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Question Number 96907 by bemath last updated on 05/Jun/20

$$\underset{\mathrm{z}=0}{\sum ^{10}}\cos {}^3\left(\frac{\pi \mathrm{z}}{3}\right)=?$$

Commented by john santu last updated on 05/Jun/20

$$\cos 3\mathrm{x}=4\cos {}^3\mathrm{x}-3\cos \mathrm{x}$$$$\cos {}^3\mathrm{x}=\frac{1}{4}[\cos 3\mathrm{x}+3\cos \mathrm{x}]$$$$\underset{\mathrm{z}=0}{\sum ^{10}}\frac{1}{4}[\cos 3\left(\frac{\pi \mathrm{z}}{3}\right)+3\cos \left(\frac{\pi \mathrm{z}}{3}\right)]$$$$=\frac{1}{4}\underset{\mathrm{z}=0}{\sum ^{10}}[\cos \pi \mathrm{z}+3\cos \left(\frac{\pi \mathrm{z}}{3}\right)]$$$$=\frac{1}{4}[\underset{\mathrm{z}=0}{\sum ^{10}}\cos \pi \mathrm{z}+3\underset{\mathrm{z}=0}{\sum ^{10}}\cos \left(\frac{\pi \mathrm{z}}{3}\right)]$$$$=\frac{1}{4}[1+3\times \frac{\cos \left(\frac{5\pi }{3}\right).\sin \left(\frac{11\pi }{6}\right)}{\sin \left(\frac{\pi }{6}\right)}]$$$$=\frac{1}{4}[1+(-\frac{3}{2})]=-\frac{1}{8}$$$$$$

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