Simplify Condition for Chord Length Equals Angle?

by DuncanM
Tags: angle, chord, condition, equals, length, simplify
 P: 90 Consider the following: On a circle of radius 1, two points are marked: P1 and P2. Two lines are drawn from the center of the circle: one from the center to P1, the other from the center to P2. The angle between these two lines is $\theta$. One more line is drawn: from P1 directly to P2. In other words, this third line is a chord on this circle. For the special condition that the length of this chord equals the angle, find a simple expression. i.e. – find a simple expression for $\theta$ given the special condition that chord length = $\theta$ = angle = $\theta$ - - - So far, all the expressions that I have worked out mix terms of $\theta$ and either sin($\theta$) or cos($\theta$); I have not been able to find an expression simply in terms of $\theta$, sin($\theta$), or cos($\theta$). For example, following is one of my approaches: Bisect the angle $\theta$, which also divides the chord in half. The chord length is $\theta$. But this value is also 2 sin($\theta$/2) Equating these two expressions: 2 sin($\theta$/2) = $\theta$ or sin($\theta$/2) = $\theta$/2 I cannot find a way to simplify this expression further. Any suggestions?