Find the principal value of :
(i) $\sin ^{-1}\left(\frac{\sqrt{3}}{2}\right)$
(ii) $\sin ^{-1}\left(\frac{1}{2}\right)$
(iii) $\cos ^{-1}\left(\frac{1}{2}\right)$
(iv) $\tan ^{-1}(1)$
(v) $\tan ^{-1}\left(\frac{1}{\sqrt{3}}\right)$
(vi) $\sec ^{-1}\left(\frac{2}{\sqrt{3}}\right)$
(vii) $\operatorname{cosec}^{-1}(\sqrt{2})$
Trigonometric Table
(i) Let $\sin ^{-1}\left(\frac{\sqrt{3}}{2}\right)=\mathrm{x}$
$\Rightarrow \frac{\sqrt{3}}{2}=\sin x$ [ We know which value of $x$ when placed in sin gives us this answer]
$\therefore \mathrm{X}=\frac{\pi}{3}$
(ii) Let $\sin ^{-1}\left(\frac{1}{2}\right)=x$
$\Rightarrow \frac{1}{2}=\sin x$ [We know which value of $x$ when put in this expression will give us this result]
$\Rightarrow \mathrm{x}=\frac{\pi}{6}$
(iii) Let $\cos ^{-1}\left(\frac{1}{2}\right)=\mathrm{x}$
$\Rightarrow \frac{1}{2}=\cos x$ [We know which value of $x$ when put in this expression will give us this result]
$\therefore \mathrm{X}=\frac{\pi}{3}$
(iv) Let $\tan ^{-1}(1)=x$
$\Rightarrow 1=\tan x$ [We know which value of $x$ when put in this expression will give us this result]
$\therefore \mathrm{X}=\frac{\pi}{4}$
(v) Let $\tan ^{-1}\left(\frac{1}{\sqrt{3}}\right)=\mathrm{x}$
$\Rightarrow \frac{1}{\sqrt{3}}=\tan x$ [We know which value of $x$ when put in this expression will give us this result]
$\therefore \mathrm{X}=\frac{\pi}{6}$
(vi) Let $\sec ^{-1}\left(\frac{2}{\sqrt{3}}\right)=x$
$\Rightarrow \frac{2}{\sqrt{3}}=\sec x$ [We know which value of $x$ when put in this expression will give us this result]
$\therefore \mathrm{X}=\frac{\pi}{6}$
(vii) Let $\operatorname{cosec}^{-1}(\sqrt{2})=x$
$\Rightarrow \sqrt{2}=\operatorname{cosec} x$
[We know which value of x when put in this expression will give us this result]
$\therefore \mathrm{X}=\frac{\pi}{4}$
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