The following diagram shows two intersecting circles of radii 4 cm and 3 cm. The centre C of the smaller circle lies on the circumference of the bigger circle. O is the centre of the bigger circle and the two circles intersect at points A and B.

Find:

(a)     \({\rm{B\hat OC}}\);

(b)     the area of the shaded region.

(a)     METHOD 1

\(2{\text{arcsin}}\left( {\frac{{1.5}}{4}} \right)\)     M1

\(\alpha  = {0.769^c}{\text{ (44.0}}^\circ {\text{)}}\)     A1

METHOD 2

using the cosine rule:

\({3^2} = {4^2} + {4^2} - 2(4)(4)\cos \alpha \)     M1

\(\alpha  = {0.769^c}{\text{ (44.0}}^\circ {\text{)}}\)     A1

[2 marks]

(b)     one segment

\({{\text{A}}_1} = \frac{1}{2} \times {4^2} \times 0.76879 - \frac{1}{2} \times {4^2}\sin (0.76879)\)     M1A1

\( = 0.58819{\text{K}}\)     (A1)

\(2{{\text{A}}_1} = 1.18{\text{ }}({\text{c}}{{\text{m}}^2})\)     A1

Note:     Award M1 only if both sector and triangle are considered.

[4 marks]

Total [6 marks]