In mathematics, the inverse trigonometric functions (occasionally called cyclometric functions^[1]) are the inverse functions of the trigonometric functions with suitably restricted domains .

The notations sin⁻¹, cos⁻¹, etc. are often used for arcsin, arccos, etc., but this convention logically conflicts with the common semantics for expressions like sin²(x), which refer to numeric power rather than function composition, and therefore may result in confusion between multiplicative inverse and compositional inverse.

In computer programming languages the functions arcsin, arccos, arctan, are usually called asin, acos, atan. Many programming languages also provide the two-argument atan2 function, which computes the arctangent of y / x given y and x, but with a range of (−π, π].

Principal values [link]

Since none of the six trigonometric functions are one-to-one, they must be restricted in order to have inverse functions. Therefore the ranges of the inverse functions are proper subsets of the domains of the original functions

For example, just as the square root function Failed to parse (Missing texvc executable; please see math/README to configure.): y = \sqrt{x}

is defined such that y2 = x, the function y = arcsin(x) is defined so that sin(y) = x. There are multiple numbers y such that sin(y) = x; for example, sin(0) = 0, but also sin(π) = 0, sin(2π) = 0, etc. It follows that the arcsine function is multivalued: arcsin(0) = 0, but also arcsin(0) = π, arcsin(0) = 2π, etc. When only one value is desired, the function may be restricted to its principal branch. With this restriction, for each x in the domain the expression arcsin(x) will evaluate only to a single value, called its principal value. These properties apply to all the inverse trigonometric functions.

The principal inverses are listed in the following table.

If x is allowed to be a complex number, then the range of y applies only to its real part.

Relationships among the inverse trigonometric functions [link]

The usual principal values of the arcsin(x) (red) and arccos(x) (blue) functions graphed on the cartesian plane.

The usual principal values of the arctan(x) and arccot(x) functions graphed on the cartesian plane.

Principal values of the arcsec(x) and arccsc(x) functions graphed on the cartesian plane.

Complementary angles:

Failed to parse (Missing texvc executable; please see math/README to configure.): \arccos x = \frac{\pi}{2} - \arcsin x

Failed to parse (Missing texvc executable; please see math/README to configure.): \arccot x = \frac{\pi}{2} - \arctan x

Failed to parse (Missing texvc executable; please see math/README to configure.): \arccsc x = \frac{\pi}{2} - \arcsec x

Negative arguments:

Failed to parse (Missing texvc executable; please see math/README to configure.): \arcsin (-x) = - \arcsin x \!

Failed to parse (Missing texvc executable; please see math/README to configure.): \arccos (-x) = \pi - \arccos x \!

Failed to parse (Missing texvc executable; please see math/README to configure.): \arctan (-x) = - \arctan x \!

Failed to parse (Missing texvc executable; please see math/README to configure.): \arccot (-x) = \pi - \arccot x \!

Failed to parse (Missing texvc executable; please see math/README to configure.): \arcsec (-x) = \pi - \arcsec x \!

Failed to parse (Missing texvc executable; please see math/README to configure.): \arccsc (-x) = - \arccsc x \!

Reciprocal arguments:

Failed to parse (Missing texvc executable; please see math/README to configure.): \arccos (1/x) \,= \arcsec x \,

Failed to parse (Missing texvc executable; please see math/README to configure.): \arcsin (1/x) \,= \arccsc x \,

Failed to parse (Missing texvc executable; please see math/README to configure.): \arctan (1/x) = \tfrac{1}{2}\pi - \arctan x =\arccot x,\text{ if }x > 0 \,

Failed to parse (Missing texvc executable; please see math/README to configure.): \arctan (1/x) = -\tfrac{1}{2}\pi - \arctan x = -\pi + \arccot x,\text{ if }x < 0 \,

Failed to parse (Missing texvc executable; please see math/README to configure.): \arccot (1/x) = \tfrac{1}{2}\pi - \arccot x =\arctan x,\text{ if }x > 0 \,

