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

• Feb 10th 2009, 09:49 PM
kayla09
[SOLVED] Complex Numbers- cos ( pi/5)
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.
• Feb 11th 2009, 12:19 AM
nzmathman
• Feb 11th 2009, 08:18 PM
kayla09
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.
• Feb 11th 2009, 11:11 PM
Cos pi/5
Hello kayla09

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

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

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

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

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

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

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

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