Chapter 6, Problem 23CLT

To determine

The product $5\sin \frac{3\pi }{4}\cos \frac{7\pi }{4}$ as a sum or difference.

Answer to Problem 23CLT

The value of $5\sin \frac{3\pi }{4}\cos \frac{7\pi }{4}=\frac{5}{2}\left(\sin \frac{5\pi }{2}-\sin \pi \right)$

Explanation of Solution

Given info:

  $5\sin \frac{3\pi }{4}\cos \frac{7\pi }{4}$

Formula used:

  $2\sin A\cos B=\sin (A+B)+\sin (A-B)$

Calculation:

We need to find product $5\sin \frac{3\pi }{4}\cos \frac{7\pi }{4}$ as a sum or difference.

We have

  $\begin{array}{l}2\sin A\cos B=\sin (A+B)+\sin (A-B)\\ 5\sin \frac{3\pi }{4}\cos \frac{7\pi }{4}=\frac{5}{2}\times 2\sin \frac{3\pi }{4}\cos \frac{7\pi }{4}\\ A=\frac{3\pi }{4}\\ B=\frac{7\pi }{4}\\ 5\sin \frac{3\pi }{4}\cos \frac{7\pi }{4}=\frac{5}{2}\left(\sin \left(\frac{3\pi }{4}+\frac{7\pi }{4}\right)+\sin \left(\frac{3\pi }{4}-\frac{7\pi }{4}\right)\right)\\ 5\sin \frac{3\pi }{4}\cos \frac{7\pi }{4}=\frac{5}{2}\left(\sin \left(\frac{10\pi }{4}\right)+\sin \left(\frac{-4\pi }{4}\right)\right)\\ 5\sin \frac{3\pi }{4}\cos \frac{7\pi }{4}=\frac{5}{2}\left(\sin \left(\frac{5\pi }{2}\right)+\sin \left(-\pi \right)\right)\\ 5\sin \frac{3\pi }{4}\cos \frac{7\pi }{4}=\frac{5}{2}\left(\sin \frac{5\pi }{2}-\sin \pi \right)\end{array}$

The value of $5\sin \frac{3\pi }{4}\cos \frac{7\pi }{4}=\frac{5}{2}\left(\sin \frac{5\pi }{2}-\sin \pi \right)$

Conclusion:

Thus,the value of $5\sin \frac{3\pi }{4}\cos \frac{7\pi }{4}=\frac{5}{2}\left(\sin \frac{5\pi }{2}-\sin \pi \right)$