\left[r(\cos\theta+i\sin\theta)\right]^n?

r^n(\cos n\theta+i\sin n\theta).

Exponential form of de Moivre?

\left(e^{i\theta}\right)^n=e^{in\theta}.

How do you find \cos n\theta in terms of \cos\theta?

Expand (\cos\theta+i\sin\theta)^n and equate real parts, then use \sin^2\theta=1-\cos^2\theta.

\cos^4\theta in multiple angles?

\frac18(\cos4\theta+4\cos2\theta+3).

How do you sum \sum r^k\cos k\theta?

Take the real part of the geometric series \sum\left(r(\cos\theta+i\sin\theta)\right)^k.

Condition for an infinite geometric series to converge?

|\rho|<1, where \rho is the common ratio (complex ratios too).

Sum to infinity of a geometric series?

\frac{a}{1-\rho} for first term a and ratio \rho, with |\rho|<1.

De Moivre's theoremAQA A-Level Further Maths: Flashcards

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State de Moivre's theorem.

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State de Moivre's theorem.: $(\cos\theta+\mathrm{i}\sin\theta)^n=\cos n\theta+\mathrm{i}\sin n\theta$ for integer $n$ .
$\left[r(\cos\theta+\mathrm{i}\sin\theta)\right]^n$ ?: $r^n(\cos n\theta+\mathrm{i}\sin n\theta)$ .
Exponential form of de Moivre?: $\left(\mathrm{e}^{\mathrm{i}\theta}\right)^n=\mathrm{e}^{\mathrm{i}n\theta}$ .
$(\cos\theta+\mathrm{i}\sin\theta)^{-n}$ ?: $\cos n\theta-\mathrm{i}\sin n\theta$ .
How do you find $\cos n\theta$ in terms of $\cos\theta$ ?: Expand $(\cos\theta+\mathrm{i}\sin\theta)^n$ and equate real parts, then use $\sin^2\theta=1-\cos^2\theta$ .
$\cos3\theta$ in terms of $\cos\theta$ ?: $4\cos^3\theta-3\cos\theta$ .
$\sin3\theta$ in terms of $\sin\theta$ ?: $3\sin\theta-4\sin^3\theta$ .
If $z=\cos\theta+\mathrm{i}\sin\theta$ , $z^n+\frac{1}{z^n}$ ?: $2\cos n\theta$ .
If $z=\cos\theta+\mathrm{i}\sin\theta$ , $z^n-\frac{1}{z^n}$ ?: $2\mathrm{i}\sin n\theta$ .
$\cos^4\theta$ in multiple angles?: $\frac18(\cos4\theta+4\cos2\theta+3)$ .
How do you sum $\sum r^k\cos k\theta$ ?: Take the real part of the geometric series $\sum\left(r(\cos\theta+\mathrm{i}\sin\theta)\right)^k$ .
Condition for an infinite geometric series to converge?: $|\rho|<1$ , where $\rho$ is the common ratio (complex ratios too).
Sum to infinity of a geometric series?: $\frac{a}{1-\rho}$ for first term $a$ and ratio $\rho$ , with $|\rho|<1$ .

Exam questions on De Moivre's theorem

The complex number $z=\cos\frac{\pi}{12}+\mathrm{i}\sin\frac{\pi}{12}$ .
Express $z^{-5}$ in the form $\cos\alpha-\mathrm{i}\sin\alpha$ , stating the exact value of $\alpha$ .2 marks
The complex number $w=1+\mathrm{i}$ .
Find $w^{10}$ in the form $a+b\mathrm{i}$ .2 marks
The equation $8x^3-6x-1=0$ is to be solved.
Use de Moivre's theorem to show that $\cos3\theta=4\cos^3\theta-3\cos\theta$ .3 marks

See the full worksheet

Written by the Exaim team, led by Shaun Daswani (Head of Upper Secondary, Improve ME Institute; MSc Financial Mathematics, Imperial College London; BSc, UCL) and Jason Daswani (operational lead, Improve ME Institute; LSE).