Angle at the centre of a circle from the ratio of two arcs and from the ratio of a segment to the rest of the circle

The diagram shows a circle with centre \(O\) and radius \(r\). The chord \(AB\) subtends an angle of \(\theta\) radians at \(O\), where \(0<\theta<\pi\), and the region between the chord and the minor arc \(AB\) is shaded.

1. The length of the minor arc \(AB\) and the length of the major arc \(AB\) are in the ratio \(2:7\). Find the exact value of \(\theta\).[2]
2. For a different value of \(\theta\), the area of the shaded region and the area of the rest of the circle are in the ratio \(1:8\). Find this value of \(\theta\), correct to 3 significant figures.[4]