[SOLVED] Complex Numbers- cos ( pi/5)

**kayla09** · February 10th 2009, 09:49 PM

I am trying to prove that cos(pi/5)= (1+sqrt(5))/4

I tried finding the 5th roots of unity and then just using the real part cis(pi/5) but then i dont know how to get it = (1+sqrt(5))/4.
I am stuck somewhere here.

**nzmathman** · February 11th 2009, 12:19 AM

See here: Trigonometry Angles--Pi/5 -- from Wolfram MathWorld

**kayla09** · February 11th 2009, 08:18 PM

i am still confused though. that website is for sin(pi/5) i let x=cos(theta) and now i have (x+y)^5 and i expanded it out. but now i am still stuck.

**Grandad** · February 11th 2009, 11:11 PM

Hello kayla09

nzmathman has shown you how to find $\sin \pi/5$ . Now use

$\cos^2(\pi/5) = 1 - \sin^2(\pi/5)$

$= 1 - \frac{5-\sqrt 5}{8}$

$= \frac{3+\sqrt 5}{8}$

$= \frac{6+2\sqrt 5}{16}$

$= \frac{1+2\sqrt 5+5}{16}$

$= \frac{(1+\sqrt 5)^2}{4^2}$

$\Rightarrow \cos(\pi/5) =\frac{1+\sqrt 5}{4}$ , taking the positive square root.

Grandad

**kayla09** · February 12th 2009, 04:44 AM

Thanks!!!

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