From the goldenratio 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,
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,
sin 36° = cos 54° = (1/2)√(3  τ).
