Originally Posted by

**Grandad** Hello AquafinaThe function $\displaystyle A(\theta)$ gives the area between any two tangents and the circle expressed as a function of $\displaystyle \theta$, the angle between the tangents.

So if we denote the angle between the tangents from $\displaystyle R$ to the circle by $\displaystyle \phi$, then the area between these tangents and the circle is $\displaystyle A(\phi)$. But this area is given by $\displaystyle B(\theta)$. Hence $\displaystyle A(\phi) = B(\theta)$.

But, from $\displaystyle \triangle ORP, \theta = \frac{\pi}{2}-\phi$. Therefore $\displaystyle A(\phi) = B(\frac{\pi}{2}-\phi)$. We can replace $\displaystyle \phi$ by any other parameter; in particular by $\displaystyle \theta$. So $\displaystyle A(\theta) = B(\frac{\pi}{2}-\theta)$.

For part (iii) I should do as you have done. The calculation is very straightforward for $\displaystyle \theta = \frac{\pi}{3}$. Why complicate it?

Grandad