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 θ = α, where α?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
  • 

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

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 α?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, ?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

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 ?then convert to?x?and?y
• e.g.? If
• 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 ?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 ?such as in

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
  • 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!