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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.

-11π/2-π/2xyy = arcsin(x)
arcsin restricts to [-1, 1] on the x-axis and [-π/2, π/2] on the y-axis.

Definition

Each inverse undoes its trigonometric counterpart on a restricted domain.

Arcsine: the angle whose sine is x.

y=sin⁡−1x is equvelent to x=sin⁡yy=\sin^{-1}x\text{ }is\text{ }equvelent\text{ }to\text{ }x=\sin y

Arccosine: the angle whose cosine is x.

y=cos⁡−1x is equvelent to x=cos⁡yy=\cos^{-1}x\text{ }is\text{ }equvelent\text{ }to\text{ }x=\cos y

Arctangent: the angle whose tangent is x.

y=tan⁡−1x is equvelent to x=tan⁡yy=\tan^{-1}x\text{ }is\text{ }equvelent\text{ }to\text{ }x=\tan y

Domain and Range

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

FunctionDomainRange
y=sin⁡−1xy=\sin^{-1}x−1≤x≤1-1\leq x\leq1−π2≤x≤π2\frac{-\pi}{2}\leq x\leq\frac{\pi}{2}
y=cos⁡−1xy=\cos^{-1}x−1≤x≤1-1\leq x\leq10≤x≤y0\leq x\leq y
y=tan⁡−1xy=\tan^{-1}x∞≤x≤−∞\infty\leq x\leq-\infty−π2≤x≤π2\frac{-\pi}{2}\leq x\leq\frac{\pi}{2}
y=cot⁡−1xy=\cot^{-1}x∞≤x≤−∞\infty\leq x\leq-\infty0≤x≤y0\leq x\leq y
y=sec⁡−1xy=\sec^{-1}xx≤−1 , x≥1x\leq-1\text{ , }x\geq10≤x≤y , y≠π20\leq x\leq y\text{ , }y\neq\frac{\pi}{2}
y=csc⁡−1xy=\csc^{-1}xx≤−1 , x≥1x\leq-1\text{ , }x\geq1−π2≤y≤π2 , y≠0\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.

sin⁡(sin⁡−1x)=x\sin\left(\sin^{-1}x\right)=x

Arcsine of sine (within principal range).

sin⁡−1(sin⁡(y))=y\sin^{-1}\left(\sin\left(y\right)\right)=y

Cosine of arccosine.

cos⁡(cos⁡−1x)=x\cos\left(\cos^{-1}x\right)=x

Arccosine of cosine.

cos⁡−1(cos⁡(y))=y\cos^{-1}\left(\cos\left(y\right)\right)=y

Tangent of arctangent.

tan⁡(tan⁡−1x)=x\tan\left(\tan^{-1}x\right)=x

Arctangent of tangent.

tan⁡−1(tan⁡(y))=y\tan^{-1}\left(\tan\left(y\right)\right)=y

Alternate Notation

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

sin⁡−1x=arcsin⁡(x)\sin^{-1}x=\arcsin\left(x\right)
cos⁡−1x=arccos⁡(x)\cos^{-1}x=\arccos\left(x\right)
tan⁡−1x=arctan⁡(x)\tan^{-1}x=\arctan\left(x\right)

Complementary Angles

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

Arccosine and arcsine.

cos⁡−1x=π2−sin−1x\cos^{-1}x=\frac{\pi}{2}-sin^{-1}x

Arccotangent and arctangent.

cot⁡−1x=π2−tan−1x\cot^{-1}x=\frac{\pi}{2}-tan^{-1}x

Arccosecant and arcsecant.

csc⁡−1x=π2−sec−1x\csc^{-1}x=\frac{\pi}{2}-sec^{-1}x

Negative Arguments

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

Arcsine is odd.

sin⁡−1(−x)=−sin⁡−1x\sin^{-1}\left(-x\right)=-\sin^{-1}x

Arccosine reflects within the principal range.

cos⁡−1(−x)=π−cos⁡−1x\cos^{-1}\left(-x\right)=\pi-\cos^{-1}x

Arctangent is odd.

tan⁡−1(−x)=−tan⁡−1x\tan^{-1}\left(-x\right)=-\tan^{-1}x

Arccotangent reflects within its range.

cot⁡−1(−x)=π−cot⁡−1x\cot^{-1}\left(-x\right)=\pi-\cot^{-1}x

Arcsecant reflects.

sec⁡−1(−x)=π−sec⁡−1x\sec^{-1}\left(-x\right)=\pi-\sec^{-1}x

Arccosecant is odd.

