Finding the cosine of 72* the hard way

• May 20th 2008, 08:25 PM
Kwerk
Finding the cosine of 72* the hard way
Alright, so let's say that w = cis(72*)

Using demoivre's theorem, w^5 = 1

What I'm hung up on, now, is showing that w^4+w^3+w^2+w+1=0, I'm supposed to use something I already know about polynomials (synthetic division?), and knowing that both w and 1 are roots of f(x) = x^5 -1, but I'm drawing a total blank.

All help is very much appreciated.
• May 20th 2008, 08:35 PM
Mathstud28
Quote:

Originally Posted by Kwerk
Alright, so let's say that w = cis(72*)

Using demoivre's theorem, w^5 = 1

What I'm hung up on, now, is showing that w^4+w^3+w^2+w+1=0, I'm supposed to use something I already know about polynomials (synthetic division?), and knowing that both w and 1 are roots of f(x) = x^5 -1, but I'm drawing a total blank.

All help is very much appreciated.

$x^5-1=(x-1)(x^4+x^3+x^2+x+1)$
• May 20th 2008, 08:50 PM
Kwerk
Okay I get that, now I have to turn that into w^2 + w^3 = (w + w^4)^2 - 2
somehow. I think I'm supposed to expand (w+w^4) ^2 into w^8+2w^5+w^2, but after than I'm lost again.
• May 20th 2008, 09:24 PM
mr fantastic
Quote:

Originally Posted by Kwerk
Alright, so let's say that w = cis(72*)

Using demoivre's theorem, w^5 = 1

What I'm hung up on, now, is showing that w^4+w^3+w^2+w+1=0, I'm supposed to use something I already know about polynomials (synthetic division?), and knowing that both w and 1 are roots of f(x) = x^5 -1, but I'm drawing a total blank.

All help is very much appreciated.

Read the attachment I have (quickly) prepared for you and work your way through it.