Write the principal value of ^ -1 (1)+ ^ -1 (- 1 2 ). - Vidyadip

Question

Write the principal value of $\tan^{-1} (1)+ \cos^{-1}\bigg(- \frac{1}{2}\bigg).$

✓

Answer

$\tan^{-1}(1) + \cos^{-1}\bigg(-\frac{1}{2}\bigg) =\tan^{-1}\bigg(\tan\frac{\pi}{4}\bigg) + \cos^{-1}\bigg(\cos\bigg(\pi -\frac{\pi}{3}\bigg)\bigg)$
$ = \tan^{-1}\bigg(\tan\frac{\pi}{4}\bigg) + \cos^{-1}\bigg(\cos\frac{2\pi}{3}\bigg)$
$ =\frac{\pi}{4} +\frac{2\pi}{3}\bigg[\because\frac{\pi}{4}\in\bigg( -\frac{\pi}{2},\frac{\pi}{2}\bigg)\text{ and }\frac{2\pi}{3}\in[0,\pi]\bigg]$
$ = \frac{3\pi +8 \pi}{12} = \frac{11\pi}{12}.$

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