Inverse trigonometric functions

Definition

Inverse trigonometric functions are the inverse operations of the trigonometric functions (sine, cosine, tangent, etc.), used to find the angle that corresponds to a given trigonometric value. These functions are commonly denoted as $\sin^{-1}(x)$, $\cos^{-1}(x)$, and $\tan^{-1}(x)$.

5 Must Know Facts For Your Next Test

1. The domain of $\sin^{-1}(x)$ is $[-1, 1]$ and its range is $[-\frac{\pi}{2}, \frac{\pi}{2}]$.
2. The domain of $\cos^{-1}(x)$ is $[-1, 1]$ and its range is $[0, \pi]$.
3. The domain of $\tan^{-1}(x)$ is all real numbers ($(-\infty, \infty)$) and its range is $(-\frac{\pi}{2}, \frac{\pi}{2})$.
4. Inverse trigonometric functions are used to solve equations where the variable is inside a trigonometric function.
5. Graphically, inverse trigonometric functions reflect their respective trigonometric function across the line $y = x$.

Review Questions

• What are the domains and ranges of $\sin^{-1}(x)$, $\cos^{-1}(x)$, and $\tan^{-1}(x)$?
• How do you use inverse trigonometric functions to find an angle given a specific sine or cosine value?
• Explain why the graph of an inverse trigonometric function reflects across the line $y = x$.

Related terms

Trigonometric Functions:

Functions such as sine ($\sin$), cosine ($\cos$), and tangent ($\tan$) that relate angles to side lengths in right triangles.

Domain:

The set of all possible input values (independent variable) for which a function is defined.

Range:

The set of all possible output values (dependent variable) of a function.