Question
If tan θ = 2, find the values of other trigonometric ratios.
✓
Answer
We know that,
$\sec ^2 \theta=1+\tan ^2 \theta$
$\Rightarrow \sec ^2 \theta=1+(2)^2$
$\Rightarrow \sec ^2 \theta=5$
$\Rightarrow \sec \theta=\sqrt{ } 5 \ldots[1]$
Also,
$\cos \theta=\frac{1}{\sec \theta}$
$\Rightarrow \cos \theta=\frac{1}{\sqrt{5}}$
Now, using
$\tan \theta=\frac{\sin \theta}{\cos \theta}$
$\Rightarrow 2=\frac{\sin \theta}{\frac{1}{\sqrt{5}}}$
$\Rightarrow \sin \theta=2 \times \frac{1}{\sqrt{5}}$
$\Rightarrow \sin \theta=\frac{2}{\sqrt{5}} \ldots[3]$
Also,
$\operatorname{cosec} \theta=\frac{1}{\sin \theta}=\frac{\sqrt{5}}{2}$
$\cot \theta=\frac{1}{\tan \theta}=\frac{1}{2}$
$\sec ^2 \theta=1+\tan ^2 \theta$
$\Rightarrow \sec ^2 \theta=1+(2)^2$
$\Rightarrow \sec ^2 \theta=5$
$\Rightarrow \sec \theta=\sqrt{ } 5 \ldots[1]$
Also,
$\cos \theta=\frac{1}{\sec \theta}$
$\Rightarrow \cos \theta=\frac{1}{\sqrt{5}}$
Now, using
$\tan \theta=\frac{\sin \theta}{\cos \theta}$
$\Rightarrow 2=\frac{\sin \theta}{\frac{1}{\sqrt{5}}}$
$\Rightarrow \sin \theta=2 \times \frac{1}{\sqrt{5}}$
$\Rightarrow \sin \theta=\frac{2}{\sqrt{5}} \ldots[3]$
Also,
$\operatorname{cosec} \theta=\frac{1}{\sin \theta}=\frac{\sqrt{5}}{2}$
$\cot \theta=\frac{1}{\tan \theta}=\frac{1}{2}$
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