opposite}{hypotenuse}=\frac{a}{c}

Cosine.

cos⁡ θ=Adjacenthypotenuse=bc\cos\text{ }\theta=\frac{Adjacent}{hypotenuse}=\frac{b}{c}

Tangent, equivalently sine over cosine.

tan⁡ θ=OppositeAdjacent=ab\tan\text{ }\theta=\frac{Opposite}{Adjacent}=\frac{a}{b}

Cotangent, the reciprocal of tangent.

cot⁡ θ=AdjacentOpposite=ba\cot\text{ }\theta=\frac{Adjacent}{Opposite}=\frac{b}{a}

Secant, the reciprocal of cosine.

sec⁡ θ=HypotenuseAdjacent=cb\sec\text{ }\theta=\frac{Hypotenuse}{Adjacent}=\frac{c}{b}

Cosecant, the reciprocal of sine.

csc⁡ θ=HypotenuseOppoie=cb\csc\text{ }\theta=\frac{Hypotenuse}{Oppoie}=\frac{c}{b}

Relations

Reciprocal and ratio identities recover the four secondary functions from sine and cosine. The Pythagorean identities follow from a2 + b2 = c2 on the unit circle. Parity (odd: sine, tangent, cotangent, cosecant; even: cosine, secant) is read off the unit-circle symmetry.

Tangent.

tan⁡ θ=sin⁡θcos⁡θ\tan\text{ }\theta=\frac{\sin\theta}{\cos\theta}

Cotangent.

cot⁡ θ=cos⁡θsin⁡θ\cot\text{ }\theta=\frac{\cos\theta}{\sin\theta}

Sine.

sin⁡ θ=1csc⁡θ\sin\text{ }\theta=\frac{1}{\csc\theta}

Cosine.

cos⁡ θ=1sec⁡θ\cos\text{ }\theta=\frac{1}{\sec\theta}

Tangent as reciprocal of cotangent.

tan⁡ θ=1cot⁡θ\tan\text{ }\theta=\frac{1}{\cot\theta}

Cotangent as reciprocal of tangent.

cot⁡  θ=1tan⁡θ\cot\text{ }\text{ }\theta=\frac{1}{\tan\theta}

Secant.

sec⁡ θ=1cos⁡θ\sec\text{ }\theta=\frac{1}{\cos\theta}

Cosecant.

csc⁡ θ=1sin⁡θ\csc\text{ }\theta=\frac{1}{\sin\theta}

Pythagorean identity.

sin⁡2 θ+cos⁡2θ=1\sin^{2}\text{ }\theta+\cos^{2}\theta=1

Divide by cos2θ for the secant-tangent form.

sec⁡2 θ−tan⁡2θ=1\sec^{2}\text{ }\theta-\tan^{2}\theta=1

Divide by sin2θ for the cosecant-cotangent form.

csc⁡2 θ+cot⁡2θ=1\csc^{2}\text{ }\theta+\cot^{2}\theta=1

Sine is odd.

sin⁡(−θ)=−sin⁡ θ\sin\left(-\theta\right)=-\sin\text{ }\theta

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

cos⁡(−θ)=−cos⁡ θ\cos\left(-\theta\right)=-\cos\text{ }\theta

Tangent is odd.

tan⁡(−θ)=−tan⁡ θ\tan\left(-\theta\right)=-\tan\text{ }\theta

Cotangent is odd.

cot⁡(−θ)=−cot⁡ θ\cot\left(-\theta\right)=-\cot\text{ }\theta

Secant follows cosine’s parity.

sec⁡(−θ)=−sec⁡ θ\sec\left(-\theta\right)=-\sec\text{ }\theta

Cosecant follows sine’s parity.

csc⁡(−θ)=−csc⁡ θ\csc\left(-\theta\right)=-\csc\text{ }\theta

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.

sin⁡(π2−θ)=cos⁡ θ\sin\left(\frac{\pi}{2-\theta}\right)=\cos\text{ }\theta

Cosine and sine.

cos⁡(π2−θ)=sin⁡ θ\cos\left(\frac{\pi}{2-\theta}\right)=\sin\text{ }\theta

Tangent and cotangent.

