Inverse Functions

Overview

Each of the trigonometric functions have an inverse function with a restricted domain.

Arcsine

The arcsine function is the inverse sine function, denoted by $\arcsin$ or $\sin^{- 1}$ . For $- 1 \leq y \leq 1$ , if $t = \arcsin y$ , then

y = \sin t and - \frac{π}{2} \leq t \leq \frac{π}{2} .

Arccosine

The arccosine function is the inverse cosine function, denoted by $\arccos$ or $\cos^{- 1}$ . For $- 1 \leq y \leq 1$ , if $t = \arccos y$ , then

y = \cos t and 0 \leq t \leq π .

Arctangent

The arctangent function is the inverse tangent function, denoted by $\arctan$ or $\tan^{- 1}$ . For $- 1 \leq y \leq 1$ , if $t = \arctan y$ , then

y = \tan t and - \frac{π}{2} < t < \frac{π}{2} .

atan2

The atan2 function denotes the $2$ -argument arctangent function. Geometrically, $atan2 (y, x)$ denotes the angle measure $- π < θ \leq π$ between the positive $x$ -axis and the ray from the origin to point $⟨ x, y ⟩$ in the Cartesian plane. That is:

atan2 (y, x) = {\begin{cases} \arctan \frac{y}{x} & if x > 0 \\ \arctan \frac{y}{x} + π & if x < 0 and y \geq 0 \\ \arctan \frac{y}{x} - π & if x < 0 and y < 0 \\ \frac{π}{2} & if x = 0 and y > 0 \\ - \frac{π}{2} & if x = 0 and y < 0 \\ undefined & if x = 0 and y = 0 \end{cases}

Let $z = x + y i$ be a complex number. Then $atan2 (y, x) = Arg (x + y i)$ .

Derivatives

The following outlines the derivatives of some of the inverse circular functions:

Function	Derivative
$\arcsin x$	$\frac{1}{\sqrt{1 - x^{2}}}$
$\arccos x$	$\frac{- 1}{\sqrt{1 - x^{2}}}$
$\arctan x$	$\frac{1}{1 + x^{2}}$