Chapter 13.7, Problem 12WE

To determine

To find: The simplified form of the trigonometric function of the single angle then given the exact value if possible.

Answer to Problem 12WE

The simplified form is $\cos \left(\frac{\pi }{4}\right)$ and the exact value is $\frac{1}{\sqrt{2}}$ .

Explanation of Solution

Given:

The given function is $\sin \frac{\pi }{3}\cos \frac{\pi }{12}-\cos \frac{\pi }{3}\sin \frac{\pi }{12}$ .

Calculation:

Consider the given function is,

  $\sin \frac{\pi }{3}\cos \frac{\pi }{12}-\cos \frac{\pi }{3}\sin \frac{\pi }{12}$

Consider the above trigonometric identity.

  $\sin \left(a-b\right)=\sin a\cos b+\sin b\cos a$

Then, the given function is written as,

  $\begin{array}{c}\sin \left(\frac{\pi }{3}-\frac{\pi }{12}\right)=\sin \frac{3\pi }{4}\\ =\sin \left(\frac{\pi }{2}+\frac{\pi }{4}\right)\\ =\cos \left(\frac{\pi }{4}\right)\\ =\frac{1}{\sqrt{2}}\end{array}$