BridgeComplex numbersFree

Complex numbers: de Moivre's theorem and roots

mathematical

Let

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

, shown on the Argand diagram.

Facts you may need

Polar form: $z = r\,\text{cis}\,\theta$ with $r = |z|$ , $\theta = \arg z$ — For $1 + \sqrt3i$: $r = 2$, $\theta = \frac{\pi}{3}$.
De Moivre: $(r\,\text{cis}\,\theta)^n = r^n\,\text{cis}\,n\theta$ — Raise the modulus, multiply the angle.
Area of a triangle with two sides $a$ , $b$ and included angle $C$ : $\frac12ab\sin C$ — Useful for triangles with a vertex at the centre.

Scholarship

Express

z^{10}

in the form

a + bi

, where

a

and

b

are exact.

Scholarship

Find all solutions of

w^3 = z

, giving your answers in polar form.

Scholarship

The solutions to (b) are plotted on an Argand diagram. Find the exact area of the triangle they form.