Edexcel A Level Further Maths: Core Pure:复习笔记7.1.1 Polar Coordinates

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 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

gEfG7l0u_7-1-1-edx-a-fm-fig1-polar

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

Exam Tip

  • Ensure your calculator is in radians mode when working with polar coordinates
  • Note how polar coordinates (r, θ) are given in the order r then θ, even thoughr = f(θ)

Worked Example

(a) Convert the polar coordinates  to Cartesian coordinates.

7-1-1-edex-alevel-fm-polar-we1-soltn-a

(b) Convert the Cartesian coordinates  to polar coordinates.

7-1-1-edex-alevel-fm-polar-we1-soltn-b

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 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

7-1-1-edx-a-fm-fig2-curvetypes

 

  • 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

DJKhN2k~_7-1-1-edx-a-fm-fig1-blank-grid

  • 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

Worked Example

On separate diagrams, sketch the graph of the following polar curves

(i)
(ii)
(iii)

7-1-1-edex-alevel-fm-polar-we2-soltn-i

7-1-1-edex-alevel-fm-polar-we2-soltn-ii

7-1-1-edex-alevel-fm-polar-we2-soltn-iii

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  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

Exam Tip

  • When converting a polar equation to a Cartesian equation, unless required by the question, do not worry about rearranging into the form
  • Make any obvious simplifications but otherwise an implicit Cartesian form is fine

Worked Example

(a) Find a Cartesian equation of the polar curve .7-1-1-edex-alevel-fm-polar-we3-soltn-a

(b) Find a polar equation in the form  for the ellipse .

7-1-1-edex-alevel-fm-polar-we3-soltn-b

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
    • 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!

Exam Tip

  • Calculators are unlikely to be able to solve these types of simultaneous equations directly
  • They may have a ‘solve’ mode you can use once the equation has been reduced to a single variable
  • However look out for when questions require exact answers

Worked Example

The diagram below shows a sketch of the polar graphs of    for .

7-1-1-sp4-intersections-we-qu

a) Find the smallest positive values of θ for which each curve crosses the pole.

ZDN-0FxF_7-1-1-edex-alevel-fm-polar-we4-soltn-a

b) For , find the points of intersection between the two curves for .

eQpNpgz7_7-1-1-edex-alevel-fm-polar-we4-soltn-b

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