Chapter 5.4, Problem 50E

Solution

To find: The value of $\sin 22.5°$ .

The correct option is (E).

Given information:

The expression is $\sin 22.5°$ .

Formula used:

Use the half-angle identity for sine in terms of cosine.

  $\sin \frac{u}{2}=\sqrt{\frac{1-\cos u}{2}}$

Calculation:

Let $22.5°=\frac{45°}{2}$

  $\sin 22.5°=\sin \frac{45°}{2}$

Here, $u=45°$ .

Simplify the given expression by using half-angle identity for sine

  $\begin{array}{c}\sin \frac{45°}{2}=\sqrt{\frac{1-\cos 45°}{2}}\\ \sin 22.5°=\sqrt{\frac{1-\frac{\sqrt{2}}{2}}{2}}\\ =\sqrt{\frac{2-\sqrt{2}}{4}}\\ =\frac{1}{2}\sqrt{2-\sqrt{2}}\end{array}$

Therefore, the correct option is (E).