Find-the-exact-value-of-cos-pi-5-

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

Find the exact value of \cos \frac{π}{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

s i n \frac{π}{10} = \frac{\sqrt{5} - 1}{4}

c o s (\frac{π}{5}) = 1 - 2 {s i n}^{2} \frac{π}{10} = 1 - 2 (\frac{6 - 2 \sqrt{5}}{16}) = \frac{\sqrt{5} + 1}{4} = \frac{ϕ}{2}

Answered by MJS_new last updated on 21/Apr/21

\cos \frac{π}{5} + i \sin \frac{π}{5} = e^{\frac{i π}{5}} is a solution 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} i

x_{4, 5} = \frac{1 + \sqrt{5}}{4} \pm \frac{\sqrt{10 - 2 \sqrt{5}}}{4} i

\cos \frac{π}{5} > 0 \Rightarrow \cos \frac{π}{5} = \frac{1 + \sqrt{5}}{4}

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