Chapter 7, Problem 28SGA

To determine

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

Answer to Problem 28SGA

  $\frac{\sqrt{2-\sqrt{2}}}{2}$

Explanation of Solution

Given information:

  $\sin 22.5°$

Calculation:

Evaluate $\alpha$ as shown below.

  $\begin{array}{c}\frac{\alpha }{2}=22.5°\\ \alpha =2\left(22.5°\right)\\ =45°\end{array}$

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

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

  $\begin{array}{c}\sin 22.5°=\sqrt{\frac{1-\cos 45°}{2}}\left[\begin{array}{l}\text{Put }\alpha =45°\\ 22.5°\text{ lies in Quadrant I, so }\sin \frac{\alpha }{2}\text{ is positive}\end{array}\right]\end{array}$

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

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

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