Graphs of Inverse Trig Functions

Arcsin (Inverse Sine): $y = \arcsin x$

Key Features

• Domain: $[-1, 1]$
• Range: $\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$
• Passes through: Origin $(0, 0)$
• Shape: S-curve contained in a box
• Increasing throughout domain
• Odd function: $\arcsin(-x) = -\arcsin(x)$

Key Points

$x$	$\arcsin x$
$-1$	$-\frac{\pi}{2}$
$0$	$0$
$1$	$\frac{\pi}{2}$

Arccos (Inverse Cosine): $y = \arccos x$

Key Features

• Domain: $[-1, 1]$
• Range: $[0, \pi]$
• Passes through: $(1, 0)$ and $(-1, \pi)$
• y-intercept: $\frac{\pi}{2}$
• Shape: Decreasing curve
• Decreasing throughout domain
• Neither even nor odd

Key Points

$x$	$\arccos x$
$-1$	$\pi$
$0$	$\frac{\pi}{2}$
$1$	$0$

Arctan (Inverse Tangent): $y = \arctan x$

Key Features

• Domain: All real numbers $(-\infty, \infty)$
• Range: $\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$
• Horizontal asymptotes: $y = \frac{\pi}{2}$ and $y = -\frac{\pi}{2}$
• Passes through: Origin $(0, 0)$
• Increasing throughout domain
• Odd function: $\arctan(-x) = -\arctan(x)$

Key Points

$x$	$\arctan x$
$-\infty$	$-\frac{\pi}{2}$ (asymptote)
$-1$	$-\frac{\pi}{4}$
$0$	$0$
$1$	$\frac{\pi}{4}$
$+\infty$	$\frac{\pi}{2}$ (asymptote)

Arccot (Inverse Cotangent): $y = \text{arccot}\, x$

Key Features

• Domain: All real numbers $(-\infty, \infty)$
• Range: $(0, \pi)$
• Horizontal asymptotes: $y = 0$ and $y = \pi$
• Decreasing throughout domain

Key Points

$x$	$\text{arccot}\, x$
$-\infty$	$\pi$ (asymptote)
$-1$	$\frac{3\pi}{4}$
$0$	$\frac{\pi}{2}$
$1$	$\frac{\pi}{4}$
$+\infty$	$0$ (asymptote)

Arcsec (Inverse Secant): $y = \text{arcsec}\, x$

Key Features

• Domain: $|x| \geq 1$, i.e., $(-\infty, -1] \cup [1, \infty)$
• Range: $\left[0, \frac{\pi}{2}\right) \cup \left(\frac{\pi}{2}, \pi\right]$
• Horizontal asymptote: $y = \frac{\pi}{2}$
• Undefined at $x = 0$

Key Points

$x$	$\text{arcsec}\, x$
$1$	$0$
$2$	$\frac{\pi}{3}$
$-1$	$\pi$
$-2$	$\frac{2\pi}{3}$

Arccsc (Inverse Cosecant): $y = \text{arccsc}\, x$

Key Features

• Domain: $|x| \geq 1$, i.e., $(-\infty, -1] \cup [1, \infty)$
• Range: $\left[-\frac{\pi}{2}, 0\right) \cup \left(0, \frac{\pi}{2}\right]$
• Horizontal asymptote: $y = 0$
• Undefined at $x = 0$
• Odd function

Key Points

$x$	$\text{arccsc}\, x$
$1$	$\frac{\pi}{2}$
$2$	$\frac{\pi}{6}$
$-1$	$-\frac{\pi}{2}$
$-2$	$-\frac{\pi}{6}$

Summary Table

Function	Domain	Range	Asymptotes
$\arcsin x$	$[-1, 1]$	$\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$	None
$\arccos x$	$[-1, 1]$	$[0, \pi]$	None
$\arctan x$	$\mathbb{R}$	$\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$	$y = \pm\frac{\pi}{2}$
$\text{arccot}\, x$	$\mathbb{R}$	$(0, \pi)$	$y = 0, \pi$
$\text{arcsec}\, x$	$\lvert x \rvert \geq 1$	$[0, \pi], x \neq \frac{\pi}{2}$	$y = \frac{\pi}{2}$
$\text{arccsc}\, x$	$\lvert x \rvert \geq 1$	$\left[-\frac{\pi}{2}, \frac{\pi}{2}\right], x \neq 0$	$y = 0$

Graph Relationships

• arcsin and arccos: $\arcsin x + \arccos x = \frac{\pi}{2}$ (vertical shift relationship)
• arctan and arccot: $\arctan x + \text{arccot}\, x = \frac{\pi}{2}$
• Reflection: $\arcsin x$ is $\arccos x$ reflected about $y = \frac{\pi}{4}$