Polar Coordinates (Edexcel A Level Further Maths): Revision Note

Intro to polar coordinates

What are polar coordinates?

• Polar coordinates are an alternative way (to Cartesian coordinates) to describe the position of a point in 2D (or 3D) space

• In 2D, the position of a point is described using an angle, θ and a distance, r

  • This is akin to “aiming in the right direction”, then “travelling so far in that direction”

• Polar coordinates generally make working with circles, spirals and similar shapes easier

  • (3D) polar coordinates are beyond the A level syllabus but they are used with objects based on spheres such as the planets in the solar system

How do I describe the position of a point using polar coordinates?

• Point P would be described by the coordinates (r, θ)

• θ is measured in radians, anti-clockwise from the initial line (equivalent to the positive x-axis)

  • Negative values of θ can be used (clockwise from the initial line)

• r is the (straight line) distance between the pole (origin) and point P

  • r is usually given as a function of θ, r = f(θ)

  • equations can be given implicitly too, e.g. r ² = f(θ)

• A half-line starts at the pole and extends outwards in the direction of θ

  • The equation of a haf-line will be of the form , where is a constant

  • The line represents positive values of r

  • Negative values of r are possible but are not included in Edexcel A level Further Mathematics

What is the connection between polar coordinates and Cartesian coordinates?

• 

• These results are not provided in the formulae booklet

  • they are easily derived from a sketch and basic trigonometry

• Be careful solving  so that θ locates point P in the correct quadrant

  • Always use a sketch to ensure θ is measured from the initial line

• Check the domain of θ to see if negative values are used

  • e.g.  0 ≤ θ < 2π  as opposed to -π ≤ θ < π

• This is very similar to the modulus-argument form of a complex number

  •  where  and 

How do I convert from polar coordinates to Cartesian coordinates?

To convert the point P(r, θ) to P(x, y)

• Find the x-coordinate using

• Find the y-coordinate using 

  • In both cases take care with which quadrant P lies in

    • A sketch is the easiest way to double check

How do I convert from Cartesian coordinates to Polar coordinates?

To convert the point P(x, y) to P(r, θ)

• Find r using Pythagoras’ theorem

• r will (generally) take the positive square root since it is a distance (from the pole)

  • (It is possible for r to be negative, depending on the nature of f(θ))

• Find θ by using a sketch in association with 

  • Use the sketch to ensure θ locates point P in the correct quadrant

  • There may be the need to add or subtract π  to get θ in the correct quadrant

Sketching curves in polar form

How do I sketch curves given in polar coordinates/polar form?

• Recognising common graphs and the style of their equations is important

• There are three basic equations to be familiar with

  •  is the equation of a half-line from the pole in the direction  radians anti-clockwise from the initial line

  •  is a circle, centre at the pole with radius 

  •  is a spiral, starting at the pole where is a positive constant

• Other common types of polar curve encountered are summarised in the diagram below

• The cardioid and one-loop limacon have a cusp at the pole

• For Rose Curves when n is even, half of the petals are where r > 0 and half of them are where r < 0

• For Rose Curves when n is odd, the petals are drawn twice – once when r > 0 and once when r < 0

  • (The positive and negative petals sit on top of each other)

  • Some graphing software will plot negative values of r with a dotted curve

How are horizontal and vertical lines described in polar coordinates?

• Straight lines have polar equations of the form 

  • For the horizontal line corresponding to ,  where k is odd

  • For the vertical line corresponding to ,  where k is an integer

• Diagonal lines are formed using other values of 

How do I plot curves given in polar coordinates/polar form?

For more unusual polar equations a table of r and θ values can be generated

• Using the table points, can be plotted and joined on polar graph paper

• Values of θ may be given, e.g.  every  radians

• Where they are not given, think about common multiples of π that suit the question

  • e.g.  if 3θ is involved in the question, or  may be suitable

• Use a calculator to find the corresponding values of r

  • Be accurate but using decimals here is fine

• It is usual for questions to only require the plotting of part of a polar curve e.g.  plotting within a domain of θ that completes a ‘loop’ e.g.  a restricted domain of θ that produces only positive values of r

• When practising problems and revising have some graphing software running so you can quickly check your sketches against an accurate diagram

Polar curve to Cartesian equation

How do I convert a polar equation to a Cartesian equation?

• For equations of the form r = f(θ) square both sides

  • Some questions may define r² rather than r

    • r² can then be replaced by x² + y²

• To eliminate θ, some manipulation and use of trigonometric identities may be needed

  • Aim to convert terms involving θ into either the form  or  then convert to x and y

• e.g.  If  then 

• Awkward powers of r may be involved but these can be manipulated into terms of r² too e.g.   

How do I convert a Cartesian equation to a polar equation?

• In general substitute  and  into the Cartesian equation and simplify/rearrange

• Trigonometric identities may be involved

• If you spot them, there may be some shortcuts e.g.  ‘hidden’ sums of  and  such as in 

Intersections of polar curves

How do I find the intersections of two curves given in polar form?

• This is essentially the same as solving simultaneous equations

  • The aim is to eliminate one of the variables (usually r) and solve for the other

  • Any previous skills used to eliminate variables may still be useful here

• The general approach is to write the two equations in the forms  and 

  • Then solve

  • If required, substitute θ into f(θ) or g(θ) to find r

• Be aware that polar curves are often given in the form 

  • Working with r² rather than r may be easier

• Skills beyond basic simultaneous equations include

  • using trigonometric identities and solving trigonometric equations

  • “squaring and adding” (this is a common technique)

    • this can produce very useful  and/or  terms!