Failed to parse (Missing texvc executable; please see math/README to configure.): \arccot (1/x) = \tfrac{3}{2}\pi - \arccot x = \pi + \arctan x,\text{ if }x < 0 \,

Failed to parse (Missing texvc executable; please see math/README to configure.): \arcsec (1/x) = \arccos x \,

Failed to parse (Missing texvc executable; please see math/README to configure.): \arccsc (1/x) = \arcsin x \,

If you only have a fragment of a sine table:

Failed to parse (Missing texvc executable; please see math/README to configure.): \arccos x = \arcsin \sqrt{1-x^2},\text{ if }0 \leq x \leq 1

Failed to parse (Missing texvc executable; please see math/README to configure.): \arctan x = \arcsin \frac{x}{\sqrt{x^2+1}}

Whenever the square root of a complex number is used here, we choose the root with the positive real part (or positive imaginary part if the square was negative real).

From the half-angle formula Failed to parse (Missing texvc executable; please see math/README to configure.): \tan \frac{\theta}{2} = \frac{\sin \theta}{1+\cos \theta} , we get:

Failed to parse (Missing texvc executable; please see math/README to configure.): \arcsin x = 2 \arctan \frac{x}{1+\sqrt{1-x^2}}

Failed to parse (Missing texvc executable; please see math/README to configure.): \arccos x = 2 \arctan \frac{\sqrt{1-x^2}}{1+x},\text{ if }-1 < x \leq +1

Failed to parse (Missing texvc executable; please see math/README to configure.): \arctan x = 2 \arctan \frac{x}{1+\sqrt{1+x^2}}

Relationships between trigonometric functions and inverse trigonometric functions [link]

Failed to parse (Missing texvc executable; please see math/README to configure.): \sin (\arccos x) = \cos(\arcsin x) = \sqrt{1-x^2}

Failed to parse (Missing texvc executable; please see math/README to configure.): \sin (\arctan x) = \frac{x}{\sqrt{1+x^2}}

Failed to parse (Missing texvc executable; please see math/README to configure.): \cos (\arctan x) = \frac{1}{\sqrt{1+x^2}}

Failed to parse (Missing texvc executable; please see math/README to configure.): \tan (\arcsin x) = \frac{x}{\sqrt{1-x^2}}

Failed to parse (Missing texvc executable; please see math/README to configure.): \tan (\arccos x) = \frac{\sqrt{1-x^2}}{x}

General solutions [link]

Each of the trigonometric functions is periodic in the real part of its argument, running through all its values twice in each interval of 2π. Sine and cosecant begin their period at 2πk − π/2 (where k is an integer), finish it at 2πk + π/2, and then reverse themselves over 2πk + π/2 to 2πk + 3π/2. Cosine and secant begin their period at 2πk, finish it at 2πk + π, and then reverse themselves over 2πk + π to 2πk + 2π. Tangent begins its period at 2πk − π/2, finishes it at 2πk + π/2, and then repeats it (forward) over 2πk + π/2 to 2πk + 3π/2. Cotangent begins its period at 2πk, finishes it at 2πk + π, and then repeats it (forward) over 2πk + π to 2πk + 2π.

This periodicity is reflected in the general inverses where k is some integer:

Failed to parse (Missing texvc executable; please see math/README to configure.): \sin(y) = x \ \Leftrightarrow\ y = \arcsin(x) + 2k\pi \text{ or } y = \pi - \arcsin(x) + 2k\pi

Failed to parse (Missing texvc executable; please see math/README to configure.): \cos(y) = x \ \Leftrightarrow\ y = \arccos(x) + 2k\pi \text{ or } y = 2\pi - \arccos(x) + 2k\pi

Failed to parse (Missing texvc executable; please see math/README to configure.): \tan(y) = x \ \Leftrightarrow\ y = \arctan(x) + k\pi

Failed to parse (Missing texvc executable; please see math/README to configure.): \cot(y) = x \ \Leftrightarrow\ y = \arccot(x) + k\pi

