Question

If $\cos \theta =\frac{9}{15}$ then find $\tan \theta$

Answer 1

$\cos \theta =\frac{9}{15}$ is positive.

$\therefore \theta$ lies in the first or fourth quadrant.

$\Rightarrow {\cos }^2\theta =\frac{81}{225}$

We know that, ${\cos }^2\theta =\frac{1}{{\sec }^2\theta }$

$\Rightarrow {\sec }^2\theta =\frac{225}{81}$

Subtracting $1$ on both sides

$\Rightarrow {\sec }^2\theta -1=\frac{225}{81}-1$

We know that, $1+{\tan }^2\theta ={\sec }^2\theta$

$\Rightarrow {\tan }^2\theta =\frac{225-81}{81}=\frac{144}{81}$

since $\tan \theta$ is negative in fourth quadrant and positive in first quadrant.

$\Rightarrow \tan \theta =\pm \frac{12}{9}=\pm \frac{4}{3}$