Write a formula for x in terms of cos θ.

Question

When two railroad tracks merge, the overlapping portions of the tracks are in the shapes of circular arcs (see figure). The radius of each arc r in feet) and the angle θ are related by x/2= 2r sin^2 (θ/2). Write a formula for x in terms of cos θ.

Answer 1

x = 2r·2sin²(θ/2) = 2r(1 - cosθ)