Question

If tanθ = 2, find the values of other trigonometric ratios.

Answer 1

tanθ = 2 (Given)

We have,

$$\begin{gathered} {\sec }^2\theta =1+{\tan }^2\theta \\ \Rightarrow \sec \theta =\sqrt{1+{\tan }^2\theta } \\ \Rightarrow \sec \theta =\sqrt{1+2^2} \\ \Rightarrow \sec \theta =\sqrt{1+4}=\sqrt{5} \end{gathered}$$
$\therefore \cos \theta =\frac{1}{\sec \theta }=\frac{1}{\sqrt{5}}$

Now,

$$\begin{gathered} \tan \theta =\frac{\sin \theta }{\cos \theta } \\ \Rightarrow \sin \theta =\tan \theta \times \cos \theta \\ \Rightarrow \sin \theta =2\times \frac{1}{\sqrt{5}}=\frac{2}{\sqrt{5}} \\ \therefore \mathrm{cosec}\theta =\frac{1}{\sin \theta }=\frac{1}{{\displaystyle \frac{2}{\sqrt{5}}}}=\frac{\sqrt{5}}{2} \end{gathered}$$
Also,

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