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Question Number 146824 by alcohol last updated on 16/Jul/21

Answered by Olaf_Thorendsen last updated on 16/Jul/21

$$1-$$$$w=e^{\frac{2i\pi }{n}}$$$$\mathrm{Let}{z}_n=1+w+w^2+...w^{n-1}$$$$z_n=1\times \frac{1-{\left(e^{\frac{2i\pi }{n}}\right)}^n}{1-e^{\frac{2i\pi }{n}}}=\frac{1-e^{2i\pi }}{1-e^{\frac{2i\pi }{n}}}=0$$$$2-$$$$z_5=1+e^{i\frac{2\pi }{5}}+e^{\frac{4i\pi }{5}}+e^{\frac{6i\pi }{5}}+e^{\frac{8i\pi }{5}}=0$$$$\Rightarrow \mathrm{Re}\left(z_5\right)=0=1+\cos \left(\frac{2\pi }{5}\right)+\cos \left(\frac{4\pi }{5}\right)+\cos \left(\frac{6\pi }{5}\right)+\cos \left(\frac{8\pi }{5}\right)$$$$3-$$$$\mathrm{A}=\cos \left(\frac{2\pi }{5}\right)+\cos \left(\frac{8\pi }{5}\right)$$$$\mathrm{A}=2\cos \left(\frac{\frac{2\pi }{5}+\frac{8\pi }{5}}{2}\right)\cos \left(\frac{\frac{2\pi }{5}-\frac{8\pi }{5}}{2}\right)$$$$\mathrm{A}=2\cos \left(\pi \right)\cos \left(\frac{3\pi }{5}\right)$$$$\mathrm{A}=2\cos \left(\frac{2\pi }{5}\right)$$$$\mathrm{A}=2({2\cos }^2\left(\frac{\pi }{5}\right)-1)={4\cos }^2\left(\frac{\pi }{5}\right)-2$$$$\mathrm{B}=\cos \left(\frac{4\pi }{5}\right)+\cos \left(\frac{6\pi }{5}\right)$$$$\mathrm{B}=-\cos \left(\frac{\pi }{5}\right)-\cos \left(\frac{\pi }{5}\right)$$$$\mathrm{B}=-2\cos \left(\frac{\pi }{5}\right)$$$$4-$$$$1+\mathrm{A}+\mathrm{B}=0=1+{4\cos }^2\left(\frac{\pi }{5}\right)-2-2\cos \left(\frac{\pi }{5}\right)$$$$\Rightarrow {4\cos }^2\left(\frac{\pi }{5}\right)-2\cos \left(\frac{\pi }{5}\right)-1=0$$$$\cos \left(\frac{\pi }{5}\right)=\frac{2\pm \sqrt{4-4\times 4\times (-1)}}{8}$$$$\cos \left(\frac{\pi }{5}\right)=\frac{1\pm \sqrt{5}}{4}$$$$\mathrm{but}\cos \left(\frac{\pi }{5}\right)>0\Rightarrow \cos \left(\frac{\pi }{5}\right)=\frac{1+\sqrt{5}}{4}$$$$$$$$5-$$$${\cos }^{3}x={\left(\frac{e^{ix}+e^{-ix}}{2}\right)}^3$$$${\cos }^{3}x=\frac{1}{8}(e^{3ix}+3e^{ix}+3e^{-ix}+e^{-3ix})$$$${\cos }^{3}x=\frac{1}{4}\cos \left(3x\right)+\frac{3}{4}\cos x$$$$\Rightarrow \cos \left(3x\right)={4\cos }^{3}x-3\cos x$$$$\cos \left(\frac{3\pi }{5}\right)={4\cos }^3\left(\frac{\pi }{5}\right)-3\cos \left(\frac{\pi }{5}\right)$$$$\cos \left(\frac{3\pi }{5}\right)=4{\left(\frac{1+\sqrt{5}}{4}\right)}^3-3\left(\frac{1+\sqrt{5}}{4}\right)$$$$\cos \left(\frac{3\pi }{5}\right)=\frac{1}{16}(1+3\sqrt{5}+15+5\sqrt{5})-3\left(\frac{1+\sqrt{5}}{4}\right)$$$$\cos \left(\frac{3\pi }{5}\right)=\frac{1-\sqrt{5}}{4}$$

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