`Pa n dQ` are two points on a circle of centre `C` and radius `alpha` . The angle `P C Q` being `2theta` , find the value of `sintheta` when the radius of the circle inscribed in the triangle `C P Q` is maximum.

Explanation

We know that r=`(triangle)/(s)`, where
`triangle` = Area of triangle CPQ and s= semiperameter of triangleCPQ
`therefore r=(alpha^(2) (sin 2 theta))/(2s)=(alpha^(2)sin 2 theta)/(2alpha+2alpha sin theta)=(alpha)/(2).(sin2theta)/(1+sin theta)`

Now for `f (theta)=(sin 2theta)/(1+sin theta)`
`f(theta)=(1+sin theta)(2cos 2theta-sin 2theta.cos theta)/(1+sin theta)^(2)`
or `2(1+sin theta)(1-2 sin^(2)theta)-2 sin theta (1-sin^(2)theta)=0`
or `2(1-sin^(2)theta)=2sin theta(1-sin theta)`
or `1-2sin^(2)theta=sin theta- sin^(2)theta`
or `sin^(2)theta=sin theta -sin^(2)theta`
or `sin^(2)theta+sin theta-1 =0`
or `sin theta=sqrt(5-1)/(2)`