# How do you find the exact values of costheta and tantheta when sintheta=-1/2?

May 10, 2018

$\cos \theta = \pm \frac{\sqrt{3}}{2}$

$\tan \theta = \pm \frac{1}{\sqrt{3}}$

#### Explanation:

Get a new triangle, question writer! Seriously, there's nothing about this question that requires using one of the two cliche triangles of trig, yet this question does.

${\cos}^{2} \theta + {\sin}^{2} \theta = 1$

${\cos}^{2} \theta = 1 - {\sin}^{2} \theta$

$\cos \theta = \pm \setminus \sqrt{1 - {\sin}^{2} \theta}$

$\cos \theta = \pm \sqrt{1 - {\left(- \frac{1}{2}\right)}^{2}} = \pm \frac{\sqrt{3}}{2}$

$\tan \theta = \frac{\sin \theta}{\cos \theta} = \frac{- \frac{1}{2}}{\pm \frac{\sqrt{3}}{2}} = \pm \frac{1}{\sqrt{3}}$

Knowing only the sine, we cannot determine the sign of the cosine or of the tangent.