TrigonometryTopic #19 of 32

Inverse Trigonometric Functions

Definitions, domains, and ranges of arcsin, arccos, arctan, and related functions.

Definitions

Inverse trig functions "undo" the regular trig functions.

Function	Notation	Meaning
Inverse Sine	$\arcsin x$ or $\sin^{-1} x$	Angle whose sine is $x$
Inverse Cosine	$\arccos x$ or $\cos^{-1} x$	Angle whose cosine is $x$
Inverse Tangent	$\arctan x$ or $\tan^{-1} x$	Angle whose tangent is $x$
Inverse Cosecant	$\text{arccsc}\, x$ or $\csc^{-1} x$	Angle whose cosecant is $x$
Inverse Secant	$\text{arcsec}\, x$ or $\sec^{-1} x$	Angle whose secant is $x$
Inverse Cotangent	$\text{arccot}\, x$ or $\cot^{-1} x$	Angle whose cotangent is $x$

Note: $\sin^{-1} x \neq \frac{1}{\sin x}$ (that would be $\csc x$ )

Domains and Ranges

Function	Domain	Range (Principal Values)
$\arcsin x$	$[-1, 1]$	$\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$
$\arccos x$	$[-1, 1]$	$[0, \pi]$
$\arctan x$	$(-\infty, \infty)$	$\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$
$\text{arccsc}\, x$	$\lvert x \rvert \geq 1$	$\left[-\frac{\pi}{2}, 0\right) \cup \left(0, \frac{\pi}{2}\right]$
$\text{arcsec}\, x$	$\lvert x \rvert \geq 1$	$\left[0, \frac{\pi}{2}\right) \cup \left(\frac{\pi}{2}, \pi\right]$
$\text{arccot}\, x$	$(-\infty, \infty)$	$(0, \pi)$

Common Values

Arcsin

$x$	$\arcsin x$
$-1$	$-\frac{\pi}{2}$
$-\frac{\sqrt{3}}{2}$	$-\frac{\pi}{3}$
$-\frac{\sqrt{2}}{2}$	$-\frac{\pi}{4}$
$-\frac{1}{2}$	$-\frac{\pi}{6}$
$0$	$0$
$\frac{1}{2}$	$\frac{\pi}{6}$
$\frac{\sqrt{2}}{2}$	$\frac{\pi}{4}$
$\frac{\sqrt{3}}{2}$	$\frac{\pi}{3}$
$1$	$\frac{\pi}{2}$

Arccos

$x$	$\arccos x$
$-1$	$\pi$
$-\frac{\sqrt{3}}{2}$	$\frac{5\pi}{6}$
$-\frac{\sqrt{2}}{2}$	$\frac{3\pi}{4}$
$-\frac{1}{2}$	$\frac{2\pi}{3}$
$0$	$\frac{\pi}{2}$
$\frac{1}{2}$	$\frac{\pi}{3}$
$\frac{\sqrt{2}}{2}$	$\frac{\pi}{4}$
$\frac{\sqrt{3}}{2}$	$\frac{\pi}{6}$
$1$	$0$

Arctan

$x$	$\arctan x$
$-\sqrt{3}$	$-\frac{\pi}{3}$
$-1$	$-\frac{\pi}{4}$
$-\frac{\sqrt{3}}{3}$	$-\frac{\pi}{6}$
$0$	$0$
$\frac{\sqrt{3}}{3}$	$\frac{\pi}{6}$
$1$	$\frac{\pi}{4}$
$\sqrt{3}$	$\frac{\pi}{3}$

Inverse Properties

Composition with Original Function

$\sin(\arcsin x) = x \quad \text{for } -1 \leq x \leq 1$

$\cos(\arccos x) = x \quad \text{for } -1 \leq x \leq 1$

$\tan(\arctan x) = x \quad \text{for all } x$

Original with Inverse (restricted domain)

$\arcsin(\sin x) = x \quad \text{for } -\frac{\pi}{2} \leq x \leq \frac{\pi}{2}$

$\arccos(\cos x) = x \quad \text{for } 0 \leq x \leq \pi$

$\arctan(\tan x) = x \quad \text{for } -\frac{\pi}{2} < x < \frac{\pi}{2}$

Relationships

$\arcsin x + \arccos x = \frac{\pi}{2}$

$\arctan x + \text{arccot}\, x = \frac{\pi}{2}$

$\text{arccsc}\, x = \arcsin\left(\frac{1}{x}\right)$

$\text{arcsec}\, x = \arccos\left(\frac{1}{x}\right)$

Derivatives (Calculus)

Function	Derivative
$\arcsin x$	$\frac{1}{\sqrt{1 - x^2}}$
$\arccos x$	$-\frac{1}{\sqrt{1 - x^2}}$
$\arctan x$	$\frac{1}{1 + x^2}$
$\text{arccsc}\, x$	$-\frac{1}{
$\text{arcsec}\, x$	$\frac{1}{
$\text{arccot}\, x$	$-\frac{1}{1 + x^2}$

Graphs of Trigonometric Functions

Graphs of Inverse Trig Functions

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