tan⁡(π2−θ)=cot⁡ θ\tan\left(\frac{\pi}{2-\theta}\right)=\cot\text{ }\theta

Cotangent and tangent.

cot⁡(π2−θ)=tan⁡ θ\cot\left(\frac{\pi}{2-\theta}\right)=\tan\text{ }\theta

Secant and cosecant.

sec⁡(π2−θ)=csc⁡ θ\sec\left(\frac{\pi}{2-\theta}\right)=\csc\text{ }\theta

Cosecant and secant.

csc⁡(π2−θ)=sec⁡ θ\csc\left(\frac{\pi}{2-\theta}\right)=\sec\text{ }\theta

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.

sin⁡(A+B)=sin⁡ A cos⁡ B+cos⁡ Asin⁡ B\sin\left(A+B\right)=\sin\text{ }A\text{ }\cos\text{ }B+\cos\text{ }A\sin\text{ }B

Sine of a difference.

sin⁡(A−B)=sin⁡ A cos⁡ B−cos⁡ Asin⁡ B\sin\left(A-B\right)=\sin\text{ }A\text{ }\cos\text{ }B-\cos\text{ }A\sin\text{ }B

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

cos⁡(A+B)=cos⁡ A cos⁡ B+sin⁡ Asin⁡ B\cos\left(A+B\right)=\cos\text{ }A\text{ }\cos\text{ }B+\sin\text{ }A\sin\text{ }B

Cosine of a difference.

cos⁡(A−B)=cos⁡ A cos⁡ B−sin⁡ Asin⁡ B\cos\left(A-B\right)=\cos\text{ }A\text{ }\cos\text{ }B-\sin\text{ }A\sin\text{ }B

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

tan⁡(A+B)=tan⁡ A+tan⁡ B1−tan⁡ Atan⁡ B\tan\left(A+B\right)=\frac{\tan\text{ }A+\tan\text{ }B}{1-\tan\text{ }A\tan\text{ }B}

Tangent of a difference.

tan⁡(A−B)=tan⁡ A−tan⁡ B1+tan⁡ Atan⁡ B\tan\left(A-B\right)=\frac{\tan\text{ }A-\tan\text{ }B}{1+\tan\text{ }A\tan\text{ }B}

sin C + sin D.

sin⁡ C+sin⁡ D=2sin⁡(C+D2)cos⁡(C−D2)\sin\text{ }C+\sin\text{ }D=2\sin\left(\frac{C+D}{2}\right)\cos\left(\frac{C-D}{2}\right)

sin C - sin D.

sin⁡ C−sin⁡ D=2cos⁡(C+D2)sin⁡(C−D2)\sin\text{ }C-\sin\text{ }D=2\cos\left(\frac{C+D}{2}\right)\sin\left(\frac{C-D}{2}\right)

cos C + cos D.

cos⁡ C+cos⁡ D=2cos⁡(C+D2)cos⁡(C−D2)\cos\text{ }C+\cos\text{ }D=2\cos\left(\frac{C+D}{2}\right)\cos\left(\frac{C-D}{2}\right)

cos C - cos D.

cos⁡ C−cos⁡ D=2sin⁡(C+D2)sin⁡(C−D2)\cos\text{ }C-\cos\text{ }D=2\sin\left(\frac{C+D}{2}\right)\sin\left(\frac{C-D}{2}\right)

2 sin A cos B.

2sin⁡ Acos⁡ B=sin⁡(A+B)+sin⁡(A−B)2\sin\text{ }A\cos\text{ }B = \sin\left(A+B\right)+\sin\left(A-B\right)

2 cos A sin B.

2cos⁡ Asin⁡ B=sin⁡(A+B)−sin⁡(A−B)2\cos\text{ }A\sin\text{ }B = \sin\left(A+B\right)-\sin\left(A-B\right)

2 cos A cos B.

2cos⁡ Acos⁡ B=cos⁡(A+B)+cos⁡(A−B)2\cos\text{ }A\cos\text{ }B = \cos\left(A+B\right)+\cos\left(A-B\right)

2 sin A sin B.

2sin⁡ Asin⁡ B=cos⁡(A−B)−cos⁡(A+B)2\sin\text{ }A\sin\text{ }B = \cos\left(A-B\right)-\cos\left(A+B\right)