# 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

#### 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.

(b) Convert the Cartesian coordinates  to polar coordinates.

### 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

• 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

#### Worked Example

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

(i)
(ii)
(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 .

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

### 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 .

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

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