1). Connect all of the vertices of a regular pentagon.
2). Use similar triangle ratios to show that the sides of the 36-72-72 isosceles triangle are related by the golden ratio.
3). Use law of cosines to prove that cos(36) = phi/2
Here’s another: write sin 3pi/5 = sin 2pi/5, expand into sines and cosines of pi/5, and divide out the sin pi/5 which should show up in every term. You’re left with a quadratic in cos pi/5.
December 3, 2010 at 11:17 am
Note also that cos(pi/5) = (1+sqrt(5))/4, which is half the golden ratio. Can you find a geometric proof?
December 3, 2010 at 11:21 am
Nice question. Thanks.![\sqrt{\frac{3+\sqrt{5}}{8}}=\frac{1+\sqrt{5}}{4}](http://s0.wp.com/latex.php?latex=%5Csqrt%7B%5Cfrac%7B3%2B%5Csqrt%7B5%7D%7D%7B8%7D%7D%3D%5Cfrac%7B1%2B%5Csqrt%7B5%7D%7D%7B4%7D&bg=ffffff&fg=8E8778&s=0&c=20201002)
Of course
January 14, 2011 at 12:29 am
1). Connect all of the vertices of a regular pentagon.
2). Use similar triangle ratios to show that the sides of the 36-72-72 isosceles triangle are related by the golden ratio.
3). Use law of cosines to prove that cos(36) = phi/2