Posts tagged 'trigonometry'

F.53 Geometric Half-Angle Trigonometry

In a unit semicircle of radius R = 1 and diameter AB = 2, a point P at central angle theta has altitude sin(theta), base projections 1 + cos(theta) and 1 - cos(theta), and chord |AP| = 2 cos(theta/2) = sqrt(2 + 2 cos(theta)), giving cos^2(theta/2) = (1 + cos(theta))/2 and sin^2(theta/2) = (1 - cos(theta))/2.

F.55 Compass Constructions and Angle-to-Side Triangle Reconstruction

Given the three interior angles (alpha, beta, gamma) of a triangle normalized to unit area K = 1 (so s = 1/r), the Conway intouch segments x = s - a = r cot(alpha/2), y = s - b = r cot(beta/2), and z = s - c = r cot(gamma/2) determine the side lengths a = y + z, b = z + x, and c = x + y = 2 s - (a + b).