The point O is the center of the circle with radius 1, and the points A and B are the points on the circumference. If the shaded segment is one third of the size of the circle, what's the value of $\theta = \angle OAB?$ I tried by subtracting the area of the triangle $OAB$ from the sector, however it ends up as an equation $\sin 2\theta = \frac{\pi}{3}-2\theta$ and I can't know how to solve this kind of equations. Any help in solving the equation or the question would be greatly appreciated.
Best Answer
As mentioned in the comments, you need numerical methods to approach an exact solution. But it is very easy to obtain a relatively tight bound on $\theta$.
Simply rewrite your equation as
$2\theta+\sin2\theta=\pi/3$
and observe that the $2\theta$ piece must be larger than the $\sin2\theta$ piece. So
$2\theta>\pi/6$
$\sin2\theta<\pi/6$
Solving these inequalities gives
$\pi/12<\theta<(1/2)\sin^{-1}(\pi/6)$
$15°<\theta<15°48'$
(The minutes in the upper bound are rounded up to assure it's an upper bound. The actual solution, rounded to the nearest minute, is $15°22'$.)