Failed to parse (Missing texvc executable; please see math/README to configure.): \sec(y) = x \ \Leftrightarrow\ y = \arcsec(x) + 2k\pi \text{ or } y = 2\pi - \arcsec (x) + 2k\pi

Failed to parse (Missing texvc executable; please see math/README to configure.): \csc(y) = x \ \Leftrightarrow\ y = \arccsc(x) + 2k\pi \text{ or } y = \pi - \arccsc(x) + 2k\pi

Derivatives of inverse trigonometric functions [link]

Main article: Differentiation of trigonometric functions

Simple derivatives for real and complex values of x are as follows:

Failed to parse (Missing texvc executable; please see math/README to configure.): \begin{align} \frac{d}{dx} \arcsin x & {}= \frac{1}{\sqrt{1-x^2}}\\ \frac{d}{dx} \arccos x & {}= \frac{-1}{\sqrt{1-x^2}}\\ \frac{d}{dx} \arctan x & {}= \frac{1}{1+x^2}\\ \frac{d}{dx} \arccot x & {}= \frac{-1}{1+x^2}\\ \frac{d}{dx} \arcsec x & {}= \frac{1}{x\,\sqrt{x^2-1}}\\ \frac{d}{dx} \arccsc x & {}= \frac{-1}{x\,\sqrt{x^2-1}} \end{align}

Only for real values of x:

Failed to parse (Missing texvc executable; please see math/README to configure.): \begin{align} \frac{d}{dx} \arcsec x & {}= \frac{1}{|x|\,\sqrt{x^2-1}}; \qquad |x| > 1\\ \frac{d}{dx} \arccsc x & {}= \frac{-1}{|x|\,\sqrt{x^2-1}}; \qquad |x| > 1 \end{align}

For a sample derivation: if Failed to parse (Missing texvc executable; please see math/README to configure.): \theta = \arcsin x \! , we get:

Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{d \arcsin x}{dx} = \frac{d \theta}{d \sin \theta} = \frac{d \theta}{\cos \theta d \theta} = \frac{1} {\cos \theta} = \frac{1} {\sqrt{1-\sin^2 \theta}} = \frac{1}{\sqrt{1-x^2}}

Expression as definite integrals [link]

Integrating the derivative and fixing the value at one point gives an expression for the inverse trigonometric function as a definite integral:

Failed to parse (Missing texvc executable; please see math/README to configure.): \begin{align} \arcsin x &{}= \int_0^x \frac {1} {\sqrt{1 - z^2}}\,dz,\qquad |x| \leq 1\\ \arccos x &{}= \int_x^1 \frac {1} {\sqrt{1 - z^2}}\,dz,\qquad |x| \leq 1\\ \arctan x &{}= \int_0^x \frac 1 {z^2 + 1}\,dz,\\ \arccot x &{}= \int_x^\infty \frac {1} {z^2 + 1}\,dz,\\ \arcsec x &{}= \int_1^x \frac 1 {z \sqrt{z^2 - 1}}\,dz, \qquad x \geq 1\\ \arcsec x &{}= \pi + \int_x^{-1} \frac 1 {z \sqrt{z^2 - 1}}\,dz, \qquad x \leq -1\\ \arccsc x &{}= \int_x^\infty \frac {1} {z \sqrt{z^2 - 1}}\,dz, \qquad x \geq 1\\ \arccsc x &{}= \int_{-\infty}^x \frac {1} {z \sqrt{z^2 - 1}}\,dz, \qquad x \leq -1 \end{align}

When x equals 1, the integrals with limited domains are improper integrals, but still well-defined.

