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.

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

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