De Moivre's Theorem gives a formula for calculating complex numbers. It enables us to connect complex numbers and trigonometry. Most importantly, it is incredibly useful for finding powers and roots of complex numbers. It can be stated in a number of ways:
\([r(cos\theta+isin\theta)]^n=r^n(cos\ n\theta+isin\ n\theta)\)
Whilst we can prove the formula using proof by induction, it becomes clear when we write it in Euler Form: \([re^{i\theta}]^n=r^ne^{in\theta}\)
On this page, you should learn to
- Find powers and roots of complex numbers using de Moivre's theorem
- Carry out a proof by induction of de Moivre's theorem
The following videos will help you understand all the concepts from this page
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Here is a quiz that practises the skills from this page
START QUIZ!
z is a complex number.
If |z| = 4 , work out
|z²| =
\(|z^{\frac{1}{2}}|\) =
Use the following property of the modulus of a complex number:
\(|z^n|={|z|}^n\)
z can be expressed in the form a + bi
If arg(z) = 60° , work out
arg(z²) = °
\(arg(z^{\frac{1}{2}})\) = °
Use the following property of the argument of a complex number:
\(arg(z^n)=n\cdot arg(z)\)
Let z = 3(cos30° + isin30°)
Find the principal argument of the following complex numbers
a) -z = °
b) iz = °
c) (1 + i)z = °
a) Consider z in Cartesian form. Multiplying by -1 will make the real and imaginary parts become negative. Note that multiplying a complex number by -1 results in a 180° rotation.
b) Multiplying a complex number by i results in a 90° anticlockwise rotation.
c) arg(1 + i) = 45°
arg((1 + i)z) = 30° + 45°
Let \(z=2(cos(\frac{\pi}{4})+isin(\frac{\pi}{4}))\)
Find the modulus of the following complex numbers
a) -z =
b) iz =
c) \((\sqrt{3}+i)z\) =
a) the modulus of -z = modulus of z
b) modulus of i = 1
\(|z\cdot w|=|z|\cdot|w|\)
|iz|=1x2=2
c) \(|\sqrt{3}+i|=\sqrt{3+1²}=2\)
\(|z\cdot w|=|z|\cdot|w|\)
\(|(\sqrt{3}+i)z|\) = 2x2=4
Let \(z=\sqrt{2}cis(\frac{\pi}{4})\)
\(z^2=acis(\frac{\pi}{b})\)
Find a and b
a =
b =
\(z^2=(\sqrt{2})^2cis(2\cdot\frac{\pi}{4})\)
Let \(z=2cis(\frac{\pi}{3})\)
\(z^3=acis(\frac{\pi}{b})\)
Find a and b
a =
b =
\(z^3=2^3cis(3\cdot\frac{\pi}{3})\)
\((\sqrt{2}cis(\frac{\pi}{4}))^8=a+bi\)
Find a and b
a =
b =
\((\sqrt{2}cis(\frac{\pi}{4}))^8\)
\(=(\sqrt{2})^8cis(8\cdot\frac{\pi}{4})\)
\(=16cis(2\pi)\)
= 16
\((\sqrt[3]{3}cis(\frac{\pi}{6}))^9=a+bi\)
Find a and b
a =
b =
\((\sqrt[3]{3}cis(\frac{\pi}{6}))^9\)
\(=({3}^\frac{1}{{3}})^9cis(9\cdot\frac{\pi}{6})\)
\(=3^3cis(\frac{3\pi}{2})\)
\(=27cis(-\frac{\pi}{2})\)
= -27i
Work out \((-1+i)^4\)
\((-1+i)^4\) =
Let z = -1 + i
\(z=\sqrt{2}cis\frac{3\pi}{4}\)
\(z^4=(\sqrt{2})^4cis(4\cdot\frac{3\pi}{4})\)
\(z^4=4cis(3\pi)\)
Which of the following Argand diagrams represents the roots to the equation \(z^3=64i\)
\(z^3=64i\)
\(z^3=64cis(\frac{\pi}{2}+2k\pi),k=0,1,2\)
\(z=4cis(\frac{\pi}{6}+\frac{2k\pi}{3}),k=0,1,2\)
\(z=4cis(\frac{\pi}{6}),4cis(\frac{5\pi}{6}),4cis(\frac{9\pi}{6})\)
Find the roots of the equation \(z^3=8i \)
Express your answers in Cartesian Form
Hint
Full Solution
\(z=-2+2\sqrt{3}i\)
a) Find |z| and arg(z)
b) Find \(z^6\) and simplify your answer
c) Given that \(w^4=z^3\) , find the values of \(w\) giving your answers in the form \(a+bi\)
Hint
Full Solution
a) By writing \(\frac{\pi}{12}=\frac{\pi}{3}-\frac{\pi}{4}\), show that \(sin(\frac{\pi}{12})=\frac{\sqrt{6}}{4}-\frac{\sqrt{2}}{4}\)
b) Work out \(cos(\frac{\pi}{12})\)
c) Hence, find the roots of the equation \(z^4=2+2\sqrt{3}i\), giving answers in the form \(z=a+ib\)
Hint
Full Solution
How much of Complex Numbers - de Moivre's Theorem have you understood?
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