Higher Engineering Mathematics

B

Geometry and trigonometry

13

Cartesian and polar co-ordinates

13.1 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.

13.2 Changing from Cartesian into

polar co-ordinates

In Fig. 13.1, if lengthsxandyare known, then the
length ofrcan be obtained from Pythagoras’ theo-
rem (see Chapter 12) sinceOPQis a right-angled
triangle. Hencer^2 =(x^2 +y^2 )

from which, r=

√

x^2 +y^2

Figure 13.1

From trigonometric ratios (see Chapter 12),

tanθ=

y

x

from which θ=tan−^1

y

x

r=

√

x^2 +y^2 andθ=tan−^1

y

x

are the two for-

mulae we need to change from Cartesian 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 lineOQ

in Fig. 13.1. 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. 13.2.

Figure 13.2

From Pythagoras’ theorem,r=

√

32 + 42 =5 (note

that−5 has no meaning in this context). By trigono-

metric ratios,θ=tan−^143 = 53. 13 ◦or 0.927 rad.

[note that 53. 13 ◦= 53. 13 ×(π/180) rad= 0 .927 rad]

Hence (3, 4) in Cartesian co-ordinates corres-

ponds to (5, 53.13◦) or (5, 0.927 rad) in polar

co-ordinates.

Problem 2. Express in polar co-ordinates the

position (−4, 3).

A diagram representing the point using the Cartesian

co-ordinates (−4, 3) is shown in Fig. 13.3.