Chapter 12

Cartesian and polar

co-ordinates

12.1 Introduction

There are two ways in which the position of a point in
a plane can be represented. These are

(a) byCartesian co-ordinates,i.e.(x,y), and

(b) bypolar co-ordinates,i.e.(r,θ), whereris a

‘radius’ from a fixed point andθis an angle from

a fixed point.

12.2 Changing from Cartesian into

polar co-ordinates

In Fig. 12.1, if lengthsx andyare known, then the
length ofrcan be obtained from Pythagoras’ theorem
(see Chapter 11) sinceOPQis a right-angled triangle.
Hencer^2 =(x^2 +y^2 )

from which,r=

√

x^2 +y^2

y

P

(^0) x Q x
r y

Figure 12.1
From trigonometric ratios (see Chapter 11),
tanθ=
y
x
from which θ=tan−^1
y
x
r=
√
x^2 +y^2 andθ=tan−^1
y
x
are the two formulae we
need to changefromCartesian to polar co-ordinates.The
angleθ, which may be expressed in degrees or radians,
mustalwaysbe measured from the positivex-axis, i.e.,
measured from the lineOQin Fig. 12.1. It is suggested
that when changing from Cartesian to polar co-ordinates
a diagram should always be sketched.
Problem 1. Change the Cartesian co-ordinates
(3, 4) into polar co-ordinates.
A diagram representing the point (3, 4) is shown in
Fig. 12.2.
P
4
3
y
0 x
r

Figure 12.2