Find, in its simplest form, the argument of ${\left( {\sin \theta  + {\text{i}}(1 - \cos \theta )} \right)^2}$ where $\theta$ is an acute angle.

${\left( {\sin \theta + {\text{i}}(1 - \cos \theta )} \right)^2} = {\sin ^2}\theta - {(1 - \cos \theta )^2} + {\text{i}}2\sin \theta (1 - \cos \theta )$     M1A1

Let $\alpha$ be the required argument.

$\tan \alpha = \frac{{2\sin \theta (1 - \cos \theta )}}{{{{\sin }^2}\theta - {{(1 - \cos \theta )}^2}}}$     M1

$= \frac{{2\sin \theta (1 - \cos \theta )}}{{(1 - {{\cos }^2}\theta ) - (1 - 2\cos \theta + {{\cos }^2}\theta )}}$     (M1)

$= \frac{{2\sin \theta (1 - \cos \theta )}}{{2\cos \theta (1 - \cos \theta )}}$     A1

$= \tan \theta$     A1

$\alpha = \theta$     A1

[7 marks]

Very few candidates scored more than the first two marks in this question. Some candidates had difficulty manipulating trigonometric identities. Most candidates did not get as far as defining the argument of the complex expression.