4 The triangle \(A B C\), shown in the diagram, is such that \(A C = 8 \mathrm {~cm} , C B = 12 \mathrm {~cm}\) and angle \(A C B = \theta\) radians.
The area of triangle \(A B C = 20 \mathrm {~cm} ^ { 2 }\).
- Show that \(\theta = 0.430\) correct to three significant figures.
- Use the cosine rule to calculate the length of \(A B\), giving your answer to two significant figures.
- The point \(D\) lies on \(C B\) such that \(A D\) is an arc of a circle centre \(C\) and radius 8 cm . The region bounded by the arc \(A D\) and the straight lines \(D B\) and \(A B\) is shaded in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{9fee4b6f-06e2-4ed8-8835-33ef33b98c94-3_424_894_1434_555}
Calculate, to two significant figures:
- the length of the \(\operatorname { arc } A D\);
- the area of the shaded region.