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Chapter 5: Problem 59

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

Step by step solution

01

Identify the values of A and B

\(\frac{3\pi}{4}\) and \(\frac{\pi}{4}\) will be identified as A and B respectively. Substituting these values into A and B in the formula.
02

Apply the Difference Identity

Substitute \(A = \frac{3\pi}{4}\) and \(B = \frac{\pi}{4}\) into the Difference Identity formula: \(-2\sin{(A + B)/2}\sin{(B - A)/2}\). This gives \(-2\sin{(\frac{3\pi}{4} + \frac{\pi}{4})/2}\sin{(\frac{3\pi}{4} - \frac{\pi}{4})/2}\).
03

Calculate the Sines

Simplify \(-2\sin{(\frac{3\pi}{4} + \frac{\pi}{4})/2}\sin{(\frac{3\pi}{4} - \frac{\pi}{4})/2}\) to \(-2\sin{(\frac{4\pi}{4}/2)}\sin{(\frac{2\pi}{4}/2)}\), which then simplifies to \(-2\sin{\frac{\pi}{2}}\sin{\frac{\pi}{4}}\). Then \(\sin{\frac{\pi}{2}} = 1\) and \(\sin{\frac{\pi}{4}} = \frac{\sqrt{2}}{2}\) are used to compute the expression.
04

Final Simplification

Substitute the calculated values of the sines, to give \(-2*1*\frac{\sqrt{2}}{2} = -\sqrt{2}\)

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