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Chapter 5: Problem 50

Use the product-to-sum formulas to rewrite the product as a sum or difference. $$7 \cos (-5 \beta) \sin 3 \beta$$

Short Answer

Expert verified
The product \(7 \cos(-5\beta) \sin(3\beta)\) can be rewritten as a difference \(3.5[\sin(-2\beta) - \sin(-8\beta)]\) using the product-to-sum formulas.

Step by step solution

01

Identify the format of the product

The given product is \(7 \cos(-5\beta) \sin(3\beta)\). It is of the format \(\cos(A) \sin(B)\), where A = -5\beta and B = 3\beta.
02

Apply the product-to-sum formula

The product-to-sum formula that matches the format of the given product is: \(\cos(A) \sin(B) = 0.5[\sin(A+B) - \sin(A-B)]\) This formula can be used to convert the product to a sum/difference. Substituting A = -5\beta and B = 3\beta into the formula, we get: \(7\cos(-5\beta) \sin(3\beta) = 0.5 \times 7[\sin(-5\beta+3\beta) - \sin(-5\beta-3\beta)]\)
03

Simplify by adding/subtracting the angles

Simplify the expressions within the sine functions: \(0.5 \times 7[\sin(-2\beta) - \sin(-8\beta)]\)
04

Multiply the overall factor

Finally, we multiply the prefactor 0.5 * 7 throughout to get the final answer: \(3.5[\sin(-2\beta) - \sin(-8\beta)]\)

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