Write the value of ^ -1 ( 350^ )- ^ -1 ( 350^ ) - Vidyadip

Question

Write the value of $\cos^{-1}(\cos350^\circ)-\sin^{-1}(\sin350^\circ)$

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Answer

$\cos^{-1}(\cos350^\circ)-\sin^{-1}(\sin350^\circ)$
$=\cos^{-1}\{\cos(360^\circ-10^\circ)\}-\sin^{-1}\{\sin(360^\circ-10^\circ)\}$
$=\cos^{-1}\{\cos10^\circ\}-\sin^{-1}\{\sin10^\circ\}$
$\{\text{Since},\cos(2\pi-\theta)=\cos\theta,\sin(2\pi-\theta)=-\sin\theta\}$
$=10^\circ-\sin^{-1}\{\sin(-10^\circ)\}$
$\{\text{Since},\cos^{-1}(\cos\theta),\text{ if }\theta\in[0,\pi]\text{ and }\sin(-\theta)=-\sin\theta\}$
$=10^\circ-(-10^\circ)$ $\Big\{\text{Since},\sin^{-1}(\sin\theta)=\theta,\text{ if }\theta\in\Big[-\frac{\pi}{2},\frac{\pi}{2}\Big]\Big\}$
$=10^\circ+10^\circ$
$=20^\circ$
Hence,
$\cos^{-1}(\cos350^\circ)-\sin^{-1}(\sin350^\circ)=20^\circ$

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