Chapter 7, Problem 27SGA

To determine

To find: the exact value of the function by using the half-angle identity.

Answer to Problem 27SGA

  $\frac{\sqrt{2-\sqrt{2}}}{2}$

Explanation of Solution

Given information:

  $\sin \frac{7\pi }{8}$

Calculation:

Evaluate $\alpha$ as shown below.

  $\begin{array}{c}\frac{\alpha }{2}=\frac{7\pi }{8}\\ \alpha =2\left(\frac{7\pi }{8}\right)\\ =\frac{7\pi }{4}\end{array}$

So the value of $\alpha$ is $\frac{7\pi }{4}$ .

Solve the function by using half-angle identity for sine as shown below.

  $\begin{array}{c}\sin \frac{\alpha }{2}=\pm \sqrt{\frac{1-\cos \alpha }{2}}\end{array}$

  $\begin{array}{c}\sin \frac{7\pi }{8}=\sqrt{\frac{1-\cos \frac{7\pi }{4}}{2}}\left[\begin{array}{l}\text{Put }\alpha =\frac{7\pi }{4}\\ \frac{7\pi }{8}\text{ lies in Quadrant II, so }\sin \frac{\alpha }{2}\text{ is positive}\end{array}\right]\end{array}$

  $\begin{array}{c}=\sqrt{\frac{1-\frac{\sqrt{2}}{2}}{2}}\end{array}$

  $\begin{array}{c}=\frac{\sqrt{2-\sqrt{2}}}{2}\end{array}$

Thus the value of given function is $\frac{\sqrt{2-\sqrt{2}}}{2}$ .