Question

If $cosec\theta =\frac{5}{3}$ then find $\tan \theta$

Answer 1

$\csc \theta =\frac{5}{3}$ is positive

$\therefore \theta$ is in first or second quadrant.

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

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

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

$\Rightarrow {\cot }^2\theta =\frac{16}{9}$

$\Rightarrow \frac{1}{{\tan }^2\theta }=\frac{16}{9}$

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

$\therefore \tan \theta =\pm \frac{3}{4}$ since $\tan \theta$ is negative in second quadrant and positive in first quadrant.