5.5: Inverse Trigonometric Functions
Now that we know the common trigonometric functions, let's look at their inverses.
Recall that if $f$ must be one-to-one for it to have an inverse.
Inverse sine function
The inverse sine function, denoted $\sin^{-1}$, has domain $[-1, 1]$ and range $\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$ where
\[\sin^{-1}x = y \text{ when } \sin y = x\]
The following cancellation properties hold:
\[\begin{align}
\sin(\sin^{-1}x) = x, &\qquad -1 \leq x \leq 1 \\
\sin^{-1}(\sin(x)) = x, &\qquad -\frac{\pi}{2}\leq x \leq \frac{\pi}{2}
\end{align}\]
Suppose $a \in [-1, 1]$. Someone says $\sin^{-1}(a) = \frac{2\pi}{3}$ is possible. Is this true?
Find each value:
\[\sin^{-1}\frac{1}{2} \qquad\qquad \sin^{-1}\left(-\frac{1}{2}\right) \qquad\qquad \sin^{-1}\frac{3}{2}\]
Find each value:
\[\sin^{-1}\left(\sin\frac{\pi}{3}\right) \qquad\qquad \sin^{-1}\left(\sin\left(-\frac{2\pi}{3}\right)\right) \]
Important: remember to always check if the input value for $\sin^{-1}$ is actually in the domain. Keep this in mind for all of the inverse functions.
Inverse cosine function
The inverse cosine function, denoted $\cos^{-1}$, has domain $[-1, 1]$ and range $\left[0, \pi\right]$ where
\[\cos^{-1}x = y \text{ when } \cos y = x\]
The following cancellation properties hold:
\[\begin{align}
\cos(\cos^{-1}x) = x, &\qquad -1 \leq x \leq 1 \\
\cos^{-1}(\cos(x)) = x, &\qquad 0\leq x \leq \pi
\end{align}\]
Find each value:
\[\cos^{-1}\frac{\sqrt{3}}{2} \qquad\qquad \cos^{-1}0 \qquad\qquad \cos^{-1}\left(-\frac{1}{2}\right)\]
Find each value:
\[\cos^{-1}\left(\cos\left(\frac{2\pi}{3}\right)\right) \qquad\qquad \cos^{-1}\left(\cos\left(-\frac{5\pi}{3}\right)\right) \]
Inverse tangent function
The inverse tangent function, denoted $\tan^{-1}$, has domain $\mathbb{R}$ and range $\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$ where
\[\tan^{-1}x = y \text{ when } \tan y = x\]
The following cancellation properties hold:
\[\begin{align}
\tan(\tan^{-1}x) = x, &\qquad x \in \mathbb{R} \\
\tan^{-1}(\tan(x)) = x, &\qquad -\frac{\pi}{2} < x < \frac{\pi}{2}
\end{align}\]
Find each value:
\[\tan^{-1} 1 \qquad\qquad \tan^{-1} \sqrt{3} \qquad\qquad \tan^{-1}0\]
Find each value:
\[\sin^{-1} \left(\tan\frac{\pi}{4}\right) \qquad\qquad \sin^{-1}\left(\cos\frac{\pi}{6}\right)\]