Basic Engineering Mathematics, Fifth Edition

Cartesian and polar co-ordinates 215

Hence, (3, 4) in Cartesian co-ordinates corre-
sponds 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 Figure 24.3.

P y

3

4

O x

r

 

Figure 24.3

From Pythagoras’ theorem,r=

√

42 + 32 = 5

By trigonometric ratios, α=tan−^1

3

4

= 36. 87 ◦or

0.644 rad

Hence,θ= 180 ◦− 36. 87 ◦= 143. 13 ◦

or θ=π− 0. 644 = 2 .498 rad

Hence,the position of pointPin polar co-ordinate
form is (5, 143.13◦) or (5, 2.498rad).

Problem 3. Express (−5,−12) in polar

co-ordinates

A sketch showing the position (−5,−12) is shown in
Figure 24.4.

y

P

12

5

O x

r





Figure 24.4

r=

√

52 + 122 =13 andα=tan−^1

12

5

= 67. 38 ◦

or 1.176 rad

Hence,θ= 180 ◦+ 67. 38 ◦= 247. 38 ◦

orθ=π+ 1. 176 = 4 .318 rad.

Thus,(−5,−12) in Cartesian co-ordinates corre-

sponds to (13, 247.38◦) or (13, 4.318rad) in polar

co-ordinates.

Problem 4. Express (2,−5) in polar co-ordinates

A sketch showing the position (2,−5) is shown in

Figure 24.5.

y

O x

5

2

r

P





Figure 24.5

r=

√

22 + 52 =

√

29 = 5 .385, correct to 3 decimal

places

α=tan−^1

5

2

= 68. 20 ◦or 1.190 rad

Hence,θ= 360 ◦− 68. 20 ◦= 291. 80 ◦

orθ= 2 π− 1. 190 = 5 .093 rad.

Thus,(2,−5) in Cartesian co-ordinates corresponds

to (5.385, 291.80◦) or (5.385, 5.093rad) in polar co-

ordinates.

Now try the following Practice Exercise

PracticeExercise 94 Changing from

Cartesianto polar co-ordinates (answers

on page 350)

In problems 1 to 8, express the given Cartesian co-

ordinates as polar co-ordinates,correct to 2 decimal

places, in both degrees and radians.

1. (3, 5) 2. (6.18, 2.35)


2. (−2, 4) 4. (−5.4, 3.7)


3. (−7,−3) 6. (−2.4,−3.6)


4. (5,−3) 8. (9.6,−12.4)