| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9795 (Pre-U Further Mathematics) |
| Session | Specimen |
| Topic | Complex numbers 2 |
| Type | De Moivre to derive trigonometric identities |
| Difficulty | Standard +0.8 Part (i) is a standard application of de Moivre's theorem requiring routine manipulation. Part (ii) requires expanding (z + z^{-1})^5 using the binomial theorem and carefully collecting terms—this demands algebraic dexterity and insight into the method, making it moderately challenging but still a recognizable Further Maths technique. |
| Spec | 4.02n Euler's formula: e^(i*theta) = cos(theta) + i*sin(theta)4.02q De Moivre's theorem: multiple angle formulae |
4 Let $z = \cos \theta + \mathrm { i } \sin \theta$.\\
(i) Use de Moivre's theorem to prove that $z ^ { n } + z ^ { - n } = 2 \cos n \theta$.\\
(ii) Deduce the identity $\cos ^ { 5 } \theta = \frac { 1 } { 16 } ( \cos 5 \theta + 5 \cos 3 \theta + 10 \cos \theta )$.
\hfill \mbox{\textit{Pre-U Pre-U 9795 Q4}}