Inverse trigonometric functions

In mathematics, the inverse trigonometric functions or cyclometric functions are the inverse functions of the trigonometric functions, though they do not meet the official definition for inverse functions as their ranges are subsets of the domains of the original functions. Since none of the six trigonometric functions are one-to-one (by failing the horizontal line test), they must be restricted in order to have inverse functions.

For example, just as the square root function $y\; =\; \backslash sqrt\{x\}$ 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.

{| class="wikitable" style="text-align:center" |- !Name !Usual notation !Definition !Domain of x for real result !Range of usual principal value
(radians) !Range of usual principal value
(degrees) |- | arcsine || y = arcsin x || x = sin y || −1 ≤ x ≤ 1 || −π/2 ≤ y ≤ π/2 || −90° ≤ y ≤ 90° |- | arccosine || y = arccos x || x = cos y || −1 ≤ x ≤ 1 || 0 ≤ y ≤ π || 0° ≤ y ≤ 180° |- | arctangent || y = arctan x || x = tan y || all real numbers || −π/2 < y < π/2 || −90° < y < 90° |- | arccotangent || y = arccot x ||x = cot y || all real numbers | 0 < y < π || 0° < y < 180° |- | arcsecant || y = arcsec x || x = sec y || x ≤ −1 or 1 ≤ x || 0 ≤ y < π/2 or π/2 < y ≤ π || 0° ≤ y < 90° or 90° < y ≤ 180° |- | arccosecant || y = arccsc x || x = csc y || x ≤ −1 or 1 ≤ x || −π/2 ≤ y < 0 or 0 < y ≤ π/2 || -90° ≤ y < 0° or 0° < y ≤ 90° |- |}

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

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].

Relationships among the inverse trigonometric functions

Complementary angles: :$\backslash arccos\; x\; =\; \backslash frac\{\backslash pi\}\{2\}\; -\; \backslash arcsin\; x$

:$\backslash arccot\; x\; =\; \backslash frac\{\backslash pi\}\{2\}\; -\; \backslash arctan\; x$

:$\backslash arccsc\; x\; =\; \backslash frac\{\backslash pi\}\{2\}\; -\; \backslash arcsec\; x$

Negative arguments: :$\backslash arcsin\; (-x)\; =\; -\; \backslash arcsin\; x\; \backslash !$ :$\backslash arccos\; (-x)\; =\; \backslash pi\; -\; \backslash arccos\; x\; \backslash !$ :$\backslash arctan\; (-x)\; =\; -\; \backslash arctan\; x\; \backslash !$ :$\backslash arccot\; (-x)\; =\; \backslash pi\; -\; \backslash arccot\; x\; \backslash !$ :$\backslash arcsec\; (-x)\; =\; \backslash pi\; -\; \backslash arcsec\; x\; \backslash !$ :$\backslash arccsc\; (-x)\; =\; -\; \backslash arccsc\; x\; \backslash !$

Reciprocal arguments: :$\backslash arccos\; x^\{-1\}\; \backslash ,=\; \backslash arcsec\; x\; \backslash ,$ :$\backslash arcsin\; x^\{-1\}\; \backslash ,=\; \backslash arccsc\; x\; \backslash ,$ :$\backslash arctan\; x^\{-1\}\; =\; \backslash tfrac\{1\}\{2\}\backslash pi\; -\; \backslash arctan\; x\; =\backslash arccot\; x,\backslash text\{\; if\; \}x\; >\; 0\; \backslash ,$ : :$\backslash arccot\; x^\{-1\}\; =\; \backslash tfrac\{1\}\{2\}\backslash pi\; -\; \backslash arccot\; x\; =\backslash arctan\; x,\backslash text\{\; if\; \}x\; >\; 0\; \backslash ,$ : :$\backslash arcsec\; x^\{-1\}\; =\; \backslash arccos\; x\; \backslash ,$ :$\backslash arccsc\; x^\{-1\}\; =\; \backslash arcsin\; x\; \backslash ,$

If you only have a fragment of a sine table:

:$\backslash arccos\; x\; =\; \backslash arcsin\; \backslash sqrt\{1-x^2\},\backslash text\{\; if\; \}0\; \backslash leq\; x\; \backslash leq\; 1$ :$\backslash arctan\; x\; =\; \backslash arcsin\; \backslash frac\{x\}\{\backslash sqrt\{x^2+1\}\}$

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 $\backslash tan\; \backslash frac\{\backslash theta\}\{2\}\; =\; \backslash frac\{\backslash sin\; \backslash theta\}\{1+\backslash cos\; \backslash theta\}$, we get:

:$\backslash arcsin\; x\; =\; 2\; \backslash arctan\; \backslash frac\{x\}\{1+\backslash sqrt\{1-x^2\}\}$

:

:$\backslash arctan\; x\; =\; 2\; \backslash arctan\; \backslash frac\{x\}\{1+\backslash sqrt\{1+x^2\}\}$

Relationships between trigonometric functions and inverse trigonometric functions

:$\backslash sin\; (\backslash arccos\; x)\; =\; \backslash cos(\backslash arcsin\; x)\; =\; \backslash sqrt\{1-x^2\}$

