Exact sine of an angle and its double, the sine rule in a triangle, and the area of an extended isosceles triangle

An acute angle \(\theta\) satisfies \(\cos\theta=\frac34\).

1. Find the exact value of
   1. \(\sin\theta\);
   2. \(\sin2\theta\).[4]

In triangle \(PQR\), shown in the diagram below, \(P\hat{Q}R=\theta\), \(P\hat{R}Q=2\theta\), \(PQ=x\) and \(PR=y\).

2. Use the sine rule to show that \(y=\frac{2x}{3}\).[2]

The side \([QR]\) is extended beyond \(R\) to a point \(S\) such that \(P\hat{S}R=\theta\), forming triangle \(PRS\), as shown in the diagram below.

3. Find the exact value of \(\sin P\hat{R}S\).[3]
4. Find the area of triangle \(PRS\) in terms of \(x\).[5]