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Chapter 5: Problem 82

Use the product-to-sum formulas to write the product as a sum or difference. $$4 \cos \frac{\pi}{3} \sin \frac{5 \pi}{6}$$

Short Answer

Expert verified
The product can be written as the difference -1.

Step by step solution

01

Identify the relevant parts

In the expression \(4 \cos \frac{\pi}{3} \sin \frac{5 \pi}{6}\), \(\frac{\pi}{3}\) corresponds to \(a\) and \(\frac{5 \pi}{6}\) corresponds to \(b\) if compared to the formula \(\cos a \sin b = \frac{1}{2}[\sin(a+b)-\sin(a-b)]\).
02

Apply the product-to-sum formula

Substitute the values of \(a\) and \(b\) into the formula \(\cos a \sin b = \frac{1}{2}[\sin(a+b)-\sin(a-b)]\) and multiply the expression by the constant, 4. You should now have \(4 \cos \frac{\pi}{3} \sin \frac{5 \pi}{6}=2[\sin(\frac{\pi}{3} + \frac{5 \pi}{6})-\sin(\frac{\pi}{3} - \frac{5 \pi}{6})]\)
03

Simplify the expression

First, simplify within the parentheses to get \(2[\sin(\frac{3 \pi}{2})-\sin(-\frac{\pi}{6})]\). Then, use the identities \(\sin(\frac{3 \pi}{2}) = -1\) and \(\sin(-\frac{\pi}{6}) = -\frac{1}{2}\) to further simplify the expression. This gives \(2[-1-(-\frac{1}{2})]\), which simplifies to \(-1\).

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Most popular questions from this chapter

Explain in your own words how knowledge of algebra is important when solving trigonometric equations.

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