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 \(2\sin \frac{3}{5}\cos \frac{4}{5}\) .