Infinite series [link]

Like the sine and cosine functions, the inverse trigonometric functions can be calculated using infinite series, as follows:

Failed to parse (Missing texvc executable; please see math/README to configure.): \begin{align} \arcsin z & {}= z + \left( \frac {1} {2} \right) \frac {z^3} {3} + \left( \frac {1 \cdot 3} {2 \cdot 4} \right) \frac {z^5} {5} + \left( \frac{1 \cdot 3 \cdot 5} {2 \cdot 4 \cdot 6 } \right) \frac{z^7} {7} + \cdots\\ & {}= \sum_{n=0}^\infty \left( \frac {(2n)!} {2^{2n}(n!)^2} \right) \frac {z^{2n+1}} {(2n+1)} ; \qquad | z | \le 1 \end{align}

Failed to parse (Missing texvc executable; please see math/README to configure.): \begin{align} \arccos z & {}= \frac {\pi} {2} - \arcsin z \\ & {}= \frac {\pi} {2} - (z + \left( \frac {1} {2} \right) \frac {z^3} {3} + \left( \frac {1 \cdot 3} {2 \cdot 4} \right) \frac {z^5} {5} + \left( \frac{1 \cdot 3 \cdot 5} {2 \cdot 4 \cdot 6 } \right) \frac{z^7} {7} + \cdots ) \\ & {}= \frac {\pi} {2} - \sum_{n=0}^\infty \left( \frac {(2n)!} {2^{2n}(n!)^2} \right) \frac {z^{2n+1}} {(2n+1)} ; \qquad | z | \le 1 \end{align}

Failed to parse (Missing texvc executable; please see math/README to configure.): \begin{align} \arctan z & {}= z - \frac {z^3} {3} +\frac {z^5} {5} -\frac {z^7} {7} +\cdots \\ & {}= \sum_{n=0}^\infty \frac {(-1)^n z^{2n+1}} {2n+1} ; \qquad | z | \le 1 \qquad z \neq i,-i \end{align}

Failed to parse (Missing texvc executable; please see math/README to configure.): \begin{align} \arccot z & {}= \frac {\pi} {2} - \arctan z \\ & {}= \frac {\pi} {2} - ( z - \frac {z^3} {3} +\frac {z^5} {5} -\frac {z^7} {7} +\cdots ) \\ & {}= \frac {\pi} {2} - \sum_{n=0}^\infty \frac {(-1)^n z^{2n+1}} {2n+1} ; \qquad | z | \le 1 \qquad z \neq i,-i \end{align}

Failed to parse (Missing texvc executable; please see math/README to configure.): \begin{align} \arcsec z & {}= \arccos {(1/z)} \\ & {}= \frac {\pi} {2} - (z^{-1} + \left( \frac {1} {2} \right) \frac {z^{-3}} {3} + \left( \frac {1 \cdot 3} {2 \cdot 4} \right) \frac {z^{-5}} {5} + \left( \frac{1 \cdot 3 \cdot 5} {2 \cdot 4 \cdot 6 } \right) \frac{z^{-7}} {7} + \cdots ) \\ & {}= \frac {\pi} {2} - \sum_{n=0}^\infty \left( \frac {(2n)!} {2^{2n}(n!)^2} \right) \frac {z^{-(2n+1)}} {(2n+1)}  ; \qquad \left| z \right| \ge 1 \end{align}

Failed to parse (Missing texvc executable; please see math/README to configure.): \begin{align} \arccsc z & {}= \arcsin {(1/z)} \\ & {}= z^{-1} + \left( \frac {1} {2} \right) \frac {z^{-3}} {3} + \left( \frac {1 \cdot 3} {2 \cdot 4 } \right) \frac {z^{-5}} {5} + \left( \frac {1 \cdot 3 \cdot 5} {2 \cdot 4 \cdot 6} \right) \frac {z^{-7}} {7} +\cdots \\ & {}= \sum_{n=0}^\infty \left( \frac {(2n)!} {2^{2n}(n!)^2} \right) \frac {z^{-(2n+1)}} {2n+1} ; \qquad \left| z \right| \ge 1 \end{align}

Leonhard Euler found a more efficient series for the arctangent, which is:

Failed to parse (Missing texvc executable; please see math/README to configure.): \arctan z = \frac{z}{1+z^2} \sum_{n=0}^\infty \prod_{k=1}^n \frac{2k z^2}{(2k+1)(1+z^2)}.

(Notice that the term in the sum for n= 0 is the empty product which is 1.)

