Question

If $3\sec \theta =5$ then find $\tan \theta$

Answer 1

$3\sec \theta =5$

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

$\Rightarrow \sec \theta =\frac{5}{3}$

Squaring both sides. We get,

$\Rightarrow {\sec }^2\theta =\frac{25}{9}$

Subtracting $1$ on both sides

$\Rightarrow {\sec }^2\theta -1=\frac{25}{9}-1$

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

$\Rightarrow {\tan }^2\theta =\frac{25-9}{9}=\frac{16}{9}$

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

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