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]