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Chapter 5: Problem 60

Use the sum-to-product formulas to find the exact value of the expression. $$\sin \frac{5 \pi}{4}-\sin \frac{3 \pi}{4}$$

Short Answer

Expert verified
The exact value of the expression \( \sin \frac{5 \pi}{4} - \sin \frac{3 \pi}{4} \) is \( -\sqrt{2} \).

Step by step solution

01

Use the sum-to-product formula

In our problem, we are asked to find the value of \( \sin \frac{5 \pi}{4} - \sin \frac{3 \pi}{4} \). This is a difference of sines in the form \( \sin A - \sin B \), so we can use the sum-to-product identity which states: \( \sin A - \sin B = 2 \cos \frac{A + B}{2} \sin \frac{A - B}{2} \)
02

Substitute given values

Substitute \( A = \frac{5 \pi}{4} \) and \( B = \frac{3 \pi}{4} \) into the sum-to-product formula. As a result, the formula becomes \( 2 \cos \frac{\frac{5 \pi}{4} + \frac{3 \pi}{4}}{2} \sin \frac{\frac{5 \pi}{4} - \frac{3 \pi}{4}}{2} \)
03

Calculate \( A + B \) and \( A - B \)

Next, calculate \( A + B = \frac{5 \pi}{4} + \frac{3 \pi}{4} = 2 \pi \), \( A - B = \frac{5 \pi}{4} - \frac{3 \pi}{4} = \frac{\pi}{2} \). Thus our expression becomes \( 2 \cos \frac{2 \pi}{2} \sin \frac{\pi}{4} = 2 \cos \pi \sin \frac{\pi}{4} \)
04

Evaluate \(\cos\) and \(\sin\) values

Evaluate \( \cos \pi \) and \( \sin \frac{\pi}{4} \). Both are common angles on the unit circle. \( \cos \pi = -1 \) and \( \sin \frac{\pi}{4} = \frac{\sqrt{2}}{2} \). So the expression becomes \( 2(-1) \frac{\sqrt{2}}{2} \)
05

Simplify the expression to find the exact value

Simplify the expression to find the exact value, \( -2 \cdot \frac{\sqrt{2}}{2} = -\sqrt{2} \)

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