Question

express the given product as a sum containing only sines or cosines.

\\( \sin ( 6 \theta ) \sin ( 7 \theta ) \\)

\\( \sin ( 6 \theta ) \sin ( 7 \theta ) = \\)
(simplify your answer, including any radicals. use integers or fractions for any numbers in the expression.)

Explanation:

Step1: Use the product - to - sum formula

The formula for \(\sin A\sin B=\frac{1}{2}[\cos(A - B)-\cos(A + B)]\). Here \(A = 6\theta\) and \(B=7\theta\).
So, \(\sin(6\theta)\sin(7\theta)=\frac{1}{2}[\cos(6\theta-7\theta)-\cos(6\theta + 7\theta)]\).

Step2: Simplify the arguments of the cosine functions

Simplify \(6\theta-7\theta=-\theta\) and \(6\theta + 7\theta = 13\theta\).
Since \(\cos(-x)=\cos x\), we have \(\sin(6\theta)\sin(7\theta)=\frac{1}{2}[\cos\theta-\cos(13\theta)]\).

Answer:

\(\frac{1}{2}\cos\theta-\frac{1}{2}\cos(13\theta)\)