Further Complex Numbers Topic Review summary

Key learning points

• de Moivre: $[r(\\cos\\theta+i\\sin\\theta)]^n=r^n(\\cos n\\theta+i\\sin n\\theta)$.
• $n$th roots: $r^{1/n}\\left(\\cos\\frac{\\theta+2k\\pi}{n}+i\\sin\\frac{\\theta+2k\\pi}{n}\\right)$.
• Express $\\sin$/$\cos$ of multiple angles via complex expansions.
• Loci and regions in the Argand diagram (circles, lines, half-planes).
• Exponential form $re^{i\\theta}$ speeds products and powers.

Worked example 1 — de Moivre

$(\\cos\\theta+i\\sin\\theta)^5$.

Working

Direct application.

Worked example 2 — Roots

Cube roots of $8$.

Working

Include all $k=0,1,2$.

Worked example 3 — Practice point 1

de Moivre: $[r(\\\\cos\\\\theta+i\\\\sin\\\\theta)]^n=r^n(\\\\cos n\\\\theta+i\\\\sin n\\\\theta)$.

Working

Attempt without notes, then check the answers PDF.

Worked example 4 — Practice point 2

$n$th roots: $r^{1/n}\\\\left(\\\\cos\\\\frac{\\\\theta+2k\\\\pi}{n}+i\\\\sin\\\\frac{\\\\theta+2k\\\\pi}{n}\\\\right)$.

Working

Attempt without notes, then check the answers PDF.

Scaffolded Learning Sheet and answers (PDFs)

This page is a short, searchable overview for revision and search engines. Use the Scaffolded Learning Sheet first, then the Answers PDF for a full written solution. The printable PDFs remain the main classroom resource.