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de Moivre's theorem — 9231 FP2
Separate modulus from argument when taking powers.
1. Understand 2. See examples 3. Check answers 4. Explain your working Notes and key methods For \(z=r(\cos\theta+i\sin\theta)\), de Moivre gives \(z^n=r^n[\cos(n\theta)+i\sin(n\theta)]\). Arguments are determined modulo 2π, so a principal argument may need reducing to the specified interval.
The real and imaginary parts are \(\displaystyle r^{n}\cos (n\theta )\) and \(\displaystyle r^{n}\sin (n\theta )\). The modulus is nonnegative; a negative real part does not make the modulus negative.
Check the conditions Use radians unless degrees are explicitly requested. For noninteger powers, a branch convention is required. Three worked examples Example 1 · de Moivre and complex powers \(\displaystyle z= 4[\cos (\frac{\pi }{5})+ i\sin (\frac{\pi }{5})]\), with \(\displaystyle n=5\).
Find the modulus of \(\displaystyle z^{n}\). Find \(\displaystyle \operatorname{Re}(z^{n})\). Find \(\displaystyle \operatorname{Im}(z^{n})\). Find \(\displaystyle |z^{n}|^{2}\). Show the reasoning (a) \(r^n=1024\)
Answer: \(1024\).
(b) \(\displaystyle 4^{5}\cos (\frac{5\pi }{5})= - 1024\)
Answer: \(-1024\).
(c) \(\displaystyle 4^{5}\sin (\frac{5\pi }{5})= 1.25403832232689\times 10^{- 13}\)
Answer: \(1.25403832232689\times10^{-13}\).
(d) \(r^{2n}=1048576\)
Answer: \(1048576\).
Example 2 · de Moivre and complex powers \(\displaystyle z= 5[\cos (\frac{\pi }{6})+ i\sin (\frac{\pi }{6})]\), with \(\displaystyle n=3\).
Find the modulus of \(\displaystyle z^{n}\). Find \(\displaystyle \operatorname{Re}(z^{n})\). Find \(\displaystyle \operatorname{Im}(z^{n})\). Find \(\displaystyle |z^{n}|^{2}\). Show the reasoning (a) \(r^n=125\)
Answer: \(125\).
(b) \(\displaystyle 5^{3}\cos (\frac{3\pi }{6})= 7.65404249467096\times 10^{- 15}\)
Answer: \(7.65404249467096\times10^{-15}\).
(c) \(\displaystyle 5^{3}\sin (\frac{3\pi }{6})= 125\)
Answer: \(125\).
(d) \(r^{2n}=15625\)
Answer: \(15625\).
Example 3 · de Moivre and complex powers \(\displaystyle z= 2[\cos (\frac{\pi }{3})+ i\sin (\frac{\pi }{3})]\), with \(\displaystyle n=4\).
Find the modulus of \(\displaystyle z^{n}\). Find \(\displaystyle \operatorname{Re}(z^{n})\). Find \(\displaystyle \operatorname{Im}(z^{n})\). Find \(\displaystyle |z^{n}|^{2}\). Show the reasoning (a) \(r^n=16\)
Answer: \(16\).
(b) \(\displaystyle 2^{4}\cos (\frac{4\pi }{3})= - 8.00000000000001\)
Answer: \(-8.00000000000001\).
(c) \(\displaystyle 2^{4}\sin (\frac{4\pi }{3})= - 13.856406460551\)
Answer: \(-13.856406460551\).
(d) \(r^{2n}=256\)
Answer: \(256\).
Common mistakes Raise the modulus to the power as well as multiplying the angle. Do not confuse an argument with the imaginary part. Practise with answer checking Original exam-style problems Write complete solutions before opening the worked solution. The marking guidance rewards reasoning as well as answers.
Next skill: Powers of sine and cosine →
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