Question

When two railroad tracks merge, the overlapping portions of the tracks are in the shapes of circular arcs. The radius of each arc r (in feet) and the angle theta are related by

x/2 = 2r sin^2 theta/2.

Write a formula for x in terms of cos theta.

Answer 1

\[\Large \frac{ x }{ 2 } = 2r \sin^2\frac{ \theta }{ 2 }\]
You could use a number of trig identities to get it in terms of cosine... http://www.sosmath.com/trig/Trig5/trig5/trig5.html

Answer 2

Use this \[\Large \sin^2 \frac{ \theta }{ 2 } = \frac{ 1-\cos \theta }{ 2 }\]