Find an expression for the area of the shaded region.

The chord [AB] divides the area of the circle in the ratio 1:7. Find the value of $\theta$ .

substitution into formula for area of triangle     A1

e.g. $\frac{1}{2}r \times r\sin \theta$

evidence of subtraction     M1

correct expression     A1     N2

e.g. $\frac{1}{2}{r^2}\theta  - \frac{1}{2}{r^2}\sin \theta$ , $\frac{1}{2}{r^2}(\theta  - \sin \theta )$

[3 marks]

evidence of recognizing that shaded area is $\frac{1}{8}$ of area of circle     M1

e.g. $\frac{1}{8}$ seen anywhere

setting up correct equation     A1

e.g. $\frac{1}{2}{r^2}(\theta - \sin \theta ) = \frac{1}{8}\pi {r^2}$

eliminating 1 variable     M1

e.g. $\frac{1}{2}(\theta - \sin \theta ) = \frac{1}{8}\pi$ , $\theta - \sin \theta  = \frac{\pi }{4}$

attempt to solve     M1

e.g. a sketch, writing $\sin x - x + \frac{\pi }{4} = 0$

$\theta  = 1.77$  (do not accept degrees)     A1     N1

[5 marks]