Question

If tan $\theta =\frac{1}{\sqrt{5}}$ and $\theta$ lies in the first quadrant, the value of $\cos \theta$ is

• A. $\frac{1}{\sqrt{6}}$
• B. $\frac{\sqrt{5}}{\sqrt{6}}$
• C. $\frac{-1}{\sqrt{6}}$
• D. $\frac{-\sqrt{5}}{\sqrt{6}}$

Answer 1

The correct option is B $\frac{\sqrt{5}}{\sqrt{6}}$

Given $\theta$ lies in first quadrant$⟹\cos \theta >0$

$\tan \theta =\frac{1}{\sqrt{5}}$

As we know that

${\sec }^2\theta =1+{\tan }^2\theta =1+(\frac{1}{\sqrt{5}}{)}^2=1+\frac{1}{5}=\frac{6}{5}$

$⟹\sec \theta =\frac{\sqrt{6}}{\sqrt{5}}$

$\cos \theta =\frac{1}{\sec \theta }=\frac{1}{\frac{\sqrt{6}}{\sqrt{5}}}=\frac{\sqrt{5}}{\sqrt{6}}$