The diagram shows right angled trapezium \(OCDF\) inside a semicircle with centre \(O\) and radius \(10\) cm such that angle \(BOC\) is \(\theta\) radians, and angle \(CDF\) and angle \(OFD\) are right angles.
Show that the perimeter, \(P\) cm, of trapezium \(OCDF\) is given by
\[P=10+30\cos\theta+10\sin\theta\][2]
Find the value of \(R\) when \(10\sin\theta+30\cos\theta\) is expressed as \(R\cos(\theta-\alpha)\), where \(R\) and \(\alpha\) are constants, and hence state the maximum perimeter of the trapezium.[3]
Show that the area, \(A\) \(\text{cm}^2\), of trapezium \(OCDF\) is given by
\[A=75\sin2\theta\][2]
The area of the trapezium varies with the value of \(\theta\). Find the value of \(\theta\) for which the area has a stationary value and determine whether this area is a maximum or a minimum.[4]