cos-pi-5-with-solution-pls-

Question Number 156126 by VIDDD last updated on 08/Oct/21

\cos \frac{π}{5} = \dots ? with solution pls

Commented by cortano last updated on 08/Oct/21

\frac{π}{5} = 36 ° \Rightarrow \cos 72 ° = 2 \cos 36 ° - 1

\Rightarrow \cos 36 ° = \sqrt{\frac{1 + \cos 72 °}{2}} = \sqrt{\frac{1 + \sin 18 °}{2}}

Answered by puissant last updated on 08/Oct/21

a = c o s \frac{π}{5}; b = c o s \frac{2 π}{5} a n d - a = c o s \frac{4 π}{5}

W e h a v e c o s \frac{2 π}{5} = 2 {c o s}^{2} \frac{π}{5} - 1

\Rightarrow {\begin{cases} b = 2 a^{2} - 1 \\ - a = 2 b^{2} - 1 \end{cases} \Rightarrow a + b = 2 (a^{2} - b^{2})

\Rightarrow a + b = 2 (a + b) (a - b)

\Rightarrow 1 = a - b \Rightarrow b = a - \frac{1}{2} . .

\Rightarrow a - \frac{1}{2} = 2 a^{2} - 1

\Rightarrow 4 a^{2} - 2 a - 1 = 0

Δ = 4 - 4 (- 1) (4) = 20 \to \sqrt{Δ} = 2 \sqrt{5} . .

a_{1} = \frac{- 2 - 2 \sqrt{5}}{8} = \frac{- 1 - \sqrt{5}}{4} < 0 (i m p o s s i b l e)

a_{2} = \frac{2 + 2 \sqrt{5}}{8} = \frac{1 + \sqrt{5}}{4} = c o s \frac{π}{5}

c o s \frac{2 π}{5} = c o s \frac{π}{5} - \frac{1}{2} = \frac{\sqrt{5} - 1}{4}

∴∵ c o s (\frac{π}{5}) = \frac{1 + \sqrt{5}}{4} a n d c o s (\frac{2 π}{5}) = \frac{\sqrt{5} - 1}{4} . .

\dots \dots \dots . L e p u i s s a n t \dots \dots \dots . .

Commented by VIDDD last updated on 09/Oct/21

t h a n k s

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