Question
Write the value of $\sin^{-1}(\sin(-600^\circ))\sin(-600^\circ).$
✓
Answer
$\sin^{-1}\{\sin(-600^\circ)\}$
$=\sin^{-1}\{\sin(-600^\circ+360\times2)\}$ $\{\text{Since},\sin(2\text{n}\pi+\theta)=\sin\theta\}$
$=\sin^{-1}\{\sin120^\circ\}$
$=180^\circ-120^\circ$
$\begin{Bmatrix}\text{Since},\sin^{-1}(\sin\theta)=\begin{cases}-\pi-\theta,&\text{ if }\theta\in\Big[\frac{-3\pi}{2},\frac{-\pi}{2}\Big]\\\theta,&\text{ if }\theta\in\Big[-\frac{\pi}{2},\frac{\\\pi}{2}\Big]\\\pi-\theta,&\text{ if }\theta\in\Big[\frac{\pi}{2},\frac{3\pi}{2}\Big]\\\pi-\theta,&\text{ if }\theta\in\Big[\frac{3\pi}{2},\frac{5\pi}{2}\Big]\end{cases}\end{Bmatrix}$
$=60^\circ$
Hence,
$\sin^{-1}\{\sin(-600^\circ)\}=60^\circ$
$=\sin^{-1}\{\sin(-600^\circ+360\times2)\}$ $\{\text{Since},\sin(2\text{n}\pi+\theta)=\sin\theta\}$
$=\sin^{-1}\{\sin120^\circ\}$
$=180^\circ-120^\circ$
$\begin{Bmatrix}\text{Since},\sin^{-1}(\sin\theta)=\begin{cases}-\pi-\theta,&\text{ if }\theta\in\Big[\frac{-3\pi}{2},\frac{-\pi}{2}\Big]\\\theta,&\text{ if }\theta\in\Big[-\frac{\pi}{2},\frac{\\\pi}{2}\Big]\\\pi-\theta,&\text{ if }\theta\in\Big[\frac{\pi}{2},\frac{3\pi}{2}\Big]\\\pi-\theta,&\text{ if }\theta\in\Big[\frac{3\pi}{2},\frac{5\pi}{2}\Big]\end{cases}\end{Bmatrix}$
$=60^\circ$
Hence,
$\sin^{-1}\{\sin(-600^\circ)\}=60^\circ$
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