$A$ is a point on the circumference of a circle.Chords $AB$ and $AC$ divide the area of the circle into three equal parts.If the angle $BAC$ is the root of the equation,$f(x)=0$,then find $f(x).$
I tried to solve it,but after some time,i got stuck.Let $O$ is the center and $r$ is the radius of the circle.Then as angle$BAC=\theta$,so angle $BOC=2\theta$,area of $BACA=\frac{\pi r^2}{2}$.Thereafter i stuck.Please help me.
Best Answer
Area is found as:
$$ =\int ^{\theta/2} _ {-\theta/2} \rho^2 d t $$
where
$$ \rho = 2 r \cos t $$
The area relation is found by integration. To solve for $\theta,$ a transcendental equation a numerical method is suitable.
Area enclosed by chords = $ r^2 ( t + \sin t) = \pi r^2 /3 $
$$ \theta + \sin \theta = \frac{\pi}{3} $$