Inverse Trigonometric Functions and Their Applications

Inverse Trigonometric Functions

Introduction to Inverse Trigonometric Functions

Inverse trigonometric functions allow us to determine the angle that corresponds to a given trigonometric value. These functions are essential in solving equations involving trigonometric expressions and in applications such as geometry, physics, and engineering.

• Inverse Sine Function (arcsin or $\sin^{-1}$)

• Inverse Cosine Function (arccos or $\cos^{-1}$)

• Inverse Tangent Function (arctan or $\tan^{-1}$)

Key objectives include understanding and using these functions, evaluating them with calculators, and finding exact values of composite functions involving inverse trigonometric functions.

Review of Inverse Functions

• Horizontal Line Test: A function is one-to-one (and thus has an inverse) if no horizontal line intersects its graph more than once.

• Graphical Relationship: If the point $(a, b)$ is on the graph of $f$, then $(b, a)$ is on the graph of $f^{-1}$. The graph of $f^{-1}$ is a reflection of the graph of $f$ about the line $y = x$.

Inverse Sine Function

Definition and Properties

The sine function $y = \sin x$ is not one-to-one over its entire domain, so its inverse is defined by restricting the domain to $-\frac{\pi}{2} \leq x \leq \frac{\pi}{2}$.

• Inverse Sine Function: Denoted $y = \sin^{-1} x$ or $y = \arcsin x$.

• Domain: $-1 \leq x \leq 1$

• Range: $-\frac{\pi}{2} \leq y \leq \frac{\pi}{2}$

• Interpretation: $y = \sin^{-1} x$ means $\sin y = x$ for $y$ in $\left[ -\frac{\pi}{2}, \frac{\pi}{2} \right]$.

Graphing the Inverse Sine Function

• To graph $y = \sin^{-1} x$, take points from the graph of $y = \sin x$ (with restricted domain) and reverse the order of the coordinates.

• Alternatively, reflect the graph of $y = \sin x$ (restricted) about the line $y = x$.

Table: Key Values of $\sin^{-1} x$

$x$	$-1$	$-\frac{\sqrt{3}}{2}$	$-\frac{\sqrt{2}}{2}$	$-\frac{1}{2}$	$0$	$\frac{1}{2}$	$\frac{\sqrt{2}}{2}$	$\frac{\sqrt{3}}{2}$	$1$
$y = \sin^{-1} x$	$-\frac{\pi}{2}$	$-\frac{\pi}{3}$	$-\frac{\pi}{4}$	$-\frac{\pi}{6}$	$0$	$\frac{\pi}{6}$	$\frac{\pi}{4}$	$\frac{\pi}{3}$	$\frac{\pi}{2}$

Example: Finding the Exact Value of an Inverse Sine Function

Find the exact value of $\sin^{-1}\left(\frac{\sqrt{3}}{2}\right)$.

1. Let $\theta = \sin^{-1}\left(\frac{\sqrt{3}}{2}\right)$.

2. Rewrite as $\sin \theta = \frac{\sqrt{3}}{2}$, where $-\frac{\pi}{2} \leq \theta \leq \frac{\pi}{2}$.

3. From the table, the angle in $\left[ -\frac{\pi}{2}, \frac{\pi}{2} \right]$ whose sine is $\frac{\sqrt{3}}{2}$ is $\frac{\pi}{3}$.

Answer: $\sin^{-1}\left(\frac{\sqrt{3}}{2}\right) = \frac{\pi}{3}$

Additional info: The same process applies for other values in the table, using the restricted range for the inverse sine function.