Kite formed by two tangents and two radii, and an angle fixed by an area condition solved numerically

From a point \(T\) outside a circle with centre \(O\) and radius \(r\), two tangents are drawn, touching the circle at \(P\) and \(Q\). Let \(\angle PTQ=\varphi\) radians. The region bounded by the line segments \(TP\) and \(TQ\) and the minor arc \(PQ\) is shaded, as shown in the diagram below.

1. Show that the area of the quadrilateral \(TPOQ\) is \(r^2\cot\frac{\varphi}{2}\).[3]
2. The area of the shaded region is one third of the area of the quadrilateral \(TPOQ\). Find \(\varphi\), giving your answer in degrees.[4]