# What is the value of costheta if sintheta=1/2 and theta is in Q2?

Dec 3, 2016

#### Answer:

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

#### Explanation:

As domain of $\theta$ is ${90}^{o} < \theta < {180}^{o}$,

while $\sin \theta$ is positive, $\cos \theta$ is negative.

As $\sin \theta = \frac{1}{2}$, and ${\sin}^{2} \theta + {\cos}^{2} \theta = 1$

we have ${\cos}^{2} \theta = 1 - {\sin}^{2} \theta$ and as $\cos \theta$ is negative

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

= $- \sqrt{1 - \frac{1}{4}} = - \sqrt{\frac{3}{4}} = - \frac{\sqrt{3}}{2}$