Alternatively, this can be expressed:

Failed to parse (Missing texvc executable; please see math/README to configure.): \arctan z = \sum_{n=0}^\infty \frac{2^{\,2n}\,(n!)^2}{\left(2n+1\right)!} \; \frac{z^{\,2n+1}}{\left(1+z^2\right)^{n+1}}

Continued fractions for arctangent [link]

Two alternatives to the power series for arctangent are these generalized continued fractions:

Failed to parse (Missing texvc executable; please see math/README to configure.): \arctan z=\cfrac{z} {1+\cfrac{(1z)^2} {3-1z^2+\cfrac{(3z)^2} {5-3z^2+\cfrac{(5z)^2} {7-5z^2+\cfrac{(7z)^2} {9-7z^2+\ddots}}}}} =\cfrac{z} {1+\cfrac{(1z)^2} {3+\cfrac{(2z)^2} {5+\cfrac{(3z)^2} {7+\cfrac{(4z)^2} {9+\ddots\,}}}}}\,

The second of these is valid in the cut complex plane. There are two cuts, from −i to the point at infinity, going down the imaginary axis, and from i to the point at infinity, going up the same axis. It works best for real numbers running from −1 to 1. The partial denominators are the odd natural numbers, and the partial numerators (after the first) are just (nz)², with each perfect square appearing once. The first was developed by Leonhard Euler; the second by Carl Friedrich Gauss utilizing the Gaussian hypergeometric series.

Indefinite integrals of inverse trigonometric functions [link]

For real and complex values of x:

Failed to parse (Missing texvc executable; please see math/README to configure.): \begin{align} \int \arcsin x\,dx &{}= x\,\arcsin x + \sqrt{1-x^2} + C\\ \int \arccos x\,dx &{}= x\,\arccos x - \sqrt{1-x^2} + C\\ \int \arctan x\,dx &{}= x\,\arctan x - \frac{1}{2}\ln\left(1+x^2\right) + C\\ \int \arccot x\,dx &{}= x\,\arccot x + \frac{1}{2}\ln\left(1+x^2\right) + C\\ \int \arcsec x\,dx &{}= x\,\arcsec x - \ln\left[x\left(1+\sqrt{{x^2-1}\over x^2}\right)\right] + C\\ \int \arccsc x\,dx &{}= x\,\arccsc x + \ln\left[x\left(1+\sqrt{{x^2-1}\over x^2}\right)\right] + C \end{align}

For real x ≥ 1:

Failed to parse (Missing texvc executable; please see math/README to configure.): \begin{align} \int \arcsec x\,dx &{}= x\,\arcsec x - \ln\left(x+\sqrt{x^2-1}\right) + C\\ \int \arccsc x\,dx &{}= x\,\arccsc x + \ln\left(x+\sqrt{x^2-1}\right) + C \end{align}

All of these can be derived using integration by parts and the simple derivative forms shown above.

Example [link]

Using Failed to parse (Missing texvc executable; please see math/README to configure.): \int u\,\mathrm{d}v = u v - \int v\,\mathrm{d}u , set

Failed to parse (Missing texvc executable; please see math/README to configure.): \begin{align} u &{}=&\arcsin x &\quad\quad\mathrm{d}v = \mathrm{d}x\\ \mathrm{d}u &{}=&\frac{\mathrm{d}x}{\sqrt{1-x^2}}&\quad\quad{}v = x \end{align}

Then

Failed to parse (Missing texvc executable; please see math/README to configure.): \int \arcsin(x)\,\mathrm{d}x = x \arcsin x - \int \frac{x}{\sqrt{1-x^2}}\,\mathrm{d}x

Substitute

Failed to parse (Missing texvc executable; please see math/README to configure.): k = 1 - x^2.\,

Then

Failed to parse (Missing texvc executable; please see math/README to configure.): \mathrm{d}k = -2x\,\mathrm{d}x

and

Failed to parse (Missing texvc executable; please see math/README to configure.): \int \frac{x}{\sqrt{1-x^2}}\,\mathrm{d}x = -\frac{1}{2}\int \frac{\mathrm{d}k}{\sqrt{k}} = -\sqrt{k}

