Inverse cosine function

Definition

The inverse cosine function, denoted as $\cos^{-1}(x)$ or $\arccos(x)$, returns the angle whose cosine is $x$. It is defined for $-1 \leq x \leq 1$ and returns values in the range $0 \leq y \leq \pi$.

5 Must Know Facts For Your Next Test

1. The domain of the inverse cosine function is $[-1, 1]$.
2. The range of the inverse cosine function is $[0, \pi]$.
3. $\cos(\cos^{-1}(x)) = x$ for all $x$ in the domain of the inverse cosine function.
4. $\cos^{-1}(\cos(y)) = y$ for all $y$ in the range of the inverse cosine function.
5. The graph of the inverse cosine function is a reflection of a portion of the cosine function across the line $y = x$.

Review Questions

• What is the domain and range of the inverse cosine function?
• Compute $\cos^{-1}(-0.5)$.
• Explain why $\cos(\cos^{-1}(x)) = x$ holds true for all valid values of $x$.

Related terms

Inverse Sine Function: The inverse sine function, denoted as $\sin^{-1}(x)$ or $\arcsin(x)$, returns the angle whose sine is $x$, with domain $[-1, 1]$ and range $[-\frac{\pi}{2}, \frac{\pi}{2}]$.

Inverse Tangent Function: The inverse tangent function, denoted as $\tan^{-1}(x)$ or $\arctan(x)$, returns the angle whose tangent is $x$, with domain all real numbers and range $(-\frac{\pi}{2}, \frac{\pi}{2})$.

Cosine Function: $\A periodic trigonometric function that describes a wave-like shape. It has a period of \(2\pi\) and ranges from -1 to 1.