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Chapter 5: Problem 101

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

Short Answer

Expert verified
-\sqrt{2}

Step by step solution

01

Identify Relevant Sum-to-Product Formula

Start by identifying the formula that applies to the given expression: \(-2 \sin \frac{a+b}{2} \sin \frac{a-b}{2}\) for the form \(\cos a - \cos b\).
02

Apply the Formula

Replace \(a=\frac{3 \pi}{4}\) and \(b=\frac{\pi}{4}\) into the formula. That gives: \(-2 \sin \frac{\frac{3 \pi}{4}+\frac{\pi}{4}}{2} \sin \frac{\frac{3 \pi}{4}-\frac{\pi}{4}}{2}\) which becomes: \(-2 \sin \frac{\pi}{2} \sin \frac{ \pi}{4}\).
03

Calculate the Values of the Sine Function

Next, calculate the values of each sine function: -2 times 1 (since \(\sin \frac{\pi}{2} = 1\)) times \(\frac{\sqrt{2}}{2}\) (since \(\sin \frac{\pi}{4} = \frac{\sqrt{2}}{2}\)).
04

Simplify the Result

Multiplying these values together gives: \(-2 \times 1 \times \frac{\sqrt{2}}{2} = -\sqrt{2}\).

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