Cartesian and polar co-ordinates 215
Hence, (3, 4) in Cartesian co-ordinates corresponds 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.
y
P
3
4
x
O
r
␪
␣
Figure 24.3
From Pythagoras’ theorem, r =
√
4 2 + 3 2 = 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 point P in polar co-ordinate
form is (5, 143.13 ◦ ) or (5, 2.498 rad).
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
x
O
r
␪
␣
Figure 24.4
r =
√
5 2 + 12 2 = 13 and α = tan −1 12
5
= 67.38 ◦
or 1.176 rad
Hence, θ = 180
◦
+ 67.38
◦
= 247.38
◦
or θ = π + 1.176 = 4.318 rad.
Thus, (−5, −12) in Cartesian co-ordinates corresponds to (13, 247.38 ◦ ) or (13, 4.318 rad) 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
x
O
5
2
r
P
␪
␣
Figure 24.5
r =
√
2 2 + 5 2 =
√
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 ◦
or θ = 2π − 1.190 = 5.093 rad.
Thus, (2, −5) in Cartesian co-ordinates corresponds
to (5.385, 291.80
◦ ) or (5.385, 5.093 rad) in polar coordinates.
Now try the following Practice Exercise
Practice Exercise 94 Changing from
Cartesian to polar co-ordinates (answers
on page 350)
In problems 1 to 8, express the given Cartesian coordinates as polar co-ordinates, correct to 2 decimal
places, in both degrees and radians.
1. (3, 5)
2. (6.18, 2.35)
3. (−2, 4)
4. (−5.4, 3.7)
5. (−7, −3)
6. (−2.4, −3.6)
7. (5, −3)
8. (9.6, −12.4)