# How do you determine costheta given sintheta=-1/5,pi<theta<(3pi)/2?

Sep 11, 2017

#### Answer:

$\cos \theta = - \frac{2 \sqrt{6}}{5}$

#### Explanation:

$\theta$ is in 3rd quadrant where both $\sin \theta \mathmr{and} \cos \theta$

is negative. $\sin \theta = \frac{1}{-} 5 \therefore {\sin}^{2} \theta = \frac{1}{25}$

$\cos \theta = \pm \sqrt{1 - {\sin}^{2} \theta} = \pm \sqrt{1 - \frac{1}{25}} = \pm \frac{\sqrt{24}}{5}$

Since $\cos \theta$ is negative in 3rd quadrant ,

$\cos \theta = - \frac{\sqrt{24}}{5} \mathmr{and} \cos \theta = - \frac{2 \sqrt{6}}{5}$ [Ans]