Proof of Cosine of 36 DegreesDate: 11/26/2001 at 21:21:03 From: John Kuli Subject: The proof of Cosine of 36 degrees. Hello! I'm in Calculus 1 at Wright State University. As a bonus question my teacher asked us to solve the following: Prove: cosine {(pi)/5} = {1+5^(1/2)}/4 I was wondering if you could help me out. Thanks. John Date: 11/26/2001 at 22:37:27 From: Doctor Paul Subject: Re: The proof of Cosine of 36 degrees. To compute cos(36) notice that cos(36) = cos(72/2) Now apply the half angle formula: cos(36) = +- sqrt[(1+cos(72))/2] Now we need to know how to compute cos(72). This has already been done in our archives, where a person asked how to compute sin(18) but sin(18) = cos(72): Finding Values by Hand http://mathforum.org/dr.math/problems/victor8.26.98.html Following the math derived in the link above, cos(72) = -0.25 + 0.25*SQRT(5) Plugging this in above (and taking the positive square root) gives: cos(36) = sqrt[(1 + (-0.25 + 0.25*SQRT(5)))/2] = sqrt[(3+sqrt(5))/8] (verify this!) How do we show this is the same as (1 + sqrt(5))/4 ? Start with something we know to be true: 8 + 16*sqrt(5) + 40 = 48 + 16*sqrt(5) then rewrite as: 1 + 2*sqrt(5) + 5 3 + sqrt(5) ----------------- = ----------- 16 8 now take the square root of both sides: 1+sqrt(5) -------- = sqrt[(3+sqrt(5))/8] 4 Thus cos(36) = sqrt[(3+sqrt(5))/8] = (1+sqrt(5))/4 You may ask, how did I know to write 8 + 16*sqrt(5) + 40 = 48 + 16*sqrt(5) ? The answer is I didn't. But I knew the two things had to be equal so I set them equal to each other and then showed they were the same (I did this on some scratch paper). Then I just copied my work in reverse order to establish the desired result. I hope this helps. Please write back if you'd like to talk about this some more - especially if you have trouble following anything I've written or if you have trouble following the math in the link I referred to. - Doctor Paul, The Math Forum http://mathforum.org/dr.math/ |
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