3.11: Tangents and sector equal to shaded region

0606/22/M/J/22 — Question 9 · 7 marks

3.11 diagram3.11 diagram
In this question all lengths are in centimetres.
The diagram shows a circle, centre OO, radius aa. The lines PTPT and QTQT are tangents to the circle at PP and QQ respectively. Angle POQPOQ is 2ϕ2\phi radians.
(a)
In the case when the area of the sector OPQOPQ is equal to the area of the shaded region, show that tan⁡ϕ=2ϕ\tan\phi = 2\phi.
[4]
(b)
In the case when the perimeter of the sector OPQOPQ is equal to half the perimeter of the shaded region, find an expression for tan⁡ϕ\tan\phi in terms of ϕ\phi.
[3]