Chapter 7.4, Problem 18E

To determine

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

Answer to Problem 18E

  $-\frac{\sqrt{2-\sqrt{3}}}{2}$

Explanation of Solution

Given information:

  $\cos \frac{7\pi }{12}$

Calculation:

Evaluate $\alpha$ as shown below.

  $\begin{array}{c}\frac{\alpha }{2}=\frac{7\pi }{12}\\ \alpha =2\left(\raisebox{1ex}{$7\pi $}\!\left/ \!\raisebox{-1ex}{$12$}\right.\right)\\ =\frac{7\pi }{6}\end{array}$

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

  $\begin{array}{c}cos\frac{\alpha }{2}=\pm \sqrt{\frac{1+\cos \alpha }{2}}\\ \cos \frac{7\pi }{12}=-\sqrt{\frac{1+\cos \frac{7\pi }{6}}{2}}\left[\begin{array}{l}\text{Put }\alpha =\frac{7\pi }{6}\\ \raisebox{1ex}{$7\pi $}\!\left/ \!\raisebox{-1ex}{$12$}\right.\text{ lies in Quadrant II, so cos}\frac{\alpha }{2}\text{ is negative}\end{array}\right]\\ =-\sqrt{\frac{1+\left(\raisebox{1ex}{$-\sqrt{3}$}\!\left/ \!\raisebox{-1ex}{$2$}\right.\right)}{2}}\\ =-\sqrt{\frac{2-\sqrt{3}}{4}}\\ =-\frac{\sqrt{2-\sqrt{3}}}{2}\end{array}$

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