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Chapter 5: Problem 53

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

Short Answer

Expert verified
\(\sin 5 \theta - \sin 3 \theta = 2 \cos(4\theta) \sin(\theta)\)

Step by step solution

01

Identifying the Appropriate Formula

In this case, we should use the sum-to-product formula for the difference of sines, which is \[\sin a - \sin b = 2 \cos \left( \frac{a+b}{2} \right) \sin \left( \frac{a-b}{2} \right) \] This formula is used to express the difference of two sine functions as a product of cosine and sine functions.
02

Applying the Formula

Let's substitute \(\sin 5 \theta\) for \( \sin a \) and \(\sin 3 \theta\) for \(\sin b\) in the formula. By applying the formula, we get \[\sin 5 \theta - \sin 3 \theta = 2 \cos \left( \frac{5 \theta + 3 \theta}{2} \right) \sin \left( \frac{5 \theta - 3 \theta}{2} \right)\]
03

Simplifying the Result

Next, simplify the terms inside the parentheses. We get \[2 \cos(4\theta) \sin(\theta)\] So, \(\sin 5\theta - \sin 3\theta\) is equivalent to \(2 \cos(4\theta) \sin(\theta)\)

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