trigonometry

134 GEOMETRY AND TRIGONOMETRY
Problem 4. Express (2, −5) in polar
co-ordinates.
A sketch showing the position (2, −5) is shown in
Fig. 13.5.
Figure 13.3
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).
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
Problem 3. Express (−5, −12) in polar
co-ordinates.
A sketch showing the position (−5, −12) is shown
in Fig. 13.4.
√
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
Figure 13.5
Thus (2, −5) in Cartesian co-ordinates corresponds
to (5.385, 291.80 ◦ ) or (5.385, 5.093 rad) in
polar co-ordinates.
Now try the following exercise.
Figure 13.4
Thus (−5, −12) in Cartesian co-ordinates corresponds
to (13, 247.38 ◦ ) or (13, 4.318 rad) in polar
co-ordinates.
Exercise 61 Further problems on changing
from Cartesian into polar co-ordinates
In Problems 1 to 8, express the given Cartesian
co-ordinates as polar co-ordinates, correct
to 2 decimal places, in both degrees and in
radians.
1. (3, 5) [(5.83, 59.04 ◦ ) or (5.83, 1.03 rad)]
[
(6.61, 20.82
2. (6.18, 2.35)
◦ ]
)or
(6.61, 0.36 rad)
[
(4.47, 116.57
3. (−2, 4)
◦ ]
)or
(4.47, 2.03 rad)