Question 6 - A-Level Maths

Exam Board	AQA
Module	FP2 (Further Pure Mathematics 2)
Year	2008
Session	January
Topic	Addition & Double Angle Formulae

1. By applying De Moivre's theorem to $( \cos \theta + \mathrm { i } \sin \theta ) ^ { 3 }$, show that $$\cos 3 \theta = \cos ^ { 3 } \theta - 3 \cos \theta \sin ^ { 2 } \theta$$
2. Find a similar expression for $\sin 3 \theta$.
3. Deduce that $$\tan 3 \theta = \frac { \tan ^ { 3 } \theta - 3 \tan \theta } { 3 \tan ^ { 2 } \theta - 1 }$$
1. Hence show that $\tan \frac { \pi } { 12 }$ is a root of the cubic equation $$x ^ { 3 } - 3 x ^ { 2 } - 3 x + 1 = 0$$
2. Find two other values of $\theta$, where $0 < \theta < \pi$, for which $\tan \theta$ is a root of this cubic equation.
Hence show that $$\tan \frac { \pi } { 12 } + \tan \frac { 5 \pi } { 12 } = 4$$