Inverse Trigonometric Functions

1. If $-1\le x\le 1$, then $\sin^{-1}x$ or $\arcsin x$ is the angle in $[-\pi/2,\pi/2]$ whose sine is equal to $x$: $\sin(\sin^{-1}x)=x$.

2. If $-\infty<x<\infty$, then $\tan^{-1}x$ or $\arctan x$ is the angle in $(-\pi/2,\pi/2)$ whose tangent is equal to $x$: $\tan(\tan^{-1}x)=x$.

3. If $-1\le x\le 1$, then $\cos^{-1}x$ or $\arccos x$ is the angle in $[0,\pi]$ whose cosine is equal to $x$: $\cos(\cos^{-1}x)=x$.

Ex 1 Find $\sin^{-1}(1/2)$ and $\sin^{-1}(-1/2)$.

Sol $\sin^{-1}(1/2)=\pi/6$ since $\sin(\pi/6)=1/2$ and $-\pi/2\le \pi/6\le \pi/2$, and

$\sin^{-1}(-1/2)=-\pi/6$ since $\sin(-\pi/6)=-1/2$ and $-\pi/2\le -\pi/6\le \pi/2$.

Ex 2 Find $\sin(\tan^{-1}(4/3))$.

Sol Let $\theta=\tan^{-1}(4/3)$, so $\tan\theta=4/3$. From the right triangle shown below, $\sin\theta=4/5$, so $\sin(\tan^{-1}(4/3))=4/5$.

Pr A Find the following, or explain why they are undefined:

a) $\sin(\sin^{-1}4/9)$ and b) $\sin(\sin^{-1}5/3)$.

Pr B Find the following, or explain why they are undefined:

a) $\tan(\tan^{-1}4)$ and b) $\tan^{-1}(tan\frac{3\pi}{5}$.

Pr 1 Find a) $\tan^{-1}1$ and b) $\tan^{-1}\sqrt{3}$.

Pr 2 Find a) $\sin^{-1}(\sqrt{3}/2)$ and b) $\sin^{-1}(\sin(3\pi/2))$.

Pr 3 Find a) $\cos^{-1}0$ and b) $\cos^{-1}(-1/2)$.

Pr 4 Find $\sin(\tan^{-1}3)$.

Pr 5 Find $\tan(\sin^{-1}(5/13))$.

Pr 6 Find $\cos(2\arcsin(3/4))$.

Pr 7 Find $\sin(2\arctan(3/4))$.

Pr 8 Find $\sin^{-1}(\sin(18\pi/5))$.

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