:$\backslash sin\; (\backslash arctan\; x)\; =\; \backslash frac\{x\}\{\backslash sqrt\{1+x^2\}\}$

:$\backslash cos\; (\backslash arctan\; x)\; =\; \backslash frac\{1\}\{\backslash sqrt\{1+x^2\}\}$

:$\backslash tan\; (\backslash arcsin\; x)\; =\; \backslash frac\{x\}\{\backslash sqrt\{1-x^2\}\}$

:$\backslash tan\; (\backslash arccos\; x)\; =\; \backslash frac\{\backslash sqrt\{1-x^2\}\}\{x\}$

General solutions

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: :$\backslash sin(y)\; =\; x\; \backslash \; \backslash Leftrightarrow\backslash \; y\; =\; \backslash arcsin(x)\; +\; 2k\backslash pi\; \backslash text\{\; or\; \}\; y\; =\; \backslash pi\; -\; \backslash arcsin(x)\; +\; 2k\backslash pi$ :$\backslash cos(y)\; =\; x\; \backslash \; \backslash Leftrightarrow\backslash \; y\; =\; \backslash arccos(x)\; +\; 2k\backslash pi\; \backslash text\{\; or\; \}\; y\; =\; 2\backslash pi\; -\; \backslash arccos(x)\; +\; 2k\backslash pi$ :$\backslash tan(y)\; =\; x\; \backslash \; \backslash Leftrightarrow\backslash \; y\; =\; \backslash arctan(x)\; +\; k\backslash pi$ :$\backslash cot(y)\; =\; x\; \backslash \; \backslash Leftrightarrow\backslash \; y\; =\; \backslash arccot(x)\; +\; k\backslash pi$ :$\backslash sec(y)\; =\; x\; \backslash \; \backslash Leftrightarrow\backslash \; y\; =\; \backslash arcsec(x)\; +\; 2k\backslash pi\; \backslash text\{\; or\; \}\; y\; =\; 2\backslash pi\; -\; \backslash arcsec\; (x)\; +\; 2k\backslash pi$ :$\backslash csc(y)\; =\; x\; \backslash \; \backslash Leftrightarrow\backslash \; y\; =\; \backslash arccsc(x)\; +\; 2k\backslash pi\; \backslash text\{\; or\; \}\; y\; =\; \backslash pi\; -\; \backslash arccsc(x)\; +\; 2k\backslash pi$

Derivatives of inverse trigonometric functions

:

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

For a sample derivation: if $\backslash theta\; =\; \backslash arcsin\; x\; \backslash !$, we get: :$\backslash frac\{d\; \backslash arcsin\; x\}\{dx\}\; =\; \backslash frac\{d\; \backslash theta\}\{d\; \backslash sin\; \backslash theta\}\; =\; \backslash frac\{1\}\; \{\backslash cos\; \backslash theta\}\; =\; \backslash frac\{1\}\; \{\backslash sqrt\{1-\backslash sin^2\; \backslash theta\}\}\; =\; \backslash frac\{1\}\{\backslash sqrt\{1-x^2\}\}$

Expression as definite integrals

Integrating the derivative and fixing the value at one point gives an expression for the inverse trigonometric function as a definite integral: : When x equals 1, the integrals with limited domains are improper integrals, but still well-defined.

Infinite series

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

:

:

:

:

:

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

:$\backslash arctan\; z\; =\; \backslash frac\{z\}\{1+z^2\}\; \backslash sum\_\{n=0\}^\backslash infty\; \backslash prod\_\{k=1\}^n\; \backslash frac\{2k\; z^2\}\{(2k+1)(1+z^2)\}.$ (Notice that the term in the sum for n= 0 is the empty product which is 1.)

Alternatively, this can be expressed: :$\backslash arctan\; z\; =\; \backslash sum\_\{n=0\}^\backslash infty\; \backslash frac\{2^\{\backslash ,2n\}\backslash ,(n!)^2\}\{\backslash left(2n+1\backslash right)!\}\; \backslash ;\; \backslash frac\{z^\{\backslash ,2n+1\}\}\{\backslash left(1+z^2\backslash right)^\{n+1\}\}$

Continued fractions for arctangent

Two alternatives to the power series for arctangent are these generalized continued fractions: :$\backslash arctan\; z=\backslash cfrac\{z\}\; \{1+\backslash cfrac\{(1z)^2\}\; \{3-z^2+\backslash cfrac\{(3z)^2\}\; \{5-3z^2+\backslash cfrac\{(5z)^2\}\; \{7-5z^2+\backslash cfrac\{(7z)^2\}\; \{9-7z^2+\backslash ddots\}\}\}\}\}\; =\backslash cfrac\{z\}\; \{1+\backslash cfrac\{(1z)^2\}\; \{3+\backslash cfrac\{(2z)^2\}\; \{5+\backslash cfrac\{(3z)^2\}\; \{7+\backslash cfrac\{(4z)^2\}\; \{9+\backslash ddots\backslash ,\}\}\}\}\}\backslash ,$
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

For real and complex values of x: :

