Polar and exponential form

Express complex numbers in modulus–argument form

r(\cos\theta+i\sin\theta)

and Euler's form

re^{i\theta}

; multiply and divide using polar form; apply De Moivre's theorem to compute powers.

Worked examples

Converting to polar form

Straightforward

Problem

Write

z = -1+\sqrt{3}\,i

in polar form.

Multiplying and dividing in polar form

Moderate

Problem

Given

z_1 = 6e^{i\pi/4}

and

z_2 = 2e^{i\pi/12}

, find

\dfrac{z_1}{z_2}

in polar form and hence in Cartesian form.

Using De Moivre's theorem to evaluate a power

Challenging

Problem

Use De Moivre's theorem,

(\cos\theta+i\sin\theta)^n = \cos(n\theta)+i\sin(n\theta)

, to compute

(1+i)^{10}

Practise

Q1·Straightforward

Find the modulus of

z = \sqrt{3}+i

Q2·Straightforward

z_1 = 4(\cos 20° + i\sin 20°)

and

z_2 = 3(\cos 70° + i\sin 70°)

. Find

|z_1 z_2|

Q3·Straightforward

Find the real part of

z = 2(\cos 90°+i\sin 90°)

Q4·Moderate

Find the argument of

z = \sqrt{3}+i

in degrees.

Q5·Moderate

z_1 = 3(\cos 40°+i\sin 40°)

and

z_2 = 2(\cos 80°+i\sin 80°)

. Find

\arg(z_1 z_2)

in degrees.

Q6·Moderate

Find the real part of

(1+i)^8

using De Moivre's theorem.

Q7·Moderate

Find the argument of

z = -\sqrt{3}+i

in degrees.

Q8·Moderate

z_1 = 10(\cos 60°+i\sin 60°)

and

z_2 = 2(\cos 10°+i\sin 10°)

. Find

\left|\dfrac{z_1}{z_2}\right|

Q9·Challenging

Find the real part of

(\cos 18°+i\sin 18°)^{10}

using De Moivre's theorem.

Q10·Challenging

Find the real part of

(1+i)^{12}

using De Moivre's theorem.

Q11·Challenging

Find the imaginary part of

(1-i)^{10}

using De Moivre's theorem.

Q12·Challenging

z = 2e^{i\pi/6}

and

w = 3e^{i\pi/3}

. Find

\text{Im}(zw)

Q13·Challenging

Use De Moivre's theorem to expand

(\cos\theta+i\sin\theta)^3

and hence prove that:

\cos 3\theta = 4\cos^3\theta - 3\cos\theta

✎ Work this one through on paper — proofs are self-assessed.