Chapter 7.4, Problem 15E

To determine

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

Answer to Problem 15E

  $\frac{\sqrt{2+\sqrt{3}}}{2}$

Explanation of Solution

Given information:

  $\sin 75°$

Calculation:

Evaluate $\alpha$ as shown below.

  $\begin{array}{c}\frac{\alpha }{2}=75°\\ \alpha =2\left(75°\right)\\ =150°\end{array}$

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

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

  $\begin{array}{c}\sin 75°=\sqrt{\frac{1-\cos 150°}{2}}\left[\begin{array}{l}\text{Put }\alpha =150°\\ 75°\text{ lies in Quadrant III, so }\sin \frac{\alpha }{2}\text{ is positive}\end{array}\right]\end{array}$

  $\begin{array}{c}=\sqrt{\frac{1-\left(-\frac{\sqrt{3}}{2}\right)}{2}}\end{array}$

  $\begin{array}{c}=\frac{\sqrt{2+\sqrt{3}}}{2}\end{array}$

Thus the value of given function is $\frac{\sqrt{2+\sqrt{3}}}{2}$ .