# Complex numbers – Further Pure Maths 2

Candidates should be able to:

• understand de Moivre’s theorem, for a positive or negative integer exponent, in terms of the geometrical effect of multiplication and division of complex numbers
• prove de Moivre’s theorem for a positive integer exponent (e.g. by induction.)
• use de Moivre’s theorem for a positive or negative rational exponent (e.g. expressing $$\cos 5\theta$$ in terms of $$\cos \theta$$ or $$\tan 5\theta$$ in terms of $$\tan \theta$$.)
– to express trigonometrical ratios of multiple angles in terms of powers of trigonometrical ratios of the fundamental angle (e.g. expressing $$\sin^6 \theta$$ in terms of $$\cos 2 \theta$$, $$\cos 4 \theta$$ and $$\cos 6 \theta$$.)
– to express powers of sini and cosi in terms of multiple angles
– in the summation of series (e.g. using the ‘C + iS’ method to sum series such as $$\sum\limits_{r=1}^{n} \binom{n}{r} \sin r\theta$$.)
– in finding and using the nth roots of unity.

Complex numbers – Further Pure Maths 2 Video (2 Parts)