The Golden Ratio and Some Useful Trigonometric Ratios

From the golden-ratio triangles we can deduce the sines and cosines of some interesting angles.

Bisect BC at P; then the right triangle ABP has angles 18° at A and 72° at P. We know that AB = τ and BP = 1/2; therefore
sin 18° = cos 72° = 1/(2τ) = (τ - 1)/2 .

From the Pythagorean theorem,

cos² 18° = 1 - sin² 18° = 1 - (τ - 1)²/4
= 1 - (2 - τ)/4
= (2 + τ)/4 .
Therefore
cos 18° = sin 72° = (1/2)√(2 + τ).
triangle ABP
Bisect AB at Q; then the right triangle ADQ has angles 36° at A and 54° at Q. We know that AD = 1 and AQ = τ/2; therefore
cos 36° = sin 54° = τ/2 .

From the Pythagorean theorem,

sin² 36° = 1 - cos² 36° = 1 - τ²/4
= 1 - (τ + 1)/4
= (3 - τ)/4 .
Therefore
sin 36° = cos 54° = (1/2)√(3 - τ).
triangle DQ

Last modified on $Date: 2007-04-19 22:46:45 -0400 (Thu, 19 Apr 2007) $

Christopher J. Henrich