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).
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).

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

Christopher J. Henrich