2025 HCI Promo Q6

The diagram above shows a unit circle with centre at the origin \(O\). The point \(A\) lies on the circumference of the circle and on the positive \(x\)-axis. A variable point \(B\) lies on the circumference of the circle such that the angle between the line segment \(OB\) and the positive \(x\)-axis is \(\theta\), where \(0<\theta<\pi\). The point \(C\) is a point of reflection of \(B\) about the \(x\)-axis. Let \(T\) denote the area of the triangle \(ABC\).

1. Show that \(T=p\sin\theta+q\sin2\theta\), where \(p\) and \(q\) are constants to be determined.[2]
2. Using calculus, find the exact value of \(\theta\) such that \(T\) is maximum as \(\theta\) varies.[4]
3. Using your answer found in part (b), find the ratio of the area of the triangle \(OBC\) to the area of the triangle \(ABC\).[2]