Inverse Trigonometric Functions – Precalculus Essentials Study Notes

Chapter 4: Trigonometric Functions

4.7 Inverse Trigonometric Functions

Inverse trigonometric functions allow us to determine the angle that corresponds to a given trigonometric value. These functions are essential for solving equations involving trigonometric expressions and for applications in geometry, calculus, and beyond.

Objectives

• Understand and use the inverse sine, cosine, and tangent functions.

• Understand and use the inverse cotangent, cosecant, and secant functions.

• Use a calculator to evaluate inverse trigonometric functions.

• Find exact values of composite functions involving inverse trigonometric functions.

Inverse Functions: General Properties

An inverse function reverses the effect of the original function. For a function to have an inverse, it must be one-to-one (no horizontal line intersects its graph more than once). The graph of an inverse function is a reflection of the original function about the line $y = x$.

The Inverse Sine Function

Definition and Properties

The inverse sine function, denoted by $\sin^{-1} x$ or $\arcsin x$, is the inverse of the restricted sine function $y = \sin x$ for $-\frac{\pi}{2} \leq x \leq \frac{\pi}{2}$. Thus,

$y = \sin^{-1} x$ means $\sin y = x$, where $-\frac{\pi}{2} \leq y \leq \frac{\pi}{2}$ and $-1 \leq x \leq 1$.

Graph of the Inverse Sine Function

The graph of $y = \sin^{-1} x$ is obtained by reflecting the graph of $y = \sin x$ (restricted to $-\frac{\pi}{2} \leq x \leq \frac{\pi}{2}$) about the line $y = x$.

Table of Sine Values

The following table lists common values of $\sin \theta$ for standard angles:

\( \theta \)	\( -\frac{\pi}{2} \)	\( -\frac{\pi}{3} \)	\( -\frac{\pi}{4} \)	\( -\frac{\pi}{6} \)	0	\( \frac{\pi}{6} \)	\( \frac{\pi}{4} \)	\( \frac{\pi}{3} \)	\( \frac{\pi}{2} \)
\( \sin \theta \)	-1	\( -\frac{\sqrt{3}}{2} \)	\( -\frac{\sqrt{2}}{2} \)	\( -\frac{1}{2} \)	0	\( \frac{1}{2} \)	\( \frac{\sqrt{2}}{2} \)	\( \frac{\sqrt{3}}{2} \)	1

Finding Exact Values of $\sin^{-1} x$

• To find $\sin^{-1} x$, determine the angle $y$ in $[-\frac{\pi}{2}, \frac{\pi}{2}]$ such that $\sin y = x$.

• Use the table above or the unit circle for reference.

The Inverse Cosine Function

Definition and Properties

The inverse cosine function, denoted by $\cos^{-1} x$ or $\arccos x$, is the inverse of the restricted cosine function $y = \cos x$ for $0 \leq x \leq \pi$. Thus,

$y = \cos^{-1} x$ means $\cos y = x$, where $0 \leq y \leq \pi$ and $-1 \leq x \leq 1$.

Graph of the Inverse Cosine Function

The graph of $y = \cos^{-1} x$ is obtained by reflecting the graph of $y = \cos x$ (restricted to $0 \leq x \leq \pi$) about the line $y = x$.

Table of Cosine Values

The following table lists common values of $\cos \theta$ for standard angles:

\( \theta \)	0	\( \frac{\pi}{6} \)	\( \frac{\pi}{4} \)	\( \frac{\pi}{3} \)	\( \frac{\pi}{2} \)	\( \frac{2\pi}{3} \)	\( \frac{3\pi}{4} \)	\( \frac{5\pi}{6} \)	\( \pi \)
\( \cos \theta \)	1	\( \frac{\sqrt{3}}{2} \)	\( \frac{\sqrt{2}}{2} \)	\( \frac{1}{2} \)	0	\( -\frac{1}{2} \)	\( -\frac{\sqrt{2}}{2} \)	\( -\frac{\sqrt{3}}{2} \)	-1

Finding Exact Values of $\cos^{-1} x$

• To find $\cos^{-1} x$, determine the angle $y$ in $[0, \pi]$ such that $\cos y = x$.

• Use the table above or the unit circle for reference.

