Trigonometry / Unit 5: Inverse Trigonometric Functions
5.3 Compositions That Simplify
Notes
Composing a function with its inverse can undo it, but only inside the allowed range.
- $\sin(\arcsin x) = x$ for $-1 \le x \le 1$. Likewise $\cos(\arccos x) = x$ and $\tan(\arctan x) = x$ for all $x$.
- $\arcsin(\sin x) = x$ only if $\displaystyle -\frac{\pi}{2} \le x \le \frac{\pi}{2}$. $\arccos(\cos x) = x$ only if $0 \le x \le \pi$.
- Otherwise, evaluate the inside first. Example: $\displaystyle \arcsin\left(\sin\frac{2\pi}{3}\right) = \arcsin\frac{\sqrt{3}}{2} = \frac{\pi}{3}$.
Practice
What is $\sin(\arcsin(0.3))$?
The input $0.3$ is in $[-1, 1]$, so the sine undoes the arcsine.
What is $\displaystyle \arcsin\left(\sin\frac{5\pi}{6}\right)$?
$\displaystyle \sin\frac{5\pi}{6} = \frac{1}{2}$ and $\displaystyle \arcsin\frac{1}{2} = \frac{\pi}{6}$. The angle $\displaystyle \frac{5\pi}{6}$ is outside the range of arcsin.
What is $\displaystyle \arccos\left(\cos\left(-\frac{\pi}{3}\right)\right)$?
$\displaystyle \cos\left(-\frac{\pi}{3}\right) = \frac{1}{2}$ and $\displaystyle \arccos\frac{1}{2} = \frac{\pi}{3}$, since the output must lie in $[0, \pi]$.
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