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Chapter 5: Problem 49

Use the product-to-sum formulas to rewrite the product as a sum or difference. $$\sin 5 \theta \sin 3 \theta$$

Short Answer

Expert verified
\[\sin 5\theta \sin 3\theta = 0.5[\cos(2\theta) - \cos(8\theta)]\]

Step by step solution

01

Recognize the formula

The given expression is in the form of \(\sin(A) \sin(B)\). The product to sum formula that corresponds to this form is: \[2\sin A \sin B =\cos(A-B)- \cos(A+B)\] where A is 5胃 and B is 3胃 in our case.
02

Apply the formula

Replace the given values in the formula which leads to: \[2 \sin(5\theta)\sin(3\theta) = \cos(5\theta - 3\theta) - \cos(5\theta + 3\theta)\]
03

Simplify the expression

On simplifying the above expression, we find: \[2 \sin(5\theta)\sin(3\theta) = \cos(2\theta) - \cos(8\theta)\]. Therefore, \[\sin(5\theta) \sin(3\theta) = 0.5[\cos(2\theta) - \cos(8\theta)]\]

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