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If \( \sin \theta=\frac{1}{\sqrt{2}} \), find all other trigonometric ratios of angle \( \theta \)
Given:
\( \sin \theta=\frac{1}{\sqrt{2}} \)
To do:
We have to find all other trigonometric ratios of angle \( \theta \).
Solution:
We know that,
$\sin ^{2} \theta+\cos ^{2} \theta=1$
$ \tan \theta=\frac{\sin \theta}{\cos \theta}$
Therefore,
$\cos \theta=\sqrt{1-\sin ^{2} \theta}$
$=\sqrt{1-(\frac{1}{\sqrt{2}})^{2}}$
$=\sqrt{1-\frac{1}{2}}$
$=\sqrt{\frac{2-1}{2}}$
$=\sqrt{\frac{1}{2}}$
$=\frac{1}{\sqrt{2}}$
$\tan \theta=\frac{\frac{1}{\sqrt{2}}}{\frac{1}{\sqrt{2}}}$
$=1$
$\cot \theta=\frac{1}{\tan \theta}$
$=\frac{1}{1}$
$=1$
$\operatorname{cosec} \theta=\frac{1}{\sin \theta}$
$=\frac{1}{\frac{1}{\sqrt{2}}}$
$=\sqrt{2}$
$\sec \theta=\frac{1}{\cos \theta}$
$=\frac{1}{\frac{1}{\sqrt{2}}}$
$=\sqrt{2}$