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• Notes

Find the Exact value of the expression $\sin ^{-1}(\sin (7 \pi / 3))$

$\sin ^{-1}(\sin (7 \pi / 3))=$

$\sin ^{-1}\left(\sin \left(2 \pi+\frac{\pi}{3}\right)\right)$

$=\sin ^{-1}\left(\sin \left(\frac{\pi}{3}\right)\right)$

$=\frac{\pi}{3}$

Find the value of $: \cos ^{-1}(\sin (5 \pi / 4))$

$\cos ^{-1}\left(\sin \left(\frac{5 \pi}{4}\right)\right)=\cos ^{-1}\left(\sin \left(\frac{4 \pi}{4}+\frac{\pi}{4}\right)\right)$

$=\cos ^{-1}\left(\sin \left(\pi+\frac{\pi}{4}\right)\right)$

$\sin (\pi+\theta)=-\sin \theta$

$=\cos ^{-1}\left(-\sin \frac{\pi}{4}\right)$

$=\cos ^{-1}\left(-\frac{1}{\sqrt{2}}\right)$

$=\pi-\cos ^{-1}\left(\frac{1}{\sqrt{2}}\right)$

$\pi-\cos ^{-1}(\theta)=\cos ^{-1}(-\theta)$

$=\pi-\frac{\pi}{4}=\frac{4 \pi}{4}-\frac{\pi}{4}=\frac{3 \pi}{4}$

Find the Exact value of the expression : $\tan \left(\sec ^{-1} 4\right)$

$=\frac{4}{1}=4$

$\sec^{-1} \sec \theta=\sec^{-1} 4$

$\theta=\sec ^{-1}(4)$

$\tan \left(\sec ^{-1}(4)\right)=\tan (\theta)$

$=\frac{\sqrt{15}}{1}$

$\tan \theta=\sqrt{15}$

Find the exact value of : $\tan \left(\sec ^{-1}(-5 / 3)\right)$

$\tan \left(\sec ^{-1}\left(\frac{-5}{3}\right)\right)=\tan \left(\pi+\sec ^{-1}\left(\frac{5}{3}\right)\right)$

$\sec ^{-1}(-\theta)=\sec ^{-1}(\theta)$

$=\tan \left(\sec ^{-1}\left(\frac{5}{3}\right)\right)$

$=\frac{5}{3}$

$=\frac{4}{3}$

$\theta=\sec^{-1} \left(\frac{5}{3}\right)$

$\tan \left(\sec ^{-1}\left(\frac{5}{3}\right)\right)=\tan (\theta)=\frac{4}{3}$

$\tan \left(\sec ^{-1}\left(\frac{-5}{3}\right)\right)=\frac{4}{3}$