What are polar coordinates?
Pick a fixed point O called the pole and a fixed ray from it called the initial line (like the positive x-axis). A point P is given by (r, θ): r = distance OP, θ = angle from the initial line to OP, measured anticlockwise.
The same point has many names: (3, 40°) = (3, 400°) = (−3, 220°). A negative r means "go the opposite way". Many courses use the principal form r ≥ 0, −π < θ ≤ π.
Converting between polar and Cartesian
Polar → Cartesian: x = r cosθ, y = r sinθ.
Cartesian → polar: r = √(x² + y²), tanθ = y/x. Always check the quadrant: for (−1, −1), tanθ = 1 but θ = 225° (or −135°), not 45°.
Equations convert too: x² + y² = 9 becomes r = 3; y = x becomes θ = 45°; x = 2 becomes r = 2 secθ.
Polar curves
- r = a: circle radius a centred at O. θ = α: a straight line through O.
- r = 2a cosθ: circle through O, centre (a, 0).
- r = a(1 + cosθ): cardioid (heart shape). r = a + b cosθ: limaçon (with an inner loop if b > a).
- r = a cos nθ: rose; n petals if n is odd, 2n petals if n is even.
- r = aθ: spiral of Archimedes.
To sketch: make a table of θ (0°, 30°, 60° …) and r, look for symmetry (cosθ only → symmetric about the initial line), and note where r = 0 (the curve passes through the pole).
Calculus with polar curves
With x = r cosθ and y = r sinθ where r = f(θ):
- dr/dθ tells if the point moves away from (positive) or towards (negative) the pole.
- Slope of the curve: dy/dx = (dy/dθ) ÷ (dx/dθ) = (r′ sinθ + r cosθ) ÷ (r′ cosθ − r sinθ).
- Area swept from θ = α to β: A = ½∫ r² dθ. It comes from adding thin sectors of area ½r²Δθ.
Try it
Draw a polar grid on paper with a compass and protractor. Plot r = 2(1 + cosθ) for θ = 0°, 60°, 90°, 120°, 180° and join the dots. Check your heart shape against the 3D scene.
Key formulas and definitions
- x = r cosθ, y = r sinθ
- r² = x² + y², tanθ = y/x (choose θ by quadrant)
- (r, θ) = (r, θ + 360°) = (−r, θ + 180°)
- Slope: dy/dx = (r′sinθ + r cosθ) / (r′cosθ − r sinθ)
- Area: A = ½ ∫_α^β r² dθ
- Rose r = a cos nθ: n petals (n odd), 2n petals (n even)
Worked examples
1. Convert (4, 60°) to Cartesian.
x = 4cos60° = 2, y = 4sin60° = 2√3 ≈ 3.46. Point (2, 3.46).
2. Convert (−3, 3) to polar with r > 0.
r = √(9 + 9) = 3√2 ≈ 4.24. tanθ = −1 and the point is in quadrant II, so θ = 135°. Answer (3√2, 135°).
3. Write x² + y² = 6x in polar form.
r² = 6r cosθ → r = 6cosθ. A circle through O with centre (3, 0).
4. How many petals does r = 3sin4θ have? r = 5cos3θ?
n = 4 is even → 8 petals. n = 3 is odd → 3 petals.
5. Find the area inside r = 2 for θ from 0 to π/2.
A = ½∫₀^{π/2} 4 dθ = 2 × π/2 = π. (A quarter of a circle of radius 2: ¼ × 4π = π. ✓)
6. Find the area enclosed by the cardioid r = 1 + cosθ.
A = ½∫₀^{2π}(1 + cosθ)² dθ = ½∫(1 + 2cosθ + cos²θ)dθ = ½(2π + 0 + π) = 3π/2.
7. For r = 2θ, is the point moving away from the pole at θ = 1? Find dy/dx there.
dr/dθ = 2 > 0, so it moves away. dy/dx = (2sin1 + 2cos1)/(2cos1 − 2sin1) = (0.841 + 0.540)/(0.540 − 0.841) ≈ −4.59.
Common mistakes
- Using θ = tan⁻¹(y/x) without checking the quadrant.
- Measuring θ clockwise instead of anticlockwise.
- Forgetting the ½ in the area formula ½∫r²dθ, or squaring only part of r.
- Saying r = a cos 2θ has 2 petals – it has 4 (n even → 2n).