csc⁡−1(−x)=−csc⁡−1x\csc^{-1}\left(-x\right)=-\csc^{-1}x

Reciprocal Arguments

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

Arcsine of 1/x.

sin⁡−1(1x)=csc⁡−1x\sin^{-1}\left(\frac{1}{x}\right)=\csc^{-1}x

Arccosine of 1/x.

cos⁡−1(1x)=sec⁡−1x\cos^{-1}\left(\frac{1}{x}\right)=\sec^{-1}x

Arctangent of 1/x, positive x.

tan⁡−1(1x)=π2−tan−1x=cot−1x   (if x>0)\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)

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

tan⁡−1(1x)=−π2−tan−1x=−π+cot−1x   (if x<0)\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)

Arccotangent of 1/x, positive x.

cot⁡−1(1x)=π2−cot−1x=tan⁡−1x   (if x>0)\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)

Arccotangent of 1/x, negative x.

cot⁡−1(1x)=−π2−cot−1x=−π+tan−1x   (if x<0)\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)

Arcsecant of 1/x.

sec⁡−1(1x)=cos⁡−1x\sec^{-1}\left(\frac{1}{x}\right)=\cos^{-1}x

Arccosecant of 1/x.

csc⁡−1(1x)=sin⁡−1x\csc^{-1}\left(\frac{1}{x}\right)=\sin^{-1}x

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

cot⁡−1(x)=sin⁡−1(1−xx)   (if 0≤x≤0)\cot^{-1}\left(x\right)=\sin^{-1}\left(\sqrt{1-x^{x}}\right)\text{ }\text{ }\text{ }\left(if\text{ }0\leq x\leq0\right)

Arctangent re-expressed as arcsine.

tan⁡−1x=sin⁡−1(x1+x2+1)\tan^{-1}x=\sin^{-1}\left(\frac{x}{1+\sqrt{x^{2}+1}}\right)

Other relationships.

Arcsine as twice an arctangent.

sin⁡−1x=2tan⁡−1(x1+1−x2)\sin^{-1}x=2\tan^{-1}\left(\frac{x}{1+\sqrt{1-x^{2}}}\right)

Arccosine as twice an arctangent.

cos⁡−1x=2tan⁡−1(1−x21+x)   (−1≤x≤1)\cos^{-1}x=2\tan^{-1}\left(\frac{\sqrt{1-x^{2}}}{1+x}\right)\text{ }\text{ }\text{ }\left(-1\leq x\leq1\right)

Arctangent as twice an arctangent of a smaller argument.

tan⁡−1x=2tan⁡−1(x1+1+x2)\tan^{-1}x=2\tan^{-1}\left(\frac{x}{1+\sqrt{1+x^{2}}}\right)

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.

sin⁡(cos⁡−1x)=cos⁡(sin⁡−1x)=1−x2\sin\left(\cos^{-1}x\right)=\cos\left(\sin^{-1}x\right)=\sqrt{1-x^{2}}

Sine of an arctangent.

sin⁡(tan⁡−1x)=x1+x2\sin\left(\tan^{-1}x\right)=\frac{x}{\sqrt{1+x^{2}}}

Cosine of an arctangent.

cos⁡(tan⁡−1x)=x1+x2\cos\left(\tan^{-1}x\right)=\frac{x}{\sqrt{1+x^{2}}}

Tangent of an arcsine.

tan⁡(sin⁡−1x)=x1−x2\tan\left(\sin^{-1}x\right)=\frac{x}{\sqrt{1-x^{2}}}

Tangent of an arccosine.

tan⁡(cos⁡−1x)=1−x2x\tan\left(\cos^{-1}x\right)=\frac{\sqrt{1-x^{2}}}{x}

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.

sin⁡(y)=x↔y=sin⁡−1x+2kπ   or   y=π−sin⁡−1x+2kπ\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

General solution of cos(y) = x.

cos⁡(y)=x↔y=cos⁡−1x+2kπ   or   y=π−cos⁡−1x+2kπ\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

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

tan⁡(y)=x↔y=tan⁡−1x+kπ\tan\left(y\right)=x\leftrightarrow y=\tan^{-1}x+k\pi

General solution of cot(y) = x.

cot⁡(y)=x↔y=cot⁡−1x+kπ\cot\left(y\right)=x\leftrightarrow y=\cot^{-1}x+k\pi

General solution of sec(y) = x.

sec⁡(y)=x↔y=sec⁡−1x+2kπ   or   y=π−sec⁡−1x+2kπ\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

General solution of csc(y) = x.

csc⁡(y)=x↔y=csc⁡−1x+2kπ   or   y=π−csc⁡−1x+2kπ\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