Chapter 14.5, Problem 69SR

To determine

To find :Find exact value of $\text{sin 22}\frac{\text{1}}{\text{2}}°.$

Answer to Problem 69SR

  $0.3825$

Explanation of Solution

Given information :

Expression $\text{sin}22\frac{1}{2}$

Formula Used:

  $2{\sin }^2\theta =1-\cos 2\theta$

Calculation :

The given equation is

  $\begin{array}{l}\text{Let}\theta =22\frac{1}{2}°\\ 2{\sin }^2\left(22\frac{1}{2}°\right)=1-\cos \left(22\frac{1}{2}°\right)\\ 2{\sin }^{2}22\frac{1}{2}°=1-\cos 45°=1-\frac{1}{\sqrt{2}}\left[\because \cos 45°=\frac{1}{\sqrt{2}}\right]\\ {\sin }^{2}22\frac{1}{2}°=\frac{1}{2}\left(1-\frac{1}{\sqrt{2}}\right)=\frac{\sqrt{2}-1}{2\sqrt{2}}=\frac{\sqrt{2}-1}{2\sqrt{2}}\cdot \frac{\sqrt{2}}{\sqrt{2}}\\ {\sin }^{2}22\frac{1}{2}°=\frac{2-\sqrt{2}}{4}\\ \Rightarrow \sin 22\frac{1}{2}°=\pm \sqrt{\frac{2-\sqrt{2}}{4}}\\ \sin 22\frac{1}{2}°=\frac{\sqrt{2-\sqrt{2}}}{2}=\frac{0.765}{2}=0.3825\end{array}$