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Q.
Principal value of $\tan ^{-1} 1+\cos ^{-1}\left(\frac{-1}{2}\right)+\sin ^{-1}\left(\frac{-1}{2}\right)$ is
equal to
Inverse Trigonometric Functions
Solution:
Iet $\tan ^{-1}(1)=\theta $
$ \Rightarrow \tan \theta=1=\tan \frac{\pi}{4}$
$\Rightarrow \theta=\frac{\pi}{4} \in\left(\frac{-\pi}{2}, \frac{\pi}{2}\right)$
$\therefore $ Principal value of $\tan ^{-1}(1)$ is $\frac{\pi}{4}$
Let $\cos ^{-1}\left(\frac{-1}{2}\right)=\phi$
$ \Rightarrow \cos \phi=\frac{-1}{2}=-\cos \frac{\pi}{3}$
$=\cos \left(\pi-\frac{\pi}{3}\right)=\cos \frac{2 \pi}{3}$
$\Rightarrow \phi=\frac{2 \pi}{3} \in[0, \pi] $
$\therefore $ Principal value of $\cos ^{-1}\left(\frac{-1}{2}\right)$ is $\frac{2 \pi}{3}$
Also, principal value of $\sin ^{-1}\left(\frac{-1}{2}\right)$ is $\left(\frac{-\pi}{6}\right)$
Principal value of $\tan ^{-1}(1)+\cos ^{-1}\left(\frac{-1}{2}\right)+\sin ^{-1}\left(\frac{-1}{2}\right)$
$=\frac{\pi}{4}+\frac{2 \pi}{3}-\frac{\pi}{6}=\frac{3 \pi}{4}$