Then

:$\backslash int\; \backslash arcsin\; x\backslash ,\backslash mathrm\{d\}x\; =\; x\; \backslash arcsin\; x\; -\; \backslash int\; \backslash frac\{x\}\{\backslash sqrt\{1-x^2\}\}\backslash ,\backslash mathrm\{d\}x$

Substitute

: $k\; =\; 1\; -\; x^2.\backslash ,$

Then

: $\backslash mathrm\{d\}k\; =\; -2x\backslash ,\backslash mathrm\{d\}x$

and

:$\backslash int\; \backslash frac\{x\}\{\backslash sqrt\{1-x^2\}\}\backslash ,\backslash mathrm\{d\}x\; =\; -\backslash frac\{1\}\{2\}\backslash int\; \backslash frac\{\backslash mathrm\{d\}k\}\{\backslash sqrt\{k\}\}\; =\; -\backslash sqrt\{k\}$

Back-substitute for x to yield

:$\backslash int\; \backslash arcsin\; x\; \backslash mathrm\{d\}x\; =\; x\; \backslash arcsin\; x\; +\; \backslash sqrt\{1-x^2\}+C$

==Two-argument variant of arctangent== 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:

:

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: :$\backslash operatorname\{atan2\}(y,\; x)=2\backslash arctan\; \backslash frac\{y\}\{\backslash sqrt\{x^2\; +\; y^2\}\; +\; x\}$ 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.

Logarithmic forms

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

:

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

Example proof

:$\backslash arcsin\; x\backslash ,=\backslash ,\backslash theta$

:$\backslash frac\{e^\{i\backslash theta\}\; -\; e^\{-i\backslash theta\}\}\{2i\}\; =\; x$

(exponential definition of sine)

Let

:$k=e^\{i\backslash ,\backslash theta\}.\; \backslash ,$

Then

:$\backslash frac\{k-\backslash frac\{1\}\{k\}\}\{2i\}\; =\; x$

:$k^2-2\backslash ,i\backslash ,k\backslash ,x-1\backslash ,=\backslash ,0$

:$k\; =\; ix\; \backslash pm\; \backslash sqrt\{1-x^2\}\; =\; e^\{i\backslash theta\}\; \backslash ,$

(the positive branch is chosen)

:$\backslash theta\; =\; \backslash arcsin\; x\; =\; -i\; \backslash ln\; \backslash left(ix\; +\; \backslash sqrt\{1-x^2\}\backslash right)\; \backslash ,$

Q.E.D.

{| style="text-align:center" |+ Inverse trigonometric functions in the complex plane | | | | | | |- |$\backslash arcsin(z)$ |$\backslash arccos(z)$ |$\backslash arctan(z)$ |$\backslash arccot(z)$ |$\backslash arcsec(z)$ |$\backslash arccsc(z)$ |}

Arctangent addition formula

:$\backslash arctan\; u\; +\; \backslash arctan\; v\; =\; \backslash arctan\; \backslash left(\; \backslash frac\{u+v\}\{1-uv\}\; \backslash right)\; \backslash pmod\; \backslash pi,\; \backslash qquad\; u\; v\; \backslash ne\; 1\; \backslash ,.$

This is derived from the tangent addition formula

:$\backslash tan\; (\; \backslash alpha\; +\; \backslash beta\; )\; =\; \backslash frac\{\backslash tan\; \backslash alpha\; +\; \backslash tan\; \backslash beta\}\; \{1\; -\; \backslash tan\; \backslash alpha\; \backslash tan\; \backslash beta\}\; \backslash ,,$

by letting :$\backslash alpha\; =\; \backslash arctan\; u\; \backslash ,,\; \backslash quad\; \backslash beta\; =\; \backslash arctan\; v\; \backslash ,.$

Application: finding the angle of 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

:$\backslash theta\; =\; \backslash arcsin\; \backslash left(\; \backslash frac\{\backslash text\{opposite\}\}\{\backslash text\{hypotenuse\}\}\; \backslash right).$

Often, the hypotenuse is unknown and would need to be calculated before using arcsine or arccosine. Arctangent comes in handy in this situation, as the length of the hypotenuse is not needed.

:$\backslash theta\; =\; \backslash arctan\; \backslash left(\; \backslash frac\{\backslash text\{opposite\}\}\{\backslash text\{adjacent\}\}\; \backslash right).$

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:

:$\backslash theta\; =\; \backslash arctan\; \backslash left(\backslash frac\{\backslash text\{opposite\}\}\{\backslash text\{adjacent\}\}\; \backslash right)\; =\; \backslash arctan\; \backslash left(\; \backslash frac\{\backslash text\{rise\}\}\{\backslash text\{run\}\}\; \backslash right)\; =\; \backslash arctan\; \backslash left(\; \backslash frac\{8\}\{20\}\; \backslash right)\; =\; 21.8^\{\backslash circ\}.$

Practical considerations

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

Trigonometric function

Tangent half-angle formula

List of trigonometric identities

Complex logarithm

Square root

Gauss's continued fraction

Inverse hyperbolic function

External links

Category:Trigonometry Category:Elementary special functions