7.1.1 Polar Coordinates

Introduction to Polar Coordinates

Polar coordinates represent points in a plane using the distance from a fixed point (called the pole) and an angle measured from a fixed direction. This system is an alternative to Cartesian coordinates $(x, y)$ .

Converting Between Polar and Cartesian Coordinates

From Polar to Cartesian:

Given polar coordinates ( $r, \theta$ ):

x = r \cos \theta, \quad y = r \sin \theta

From Cartesian to Polar:

Given Cartesian coordinates $(x, y)$ :

r = \sqrt{x^2 + y^2}, \quad \theta = \tan^{-1}\left(\frac{y}{x}\right)

Sketching Polar Curves

To sketch polar curves $r = f(\theta)$

Generate a table of values for $\theta$ and calculate $r$
Plot the points $(r, \theta)$ in polar coordinates.
Connect the points smoothly, considering symmetry and periodicity.

Common Polar Curves

Circle: $r = a$

A circle of radius $a$ centered at the pole.
Symmetric about all axes.

Line: $r = p \sec(\alpha - \theta)$

Represents a straight line.
$\alpha$ is the angle of inclination of the line.

Cardioid: $r = a(1 \pm \cos \theta)$

A heart-shaped curve.
Symmetric about the polar axis ( $\theta = 0$ )

Rose Curve: $r = a \cos n\theta$ or $r = a \sin n\theta$

Petal-shaped curve.
$n$ determines the number of petals ( $n$ petals if $n$ is odd, $2n$ petals if $n$ is even).

Lemniscate: $r^2 = a^2 \cos 2\theta$

Figure-eight curve.
Symmetric about the polar axis.

Spiral of Archimedes: $r = k\theta$

A spiral curve.
Expands outward as $\theta$ increases.

Worked Examples

infoNote

Example 1: Convert Between Polar and Cartesian Coordinates

Step 1: Convert Polar to Cartesian:

Given $(r, \theta) = (4, \frac{\pi}{3})$

x = r \cos \theta = 4 \cos \frac{\pi}{3} = 4 \times \frac{1}{2} = :highlight[2]

y = r \sin \theta = 4 \sin \frac{\pi}{3} = 4 \times \frac{\sqrt{3}}{2} = :highlight[2\sqrt{3}]

Cartesian coordinates: $(x, y) = :success[(2, 2\sqrt{3})]$

Step 2: Convert Cartesian to Polar:

Given $(x, y) = (3, 3)$

r = \sqrt{x^2 + y^2} = \sqrt{3^2 + 3^2} = \sqrt{18} = :highlight[3\sqrt{2}]

\theta = \tan^{-1}\left(\frac{y}{x}\right) = \tan^{-1}(1) = :highlight[\frac{\pi}{4}]

Polar coordinates: $(r, \theta) = :success[(3\sqrt{2}, \frac{\pi}{4})]$

infoNote

Example 2: Sketch $r = 2 + 2\cos\theta$

Step 1: Understand the curve:

This is a cardioid, symmetric about the polar axis.

Step 2: Generate values for $\theta$ :

$\theta = 0: \, r = 2 + 2\cos 0 = :highlight[4]$
$\theta = \frac{\pi}{2}: \, r = 2 + 2\cos \frac{\pi}{2} = :highlight[2]$
$\theta = \pi: \, r = 2 + 2\cos \pi = :highlight[0]$
$\theta = \frac{3\pi}{2}: \, r = 2 + 2\cos \frac{3\pi}{2} = :highlight[2]$

Step 3: Sketch the curve:

Plot the points and connect smoothly, ensuring symmetry about $\theta = 0$

infoNote

Example 3: Sketch $r^2 = 9\cos 2\theta$

Step 1: Understand the curve:

This is a lemniscate, symmetric about the polar axis.

Step 2: Generate values for $\theta$ :

$\theta = 0: \, r^2 = 9\cos 0 = 9 \implies r = :highlight[\pm 3]$
$\theta = \frac{\pi}{4}: \, r^2 = 9\cos \frac{\pi}{2} = 0 \implies r = :highlight[0]$
$\theta = \frac{\pi}{2}: \, r^2 = 9\cos \pi = -9$ (not valid for real $r$ )

Step 3: Sketch the curve:

Plot points for valid values of r and ensure figure-eight symmetry.

