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Question Number 139024 by bramlexs22 last updated on 21/Apr/21

$$\mathrm{Find}\mathrm{the}\mathrm{exact}\mathrm{value}\mathrm{of}\cos \frac{\pi }{5}$$$$$$

Commented by EDWIN88 last updated on 21/Apr/21

$$\frac{\sqrt{5}+1}{4}$$

Answered by Dwaipayan Shikari last updated on 21/Apr/21

$$sin\frac{\pi }{10}=\frac{\sqrt{5}-1}{4}$$$$cos\left(\frac{\pi }{5}\right)=1-2{sin}^2\frac{\pi }{10}=1-2\left(\frac{6-2\sqrt{5}}{16}\right)=\frac{\sqrt{5}+1}{4}=\frac{\varphi }{2}$$

Answered by MJS_new last updated on 21/Apr/21

$$\cos \frac{\pi }{5}+\mathrm{i}\sin \frac{\pi }{5}={\mathrm{e}}^{\frac{\mathrm{i}\pi }{5}}\mathrm{is}\mathrm{a}\mathrm{solution}\mathrm{of}$$$$x^5+1=0$$$$(x+1)(x^4-x^3+x^2-x+1)=0$$$$x_1=-1$$$$x^4-x^3+x^2-x+1=0$$$$(x^2-\frac{1-\sqrt{5}}{2}x+1)(x^2-\frac{1+\sqrt{5}}{2}x+1)=0$$$$x_{2,3}=\frac{1-\sqrt{5}}{4}\pm \frac{\sqrt{10+2\sqrt{5}}}{4}\mathrm{i}$$$$x_{4,5}=\frac{1+\sqrt{5}}{4}\pm \frac{\sqrt{10-2\sqrt{5}}}{4}\mathrm{i}$$$$\cos \frac{\pi }{5}>0\Rightarrow \cos \frac{\pi }{5}=\frac{1+\sqrt{5}}{4}$$

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