Chapter 13.8, Problem 31WE

To determine

To find: The value of $\cos \frac{\alpha }{2}$ .

Answer to Problem 31WE

The value of $\cos \frac{\alpha }{2}$ is $-\frac{5\sqrt{34}}{34}$ .

Explanation of Solution

Given information:

Given that $\cos \alpha =\frac{8}{17}$ .,and $180\circ < \alpha < {360}^{\circ }$

Calculation:

The formula for half angle cosine is

  $\cos \frac{\alpha }{2}=\pm \sqrt{\frac{1+\cos \alpha }{2}}$ (1)

Given, $\cos \alpha =\frac{8}{17}$

It also given that $180\circ < \alpha < {360}^{\circ }$ , is it means $90\circ < \alpha < {180}^{\circ }$ that $\frac{\alpha }{2}$ lies in second quadrant and the value is always negative of cosine in second quadrant.

Substitute the value of $\cos \alpha$ in equation $\left(1\right)$ .

  $\begin{array}{c}\cos \frac{\alpha }{2}=-\sqrt{\frac{1+\cos \alpha }{2}}\\ =-\sqrt{\frac{1+\frac{8}{17}}{2}}\\ =-\sqrt{\frac{25}{34}}\\ =-\frac{5}{\sqrt{34}}\end{array}$

Rationalize the denominator.

  $\begin{array}{c}-\frac{5}{\sqrt{34}}=-\frac{5}{\sqrt{34}}.\frac{\sqrt{34}}{\sqrt{34}}\\ =-\frac{5\sqrt{34}}{34}\end{array}$

Therefore, the value of

  $\cos \frac{\alpha }{2}$ is $-\frac{5\sqrt{34}}{34}$ .