show that $3{\sin ^2}\theta  - 7\sin \theta  + 2 = 0$;

find the smallest possible positive value of $\theta$.

attempting to form $(3\cos \theta  + 6)(\cos \theta  - 2) + 7(1 + \sin \theta ) = 0$     M1

$3{\cos ^2}\theta  - 12 + 7\sin \theta  + 7 = 0$     A1

$3\left( {1 - {{\sin }^2}\theta } \right) + 7\sin \theta  - 5 = 0$     M1

$3{\sin ^2}\theta  - 7\sin \theta  + 2 = 0$     AG

[3 marks]

attempting to solve algebraically (including substitution) or graphically for $\sin \theta$     (M1)

$\sin \theta  = \frac{1}{3}$     (A1)

$\theta  = 0.340{\text{ }}( = 19.5^\circ )$     A1

[3 marks]

Part (a) was very well done. Most candidates were able to use the scalar product and ${\cos ^2}\theta  = 1 - {\sin ^2}\theta$ to show the required result.

Part (b) was reasonably well done. A few candidates confused ‘smallest possible positive value’ with a minimum function value. Some candidates gave $\theta  = 0.34$ as their final answer.