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Chapter 5: Problem 7

Find the exact value of each expression. (a) \(\cos \left(\frac{\pi}{4}+\frac{\pi}{3}\right)\) (b) \(\cos \frac{\pi}{4}+\cos \frac{\pi}{3}\)

Short Answer

Expert verified
(a) \(\frac{\sqrt{2}-\sqrt{6}}{4}\) (b) \(\frac{\sqrt{2}+1}{2}\)

Step by step solution

01

Solve \(\cos \left(\frac{\pi}{4}+\frac{\pi}{3}\right)\)

The identity for \(\cos(a + b)\) is \(\cos a \cos b - \sin a \sin b\). Applying this to our case we get:\(\cos \left(\frac{\pi}{4}+\frac{\pi}{3}\right) = \cos \frac{\pi}{4} \cos \frac{\pi}{3} - \sin \frac{\pi}{4} \sin \frac{\pi}{3}\). The values for \(\cos \frac{\pi}{4}, \cos \frac{\pi}{3}, \sin \frac{\pi}{4}\), and \(\sin \frac{\pi}{3}\) are \(\frac{\sqrt{2}}{2}, \frac{1}{2}, \frac{\sqrt{2}}{2}\), and \(\frac{\sqrt{3}}{2}\) respectively. Substituting these values, we get: \(\frac{\sqrt{2}}{2} * \frac{1}{2} - \frac{\sqrt{2}}{2} * \frac{\sqrt{3}}{2} = \frac{\sqrt{2}}{4} - \frac{\sqrt{6}}{4} = \frac{\sqrt{2}-\sqrt{6}}{4}\)
02

Solve \(\cos \frac{\pi}{4}+\cos \frac{\pi}{3}\)

This is the direct sum of the cosine of two angles. So, \(\cos \frac{\pi}{4}+\cos \frac{\pi}{3} = \frac{\sqrt{2}}{2} + \frac{1}{2} = \frac{\sqrt{2}+1}{2}\)

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