Polar Coordinates Topic Review summary

Key learning points

• $x=r\\cos\\theta$, $y=r\\sin\\theta$, $r^2=x^2+y^2$.
• Sketch by plotting key $\\theta$ values and using symmetry.
• Tangents: $\\frac{dy}{dx}=\\frac{\\frac{dy}{d\\theta}}{\\frac{dx}{d\\theta}}$.
• Area of a sector-like region $\\frac12\\int_{\\alpha}^{\\beta} r^2\\,d\\theta$.
• Common curves: cardioid, rose, limaçon — recognise $r=a(1\\pm\\cos\\theta)$ etc.

Worked example 1 — Convert

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

Working

Multiply by $r$.

Worked example 2 — Area

Area element in polar form.

Working

Integrate between limits.

Worked example 3 — Practice point 1

$x=r\\\\cos\\\\theta$, $y=r\\\\sin\\\\theta$, $r^2=x^2+y^2$.

Working

Attempt without notes, then check the answers PDF.

Worked example 4 — Practice point 2

Sketch by plotting key $\\\\theta$ values and using symmetry.

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.