Back-substitute for x to yield

Failed to parse (Missing texvc executable; please see math/README to configure.): \int \arcsin(x)\, \mathrm{d}x = x \arcsin x + \sqrt{1-x^2}+C

Two-argument variant of arctangent [link]

The two-argument atan2 function computes the arctangent of y / x given y and x, but with a range of (−π, π]. In other words, atan2(y, x) is the angle between the positive x-axis of a plane and the point (x, y) on it, with positive sign for counter-clockwise angles (upper half-plane, y > 0), and negative sign for clockwise angles (lower half-plane, y < 0). It was first introduced in many computer programming languages, but it is now also common in other fields of science and engineering.

In terms of the standard arctan function, that is with range of (−π/2, π/2), it can be expressed as follows:

Failed to parse (Missing texvc executable; please see math/README to configure.): \operatorname{atan2}(y, x) = \begin{cases} \arctan(\frac y x) & \qquad x > 0 \\ \pi + \arctan(\frac y x) & \qquad y \ge 0 , x < 0 \\ -\pi + \arctan(\frac y x) & \qquad y < 0 , x < 0 \\ \frac{\pi}{2} & \qquad y > 0 , x = 0 \\ -\frac{\pi}{2} & \qquad y < 0 , x = 0 \\ \text{undefined} & \qquad y = 0, x = 0 \end{cases}

It also equals the principal value of the argument of the complex number x + iy.

This function may also be defined using the tangent half-angle formulae as follows:

Failed to parse (Missing texvc executable; please see math/README to configure.): \operatorname{atan2}(y, x)=2\arctan \frac{y}{\sqrt{x^2 + y^2} + x}

provided that either x > 0 or y ≠ 0. However this fails if given x ≤ 0 and y = 0 so the expression is unsuitable for computational use.

The above argument order (y, x) seems to be the most common, and in particular is used in ISO standards such as the C programming language, but a few authors may use the opposite convention (x, y) so some caution is warranted. These variations are detailed at Atan2.

Logarithmic forms [link]

These functions may also be expressed using complex logarithms. This extends in a natural fashion their domain to the complex plane.

Failed to parse (Missing texvc executable; please see math/README to configure.): \begin{align} \arcsin x &{}= -i\,\ln\left(i\,x+\sqrt{1-x^2}\right) &{}= \arccsc \frac{1}{x}\\[10pt] \arccos x &{}= -i\,\ln\left(x+i\,\sqrt{1-x^2}\right) = \frac{\pi}{2}\,+i\ln\left(i\,x+\sqrt{1-x^2}\right) = \frac{\pi}{2}-\arcsin x &{}= \arcsec \frac{1}{x}\\[10pt] \arctan x &{}= \tfrac{1}{2}i\left(\ln\left(1-i\,x\right)-\ln\left(1+i\,x\right)\right) &{}= \arccot \frac{1}{x}\\[10pt] \arccot x &{}= \tfrac{1}{2}i\left(\ln\left(1-\frac{i}{x}\right)-\ln\left(1+\frac{i}{x}\right)\right) &{}= \arctan \frac{1}{x}\\[10pt] \arcsec x &{}= -i\,\ln\left(i\,\sqrt{1-\frac{1}{x^2}}+\frac{1}{x}\right) = i\,\ln\left(\sqrt{1-\frac{1}{x^2}}+\frac{i}{x}\right)+\frac{\pi}{2} = \frac{\pi}{2}-\arccsc x &{}= \arccos \frac{1}{x}\\[10pt] \arccsc x &{}= -i\,\ln\left(\sqrt{1-\frac{1}{x^2}}+\frac{i}{x}\right) &{}= \arcsin \frac{1}{x} \end{align}

Elementary proofs of these relations proceed via expansion to exponential forms of the trigonometric functions.

