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,

$\cos \theta=\sqrt{1-\sin ^{2} \theta}$

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

$=\sqrt{1-\frac{1}{2}}$

$=\sqrt{\frac{2-1}{2}}$

$=\sqrt{\frac{1}{2}}$

$=\frac{1}{\sqrt{2}}$
$\tan \theta=\frac{\frac{1}{\sqrt{2}}}{\frac{1}{\sqrt{2}}}$

$=1$

$\cot \theta=\frac{1}{\tan \theta}$

$=\frac{1}{1}$

$=1$ 

$\operatorname{cosec} \theta=\frac{1}{\sin \theta}$

$=\frac{1}{\frac{1}{\sqrt{2}}}$

$=\sqrt{2}$

$\sec \theta=\frac{1}{\cos \theta}$

$=\frac{1}{\frac{1}{\sqrt{2}}}$

$=\sqrt{2}$ 

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Updated on: 10-Oct-2022

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