Question

If $\csc \theta =\frac{5}{3}$ then find $\cos \theta$

Answer 1

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

$\therefore \theta$ is positive in the first and second quadrant.

We know that, $\csc \theta =\frac{1}{\sin \theta }$

$\Rightarrow \frac{1}{\sin \theta }=\frac{5}{3}$

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

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

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

We know that, ${\sin }^2\theta +{\cos }^2\theta =1$

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

$\Rightarrow {\cos }^2\theta =\frac{16}{25}$

Since $\cos \theta$ is negative in second quadrant and positive in first quadrant.

$\therefore \cos \theta =\pm \frac{4}{5}$