Polar Coordinates summary

Key learning points

• $x=r\cos\theta$, $y=r\sin\theta$, $r^2=x^2+y^2$.
• Sketch using symmetry and strategic $\theta$ values.
• Polar area formulae appear on some syllabuses — use the form you are taught.
• Tangents via parametric derivatives with parameter $\theta$.

Worked example 1 — Convert

Convert $r=2\cos\theta$ to Cartesian.

Working

Use $r^2=2x$ after multiplying by $r$.

Worked example 2 — , ,

Use this method from the Polar Coordinates sheet: $x=r\cos\theta$, $y=r\sin\theta$, $r^2=x^2+y^2$.

Working

Each key learning point matches a question type on the printable worksheet.

Worked example 3 — Sketch using symmetry and strategic values

Use this method from the Polar Coordinates sheet: Sketch using symmetry and strategic $\theta$ values.

Working

Each key learning point matches a question type on the printable worksheet.

Worked example 4 — Polar area formulae appear on some syllabuses

Use this method from the Polar Coordinates sheet: Polar area formulae appear on some syllabuses — use the form you are taught.

Working

Each key learning point matches a question type on the printable worksheet.

Worked example 5 — Tangents via parametric derivatives with parameter

Use this method from the Polar Coordinates sheet: Tangents via parametric derivatives with parameter $\theta$.

Working

Each key learning point matches a question type on the printable worksheet.

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.