Complex numbers in polar form

Warm-up

Three quick questions from earlier lessons.

Discover

Plot $z = 3 + 4i$ as the point $(3, 4)$ . How far is it from 0?

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Key idea

A complex number is a point of the plane, so it has polar coordinates too.

z = a + bi = r(\cos\theta + i\sin\theta)

, with modulus

r = |z| = \sqrt{a^2 + b^2}

and argument

\theta

, where

\tan\theta = \dfrac{b}{a}

in the quadrant of

(a, b)

. Back:

a = r\cos\theta

,

b = r\sin\theta

.

Worked example

Write $z = -2 + 2\sqrt{3}\,i$ in polar form.

r = 4

√(4 + 12).

\theta = \pi - \dfrac{\pi}{3} = \dfrac{2\pi}{3}

tan θ = −√3; the point is left and up: quadrant II.

z = 4\left(\cos\dfrac{2\pi}{3} + i\sin\dfrac{2\pi}{3}\right)

Put together.

Your turn, guided

Your turn: write $2(\cos 60^\circ + i\sin 60^\circ)$ as $a + bi$ .

a = 1

2 cos 60° = 1.

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

2 sin 60° = √3: type z.

Practice ★

Find the exact modulus $|z|$ of $z = 3 - 8i$ .
Find the exact modulus $|z|$ of $z = 2 - 2i$ .
Find the exact modulus $|z|$ of $z = 8 - 9i$ .

Watch out!

Practice ★★

Find the argument of $z = -6 + 1i$ , between $0^\circ$ and $360^\circ$ , to 1 decimal place.
Find the argument of $z = -5 + 1i$ , between $0^\circ$ and $360^\circ$ , to 1 decimal place.

Write $z = 3\left(\cos 120^\circ + i\sin 120^\circ\right)$ in the form $a + bi$ (exact).
Write $z = 2\left(\cos 120^\circ + i\sin 120^\circ\right)$ in the form $a + bi$ (exact).
Write $z = 2\left(\cos 60^\circ + i\sin 60^\circ\right)$ in the form $a + bi$ (exact).

Exit ticket

Three questions to check you have got it.

Write $z = 3\left(\cos 120^\circ + i\sin 120^\circ\right)$ in the form $a + bi$ (exact).
Write $z = 2\left(\cos 120^\circ + i\sin 120^\circ\right)$ in the form $a + bi$ (exact).
Write $z = 2\left(\cos 60^\circ + i\sin 60^\circ\right)$ in the form $a + bi$ (exact).

Summary card

Polar form of a complex number

$z = r(\cos\theta + i\sin\theta)$ .
$r = \sqrt{a^2 + b^2}$ ; $\tan\theta = \dfrac{b}{a}$ , quadrant of $(a, b)$ .
$a = r\cos\theta$ , $b = r\sin\theta$ .

Watch out: plot the point before choosing $\theta$ .

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