Inverse functions

An inverse trigonometric function returns an angle. For example, $\arccos \frac{3}{5}$ is the angle whose cosine is $\frac{3}{5}$. Its range is restricted so the output is unique: $\arcsin x\in [-\frac{\pi }{2},\frac{\pi }{2}]$, $\arccos x\in [0,\pi ]$, and $\arctan x\in (-\frac{\pi }{2},\frac{\pi }{2})$. To evaluate expressions like $\sin (\arccos \frac{3}{5})$, let $\theta =\arccos \frac{3}{5}$, so $\cos \theta =\frac{3}{5}$. Then use ${\sin }^2\theta +{\cos }^2\theta =1$ and the sign of $\theta$ to find $\sin \theta$. For $\cos (\arcsin x)$, set $\theta =\arcsin x$; since $\theta \in [-\frac{\pi }{2},\frac{\pi }{2}]$, $\cos \theta \ge 0$, giving $\sqrt{1-x^2}$.