Express a complex number in Cartesian form 𝑧 = 𝑎 + 𝑏𝑖 and polar form. 𝑧 = 𝑟 (cos(𝜃) + 𝑖 sin(𝜃)) or 𝑧 = 𝑟 cis(𝜃)

A complex number is given by \(z = 3(\cos(\frac{2\pi}{3}) + i\sin(\frac{2\pi}{3}))\). Express \(z\) in Cartesian form \(a + bi\), showing exact values.

A complex number \(z\) is represented in the complex plane as a point that is obtained by rotating the point corresponding to \(3 + 4i\) anticlockwise about the origin through an angle of \(\frac{\pi}{3}\) radians. (a) Express the original complex number \(3 + 4i\) in polar form \(r \operatorname{cis}(\theta)\). (1 mark) (b) Express the rotated complex number in polar form. (1 mark) (c) Express the rotated complex number in Cartesian form \(a + bi\). (1 mark)

The complex number \(z\) is shown on the Argand diagram below. Express \(z\) in Cartesian form \(a + bi\).

Consider the complex number \(z = 4\sqrt{3} - 4i\). (a) Express \(z\) in polar form \(r \operatorname{cis}(\theta)\), where \(r > 0\) and \(-\pi < \theta \leq \pi\). (2 marks) (b) Hence, or otherwise, find \(z^3\) and express your answer in Cartesian form \(a + bi\). (2 marks) (c) Express \(\frac{16}{z}\) in Cartesian form. (1 mark)