1.5 Polar Coordinates

2026 Syllabus Objectives

  1. Understand the relations between Cartesian and polar coordinates, and convert equations of curves from Cartesian to polar form and vice versa.

    • Note: The convention r≥0r \geq 0 will be used.

  2. Sketch simple polar curves, for 0≤θ<2π0 \leq \theta < 2\pi or −π<θ≤π-\pi < \theta \leq \pi or a subset of either of these intervals.

    • Note: Detailed plotting of curves will not be required, but sketches will generally be expected to show significant features, such as symmetry, coordinates of intersections with the initial line, the form of the curve at the pole and least/greatest values of rr.

  3. Recall the formula 12∫r2 dθ\frac{1}{2} \int r^2 \, d\theta for the area of a sector, and use this formula in simple cases.

📐 The Polar Coordinate System

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What are Polar Coordinates?

Rather than using Cartesian coordinates (x,y)(x, y) to describe the position of a point, we use polar coordinates (r,θ)(r, \theta).

Key Components:

  • Pole (O): The origin in the polar coordinate system
  • Initial Line: A line in a fixed direction (conventionally the xx-axis) from which angles are measured
  • rr: The distance of point PP from the pole
  • θ\theta: The angle that line OPOP makes with the initial line

🔑 Sign Conventions

  • If θ\theta is measured anticlockwise from the initial line → θ\theta is positive
  • If θ\theta is measured clockwise from the initial line → θ\theta is negative
  • The initial line (positive xx-axis) is where θ=0\theta = 0
  • The positive yy-axis is where θ=π2\theta = \frac{\pi}{2}

Important Convention: Throughout this topic, we use r≥0r \geq 0.

🔄 Converting Between Cartesian and Polar Coordinates

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From Polar to Cartesian

Using a right-angled triangle, we can derive the fundamental relationships:

x=rcos⁡θx = r\cos\theta

y=rsin⁡θy = r\sin\theta

x2+y2=r2x^2 + y^2 = r^2

From Cartesian to Polar

To convert from Cartesian to polar form:

r2=x2+y2r^2 = x^2 + y^2

tan⁡θ=yx\tan\theta = \frac{y}{x}

Polar Functions

Instead of writing functions in the form y=f(x)y = f(x), we work with functions in the form:

r=f(θ)r = f(\theta)

The parametric polar form is:

x=f(θ)cos⁡θ,y=f(θ)sin⁡θx = f(\theta)\cos\theta, \quad y = f(\theta)\sin\theta

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