Products and powers in polar form (De Moivre)

Warm-up

Three quick questions from earlier lessons.

Discover

Multiplying by $i$ turns a point a quarter turn around 0: $i \cdot 1 = i$ , $i \cdot i = -1$ . What does multiplying by $2i$ do to the length and the angle of $z$ ?

Doubles the length and adds $90^\circ$
Adds 2 to the length and doubles the angle

Key idea

In polar form, products and powers are easy.

r_1(\cos\alpha + i\sin\alpha) \cdot r_2(\cos\beta + i\sin\beta) = r_1 r_2\big(\cos(\alpha + \beta) + i\sin(\alpha + \beta)\big)

: multiply the moduli, add the arguments. De Moivre:

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

.

Worked example

Compute $\left(2(\cos 20^\circ + i\sin 20^\circ)\right)^3$ .

2^3 = 8

Modulus to the power 3.

3 \cdot 20^\circ = 60^\circ

Argument times 3.

8(\cos 60^\circ + i\sin 60^\circ) = 4 + 4\sqrt{3}\,i

cos 60° = 1/2, sin 60° = √3/2.

Your turn, guided

Your turn: $3(\cos 50^\circ + i\sin 50^\circ) \cdot 2(\cos 40^\circ + i\sin 40^\circ)$ .

6(\cos 90^\circ + i\sin 90^\circ)

3 · 2 and 50° + 40°.

= 6i

cos 90° = 0, sin 90° = 1: type the result.

Practice ★

$z_1 = 2\left(\cos 40^\circ + i\sin 40^\circ\right)$ and $z_2 = 4\left(\cos 55^\circ + i\sin 55^\circ\right)$ . Find $z_1 z_2$ in polar form.
$z_1 = 3\left(\cos 50^\circ + i\sin 50^\circ\right)$ and $z_2 = 2\left(\cos 55^\circ + i\sin 55^\circ\right)$ . Find $z_1 z_2$ in polar form.
$z_1 = 5\left(\cos 30^\circ + i\sin 30^\circ\right)$ and $z_2 = 6\left(\cos 45^\circ + i\sin 45^\circ\right)$ . Find $z_1 z_2$ in polar form.

Watch out!

Practice ★★

Use De Moivre's theorem to write $\left(\cos 60^\circ + i\sin 60^\circ\right)^{ 2 }$ in the form $a + bi$ (exact).
Use De Moivre's theorem to write $\left(\cos 60^\circ + i\sin 60^\circ\right)^{ 4 }$ in the form $a + bi$ (exact).
Use De Moivre's theorem to write $\left(\cos 30^\circ + i\sin 30^\circ\right)^{ 6 }$ in the form $a + bi$ (exact).

Exit ticket

Three questions to check you have got it.

Use De Moivre's theorem to write $\left(\cos 60^\circ + i\sin 60^\circ\right)^{ 2 }$ in the form $a + bi$ (exact).
Use De Moivre's theorem to write $\left(\cos 60^\circ + i\sin 60^\circ\right)^{ 4 }$ in the form $a + bi$ (exact).
Use De Moivre's theorem to write $\left(\cos 30^\circ + i\sin 30^\circ\right)^{ 6 }$ in the form $a + bi$ (exact).

Summary card

Products and powers in polar form

Product: multiply moduli, add arguments.
De Moivre: $(r\,\mathrm{cis}\,\theta)^n = r^n\,\mathrm{cis}(n\theta)$ .

Watch out: do not raise the real and imaginary parts separately.

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