Note Summary

infoNote

Common Mistakes:

Incorrect conversion between coordinates: Forgetting $r \geq 0$ or miscalculating trigonometric values can lead to errors.
Plotting without checking symmetry: Many polar curves are symmetric; neglecting this wastes effort and may produce inaccuracies.
Forgetting the square root in $r^2$ : For equations like $r^2 = f(\theta)$ , always take both positive and negative roots.
Mismanaging periodicity of $\theta$ : For trigonometric functions, $\theta$ often cycles every $2\pi$ , leading to repeated points.
Overlooking the domain of $\theta$ : Ensure $\theta$ is within its given range (e.g., $[0, 2\pi]$ ).

infoNote

Key Formulas:

Polar to Cartesian:

x = r \cos \theta, \quad y = r \sin \theta

Cartesian to Polar:

r = \sqrt{x^2 + y^2}, \quad \theta = \tan^{-1}\left(\frac{y}{x}\right)

Cardioid:

r = a(1 \pm \cos \theta), \quad r = a(1 \pm \sin \theta)

Lemniscate:

r^2 = a^2 \cos 2\theta, \quad r^2 = a^2 \sin 2\theta

Spiral of Archimedes:

r = k\theta

Polar Coordinates Simplified Revision Notes for A-Level Edexcel Further Maths Core Pure

7.1.1 Polar Coordinates

Introduction to Polar Coordinates

Converting Between Polar and Cartesian Coordinates

From Polar to Cartesian:

From Cartesian to Polar:

Sketching Polar Curves

Common Polar Curves

Circle: $r = a$

Line: $r = p \sec(\alpha - \theta)$

Cardioid: $r = a(1 \pm \cos \theta)$

Rose Curve: $r = a \cos n\theta$ or $r = a \sin n\theta$

Lemniscate: $r^2 = a^2 \cos 2\theta$

Spiral of Archimedes: $r = k\theta$

Worked Examples

Example 1: Convert Between Polar and Cartesian Coordinates

Example 2: Sketch $r = 2 + 2\cos\theta$

Example 3: Sketch $r^2 = 9\cos 2\theta$

Note Summary

Common Mistakes:

Key Formulas:

500K+ Students Use These Powerful Tools to Master Polar Coordinates For their A-Level Exams.

Other Revision Notes related to Polar Coordinates you should explore

Polar Coordinates Simplified Revision Notes for A-Level Edexcel Further Maths Core Pure

7.1.1 Polar Coordinates

Introduction to Polar Coordinates

Converting Between Polar and Cartesian Coordinates

From Polar to Cartesian:

From Cartesian to Polar:

Sketching Polar Curves

Common Polar Curves

Circle: r=ar = ar=a

Line: r=psec⁡(α−θ)r = p \sec(\alpha - \theta)r=psec(α−θ)

Cardioid: r=a(1±cos⁡θ)r = a(1 \pm \cos \theta)r=a(1±cosθ)

Rose Curve: r=acos⁡nθr = a \cos n\thetar=acosnθ or r=asin⁡nθr = a \sin n\thetar=asinnθ

Lemniscate: r2=a2cos⁡2θr^2 = a^2 \cos 2\thetar2=a2cos2θ

Spiral of Archimedes: r=kθr = k\thetar=kθ

Worked Examples

Example 1: Convert Between Polar and Cartesian Coordinates

Example 2: Sketch r=2+2cos⁡θr = 2 + 2\cos\thetar=2+2cosθ

Example 3**: Sketch** r2=9cos⁡2θr^2 = 9\cos 2\thetar2=9cos2θ

Note Summary

Common Mistakes:

Key Formulas:

500K+ Students Use These Powerful Tools to Master Polar Coordinates For their A-Level Exams.

Other Revision Notes related to Polar Coordinates you should explore

Circle: $r = a$

Line: $r = p \sec(\alpha - \theta)$

Cardioid: $r = a(1 \pm \cos \theta)$

Rose Curve: $r = a \cos n\theta$ or $r = a \sin n\theta$

Lemniscate: $r^2 = a^2 \cos 2\theta$

Spiral of Archimedes: $r = k\theta$

Example 2: Sketch $r = 2 + 2\cos\theta$

Example 3: Sketch $r^2 = 9\cos 2\theta$