Example proof [link]

Failed to parse (Missing texvc executable; please see math/README to configure.): \arcsin x = \theta \,

Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{e^{i\theta} - e^{-i\theta}}{2i} = x

(exponential definition of sine)

Let

Failed to parse (Missing texvc executable; please see math/README to configure.): k=e^{i\,\theta}. \,

Then

Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{k-\frac{1}{k}}{2i} = x

Failed to parse (Missing texvc executable; please see math/README to configure.): k^2-2\,i\,k\,x-1\,=\,0

Failed to parse (Missing texvc executable; please see math/README to configure.): k = ix \pm \sqrt{1-x^2} = e^{i\theta} \,

(the positive branch is chosen)

Failed to parse (Missing texvc executable; please see math/README to configure.): \theta = \arcsin x = -i \ln \left(ix + \sqrt{1-x^2}\right) \,

Inverse trigonometric functions in the complex plane

Failed to parse (Missing texvc executable; please see math/README to configure.): \arcsin(z)	Failed to parse (Missing texvc executable; please see math/README to configure.): \arccos(z)	Failed to parse (Missing texvc executable; please see math/README to configure.): \arctan(z)	Failed to parse (Missing texvc executable; please see math/README to configure.): \arccot(z)	Failed to parse (Missing texvc executable; please see math/README to configure.): \arcsec(z)	Failed to parse (Missing texvc executable; please see math/README to configure.): \arccsc(z)

Arctangent addition formula [link]

Failed to parse (Missing texvc executable; please see math/README to configure.): \arctan u + \arctan v = \arctan \left( \frac{u+v}{1-uv} \right) \pmod \pi, \qquad u v \ne 1 \,.

This is derived from the tangent addition formula

Failed to parse (Missing texvc executable; please see math/README to configure.): \tan ( \alpha + \beta ) = \frac{\tan \alpha + \tan \beta} {1 - \tan \alpha \tan \beta} \,,

by letting

Failed to parse (Missing texvc executable; please see math/README to configure.): \alpha = \arctan u \,, \quad \beta = \arctan v \,.

Application: finding the angle of a right triangle [link]

A right triangle.

Inverse trigonometric functions are useful when trying to determine the remaining two angles of a right triangle when the lengths of the sides of the triangle are known. Recalling the right-triangle definitions of sine, for example, it follows that

Failed to parse (Missing texvc executable; please see math/README to configure.): \theta = \arcsin \left( \frac{\text{opposite}}{\text{hypotenuse}} \right).

Often, the hypotenuse is unknown and would need to be calculated before using arcsine or arccosine using the Pythagorean Theorem: Failed to parse (Missing texvc executable; please see math/README to configure.): a^2+b^2=h^2

where Failed to parse (Missing texvc executable; please see math/README to configure.): h
is the length of the hypotenuse. Arctangent comes in handy in this situation, as the length of the hypotenuse is not needed.

Failed to parse (Missing texvc executable; please see math/README to configure.): \theta = \arctan \left( \frac{\text{opposite}}{\text{adjacent}} \right).

For example, suppose a roof drops 8 feet as it runs out 20 feet. The roof makes an angle θ with the horizontal, where θ may be computed as follows:

Failed to parse (Missing texvc executable; please see math/README to configure.): \theta = \arctan \left(\frac{\text{opposite}}{\text{adjacent}} \right) = \arctan \left( \frac{\text{rise}}{\text{run}} \right) = \arctan \left( \frac{8}{20} \right) = 21.8^{\circ}.

Practical considerations [link]

For angles near 0 and π, arccosine is ill-conditioned and will thus calculate the angle with reduced accuracy in a computer implementation (due to the limited number of digits). Similarly, arcsine is inaccurate for angles near −π/2 and π/2. To achieve full accuracy for all angles, arctangent or atan2 should be used for the implementation.

See also [link]

• Trigonometric function
• Tangent half-angle formula
• List of trigonometric identities
• Complex logarithm
• Argument (complex analysis)
• Square root
• Gauss's continued fraction
• Inverse hyperbolic function
• List of integrals of inverse trigonometric functions

References [link]

External links [link]

• Weisstein, Eric W., "Inverse Trigonometric Functions" from MathWorld.
• Weisstein, Eric W., "Inverse Tangent" from MathWorld.