Chapter 4.7, Problem 34ES

A segment of a circle is the region enclosed by an arc and its chord (Figure Ex-34). If $r$ is the radius of the circle and $\theta$ the angle subtended at the center of the circle, then it can be shown that the area $A$ of the segment is $A=\frac{1}{2}{r}^2\left(\theta -\sin \theta \right),$ where $\theta$ is in radians. Find the value of $\theta$ for which the area of the segment is one-fourth the area of the circle. Give $\theta$ to the nearest degree.