Chapter 13.7, Problem 21WE

To determine

To Find: The simplified form of function.

Answer to Problem 21WE

The simplified form of the given function is $\cos \theta$ .

Explanation of Solution

Given:

The given identity is $\sin \left(\frac{\pi }{6}+\theta \right)+\sin \left(\frac{\pi }{6}-\theta \right)$ .

Calculation:

Consider the function is,

  $\sin \left(\frac{\pi }{6}+\theta \right)+\sin \left(\frac{\pi }{6}-\theta \right)$

Consider the trigonometric identities.

  $\begin{array}{l}\sin \left(a+b\right)=\sin a\cos b+\cos a\sin b\\ \sin \left(a-b\right)=\sin a\cos b-\cos a\sin b\end{array}$

From, the given identity,

  $\begin{array}{c}\sin \left(\frac{\pi }{6}+\theta \right)+\sin \left(\frac{\pi }{6}-\theta \right)=\sin \frac{\pi }{6}\cos \theta +\sin \theta \cos \frac{\pi }{6}+\sin \frac{\pi }{6}\cos \theta -\sin \theta \cos \frac{\pi }{6}\\ =2\sin \frac{\pi }{6}\cos \theta \\ =2\left(\frac{1}{2}\right)\cos \theta \\ =\cos \theta \end{array}$