Chapter 13.7, Problem 13OE

To determine

To simplify: The each expression to a trigonometric function of a single angle and hence give the exact value of the function.

Answer to Problem 13OE

The simplified form of expression is $\sin \pi$ and exact value is $0$ .

Explanation of Solution

Given:

The given expression is as:

  $\sin \frac{3\pi }{4}\cos \frac{\pi }{4}+\cos \frac{3\pi }{4}\sin \frac{\pi }{4}$ .

The trigonometric additional formula for $\sin$ function is given as:

  $\sin \left(x+y\right)=\sin x\cos y+\cos x\sin y$ .

Using the above formula to solve the above expression is given as:

  $\begin{array}{c}\sin \frac{3\pi }{4}\cos \frac{\pi }{4}+\cos \frac{3\pi }{4}\sin \frac{\pi }{4}=\sin \left(\frac{3\pi }{4}+\frac{\pi }{4}\right)\\ \sin \frac{3\pi }{4}\cos \frac{\pi }{4}+\cos \frac{3\pi }{4}\sin \frac{\pi }{4}=\sin \pi \\ \sin \frac{3\pi }{4}\cos \frac{\pi }{4}+\cos \frac{3\pi }{4}\sin \frac{\pi }{4}=0\left[\because \sin n\pi =0\forall n\in N\right]\end{array}$ .

Therefore the simplified form of expression is $\sin \pi$ and exact value is $0$ .