The Inverse Tangent Function

Definition and Properties

The inverse tangent function, denoted by $\tan^{-1} x$ or $\arctan x$, is the inverse of the restricted tangent function $y = \tan x$ for $-\frac{\pi}{2} < x < \frac{\pi}{2}$. Thus,

$y = \tan^{-1} x$ means $\tan y = x$, where $-\frac{\pi}{2} < y < \frac{\pi}{2}$ and $-\infty < x < \infty$.

Graph of the Inverse Tangent Function

The graph of $y = \tan^{-1} x$ is obtained by reflecting the graph of $y = \tan x$ (restricted to $-\frac{\pi}{2} < x < \frac{\pi}{2}$) about the line $y = x$.

Table of Tangent Values

The following table lists common values of $\tan \theta$ for standard angles:

\( \theta \)	\( -\frac{\pi}{3} \)	\( -\frac{\pi}{4} \)	\( -\frac{\pi}{6} \)	0	\( \frac{\pi}{6} \)	\( \frac{\pi}{4} \)	\( \frac{\pi}{3} \)
\( \tan \theta \)	$-\sqrt{3}$	$-1$	$-\frac{\sqrt{3}}{3}$	0	$\frac{\sqrt{3}}{3}$	1	$\sqrt{3}$

Finding Exact Values of $\tan^{-1} x$

• To find $\tan^{-1} x$, determine the angle $y$ in $(-\frac{\pi}{2}, \frac{\pi}{2})$ such that $\tan y = x$.

• Use the table above or the unit circle for reference.

The Inverse Cotangent, Cosecant, and Secant Functions

Definitions

• Inverse cotangent ($\cot^{-1} x$): Inverse of $y = \cot x$ for $0 < x < \pi$.

• Inverse cosecant ($\csc^{-1} x$): Inverse of $y = \csc x$ for $y \leq -\frac{\pi}{2}$ or $y \geq \frac{\pi}{2}$, $x \leq -1$ or $x \geq 1$.

• Inverse secant ($\sec^{-1} x$): Inverse of $y = \sec x$ for $0 \leq y < \frac{\pi}{2}$ or $\frac{\pi}{2} < y \leq \pi$, $x \leq -1$ or $x \geq 1$.

Graphs of the Inverse Trigonometric Functions

The graphs of all six inverse trigonometric functions illustrate their domains and ranges, which are essential for understanding their behavior and for solving equations.

Inverse Properties

Inverse trigonometric functions have important properties that relate them to their original trigonometric functions:

• $\sin(\sin^{-1} x) = x$ for every $x$ in $[-1, 1]$

• $\sin^{-1}(\sin x) = x$ for every $x$ in $\left[ -\frac{\pi}{2}, \frac{\pi}{2} \right]$

• $\cos(\cos^{-1} x) = x$ for every $x$ in $[-1, 1]$

• $\cos^{-1}(\cos x) = x$ for every $x$ in $[0, \pi]$

• $\tan(\tan^{-1} x) = x$ for every real number $x$

• $\tan^{-1}(\tan x) = x$ for every $x$ in $\left( -\frac{\pi}{2}, \frac{\pi}{2} \right)$

Evaluating Compositions and Calculator Use

• To evaluate expressions like $\cos(\cos^{-1} 0.7)$, use the property $\cos(\cos^{-1} x) = x$ (if $x$ is in the domain).

• For values outside the domain, the expression is not defined (e.g., $\cos^{-1}(-1.2)$ is undefined).

• Calculators can be used to approximate values of inverse trigonometric functions for non-exact values.

Summary Table: Domains and Ranges of Inverse Trigonometric Functions

Function	Domain	Range
$\sin^{-1} x$	$[-1, 1]$	$\left[ -\frac{\pi}{2}, \frac{\pi}{2} \right]$
$\cos^{-1} x$	$[-1, 1]$	$[0, \pi]$
$\tan^{-1} x$	$(-\infty, \infty)$	$\left( -\frac{\pi}{2}, \frac{\pi}{2} \right)$
$\cot^{-1} x$	$(-\infty, \infty)$	$(0, \pi)$
$\sec^{-1} x$	$(-\infty, -1] \cup [1, \infty)$	$[0, \frac{\pi}{2}) \cup (\frac{\pi}{2}, \pi]$
$\csc^{-1} x$	$(-\infty, -1] \cup [1, \infty)$	$\left[ -\frac{\pi}{2}, 0 \right) \cup \left( 0, \frac{\pi}{2} \right]$