Chord dividing circle , function Two chords $PA$ and $PB$ divide circle into three parts. The angle $PAB$ is a root of $f(x)=0$.
Find $f(x)$.
Clearly , $PA$ and $PB$ divides circle into three parts means it divides it into $3$ parts of equal areas
How can i find $f(x)$ then ?
thanks
 A: You may assume your circle to be the unit circle in the $(x,y)$-plane and $P=(1,0)$. If the three parts have to have equal area then $A=\bigl(\cos(2\phi),\sin(2\phi)\bigr)$ and $B=\bigl(\cos(2\phi),-\sin(2\phi)\bigr)$ for some $\phi\in\ ]0,{\pi\over2}[\ $. Calculating the area over the segment $PA$ gives the condition
$$2\Bigl({\phi\over2}-{1\over2}\cos\phi\sin\phi\Bigr)={\pi\over3}\ ,$$
or $f(\phi):=\phi-\cos\phi\sin\phi-{\pi\over 3}=0$. This equation has to be solved numerically. One finds $\phi\doteq1.30266$.
A: Hint:  If you look up circular segment in Wikipedia, you should be able to write the area of the two segments cut off as a function of the angle between the chord and the tangent.  That angle is $\theta/2$ using the figure in the article.  Then the area of the circle that is left is what you want.  The question is whether you can write this area as a function of the angle between the chords (one variable) instead of the angles between the chords and their respective tangents (two variables).
