Write down the value of $\tan \theta$ .

Find the value of

(i)     $\sin 2\theta$ ;

(ii)    $\cos 2\theta$ .

$\tan \theta = \frac{3}{4}$ (do not accept $\frac{3}{4}x$ )     A1     N1

[1 mark]

(i) $\sin \theta = \frac{3}{5}$ , $\cos \theta = \frac{4}{5}$     (A1)(A1)

correct substitution     A1

e.g. $\sin 2\theta = 2\left( {\frac{3}{5}} \right)\left( {\frac{4}{5}} \right)$

$\sin 2\theta = \frac{{24}}{{25}}$     A1     N3

(ii) correct substitution     A1

e.g. $\cos 2\theta = 1 - 2{\left( {\frac{3}{5}} \right)^2}$ ,  ${\left( {\frac{4}{5}} \right)^2} - {\left( {\frac{3}{5}} \right)^2}$

$\cos 2\theta = \frac{7}{{25}}$     A1     N1

[6 marks]

Many candidates drew a diagram to correctly find $\tan \theta$ , although few recognized that a line through the origin can be expressed as $y = x\tan \theta$ , with gradient $\tan \theta$ , which is explicit in the syllabus.

A surprising number were unable to find the ratios for $\sin \theta$ and $\cos \theta$ from $\tan \theta$ . It was not uncommon for candidates to use unreasonable values, such as $\sin \theta = 3$ and $\cos \theta = 4$ , or to write nonsense such as $2\sin \frac{3}{5}\cos \frac{4}{5}$ .