# Given sintheta=-8/17 and 180<theta<270, how do you find cos(theta/2)?

Aug 31, 2016

$\cos \left(\frac{\theta}{2}\right) = - \frac{1}{\sqrt{17}}$.

#### Explanation:

$180 < , \theta < , 270$.

$\Rightarrow \cos \theta < 0 , \mathmr{and} , \therefore , \cos \theta = - \sqrt{1 - {\sin}^{2} \theta} ,$

where $\sin \theta = - \frac{8}{17}$

$\therefore \cos \theta = - \sqrt{1 - {\left(- \frac{8}{17}\right)}^{2}} = - \frac{15}{17}$.

$\text{Now} , \cos \theta = 2 {\cos}^{2} \left(\frac{\theta}{2}\right) - 1 \Rightarrow - \frac{15}{17} = 2 {\cos}^{2} \left(\frac{\theta}{2}\right) - 1$

$\Rightarrow 2 {\cos}^{2} \left(\frac{\theta}{2}\right) = 1 - \frac{15}{17} = \frac{2}{17}$

$\Rightarrow {\cos}^{2} \left(\frac{\theta}{2}\right) = \frac{1}{17}$

$\Rightarrow \cos \left(\frac{\theta}{2}\right) = \pm \frac{1}{\sqrt{17}}$

$\text{But} , 180 < , \theta < , 270 \Rightarrow \frac{180}{2} < , \frac{\theta}{2} < , \frac{270}{2} ,$

i.e., $\cos \left(\frac{\theta}{2}\right) < 0$.

Therefore, $\cos \left(\frac{\theta}{2}\right) = - \frac{1}{\sqrt{17}}$.

Enjoy Maths.!