Polar co-ordinates - Learning the conversion between Cartesian and polar systems

Q: What are polar coordinates (r,θ) used for?

They describe the position of a point in the plane uniquely by giving a distance r from the pole along a radius and an angle θ from the initial line.

Q: How do you convert polar coordinates to Cartesian coordinates?

Use x = r cosθ and y = r sinθ, taking care with the angle so the correct quadrant is obtained.

Q: How do you convert Cartesian coordinates to polar coordinates?

Use r² = x² + y² and tanθ = y/x, and apply the angle carefully because incorrect θ can result for points not in the first quadrant.

Oct 13, 2026

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Explains how to specify the position of a point in the plane using polar coordinates (r,θ) and how this relates to Cartesian coordinates (x,y). Defines the pole and initial line, the conventions for measuring θ in radians and for restricting θ and r, and emphasizes the importance of careful angle handling. Covers conversions both ways: x=r cosθ and y=r sinθ, plus r²=x²+y² and tanθ=y/x, and shows how to convert polar curve equations into Cartesian form while identifying mismatching curve portions.

LanguageEnglish

CategoryGeneral

FormatPDF

Pages11

Reading Time4 min

AccessFree to download

Introduction

The polar co-ordinate system

Conversion between the two systems

Some equations of curves in polar co-ordinates

Polar co-ordinates mc-TY-polar-2009-1 The (x, y) co-ordinates of a point in the plane are called its Cartesian co-ordinates. But thereis another way to specify the position of a point, and that is to use polar co-ordinates (r,θ) . In this unit we explain how to convert from Cartesian co-ordinates to polar co-ordinates, and back again. In order to master the techniques explained here it is vital that you undertake plenty of practice exercises so that they become second nature. After reading this text, and/or viewing the video tutorial on this topic, you should be able to: • plot a point on a diagram when given its polar co-ordinates; • convert the polar co-ordinates of a point into Cartesian co-ordinates; • convert the Cartesian co-ordinates of a point into polar co-ordinates; • convert the polar co-ordinate equation of a curve into a Cartesian co-ordinate equation; • identify new portions of a curve which satisfy the Cartesian co-ordinate equation but which do not satisfy the polar co-ordinate equation. Contents 1. Introduction 2 2. The polar co-ordinate system 2 3. Conversion between the two systems 3 4. Some equations of curves in polar co-ordinates 7 [www.mathcentre.ac.uk 1](www.mathcentre.ac.uk 1) 􀀍c mathcentre 2009 1. Introduction The position of a point in a plane can be described using Cartesian, or rectangular, co-ordinates. In the Cartesian system we ﬁx an origin, a point O, and then take two lines which are perpendicular to each other passing through O . These two lines are the axes. The x-axis is horizontal, and the y-axis is vertical. The position of any point P in the plane is then described by noting its displacement from O in the direction of the x-axis, the x co-ordinate, and also its displacement from O in the direction of the y-axis, the y co-ordinate. |
y | | P (x, y)
| | --- | --- | --- | | | | | | O | x | | Notice that we talk about the displacement, and not the distance. This is because, in this system, we need to use the directions indicated by the arrows to help us to be able to specify a point uniquely. Thus moving up from the origin is a positive displacement, whereas moving down from the origin is a negative displacement. Similarly, moving to the right from the origin gives a positive displacement, and moving to the left gives a negative displacement. The polar co-ordinate system is another system for specifying a point in the plane uniquely. 2. The polar co-ordinate system For polar co-ordinates we take an origin (or pole) O, and a ﬁxed initial line OA. A point P is then described by specifying a distance r, the distance O to P along the radius direction, and the angle θ we have had to turn from the initial line to be looking along the radius direction. The point then has polar co-ordinates (r,θ) . P (r, θ) O A [www.mathcentre.ac.uk 2](www.mathcentre.ac.uk 2) 􀀍c mathcentre 2009 There are similar conventions about directions with polar co-ordinates as there are with Cartesian co-ordinates. The conventions are: 1. θ is measured in radians; 2. θ is measured from the initial line, and is measured as positive in an anticlockwise direction from the initial line, or negative in a clockwise direction from the initial line; θ positive O θ negative A 3. the values of θ are restricted to −π < θ ≤ π; 4. the values of r are restricted to r > 0 (strictly speaking, the origin O itself is excluded from the plane when the points are described using polar co-ordinates, although we will sometimes disregard this when drawing graphs) . Key Point The polar co-ordinates of a point P with respect to a pole O and initial line OA are the numbers (r,θ), where r is the length of the line OP and θ is the angle between OA and OP . We always take r > 0, and we always take −π < θ ≤ π . 3. Conversion between the two systems Consider a point P with Cartesian co-ordinates (x, y) and polar co-ordinates (r,θ) . We shall identify the pole with the or

FAQs

What are polar coordinates (r,θ) used for?

They describe the position of a point in the plane uniquely by giving a distance r from the pole along a radius and an angle θ from the initial line.

How do you convert polar coordinates to Cartesian coordinates?

Use x = r cosθ and y = r sinθ, taking care with the angle so the correct quadrant is obtained.

How do you convert Cartesian coordinates to polar coordinates?

Use r² = x² + y² and tanθ = y/x, and apply the angle carefully because incorrect θ can result for points not in the first quadrant.

Oct 13, 2026

Upload by asba