Geometry and trigonometry

13
Cartesian and polar co-ordinates
B
 y
13.1 Introduction r = x 2 + y2 and θ = tan−1 are the two for-
x
mulae we need to change from Cartesian to polar
There are two ways in which the position of a point co-ordinates. The angle θ, which may be expressed
in a plane can be represented. These are in degrees or radians, must always be measured from
(a) by Cartesian co-ordinates, i.e. (x, y), and the positive x-axis, i.e. measured from the line OQ
in Fig. 13.1. It is suggested that when changing from
(b) by polar co-ordinates, i.e. (r, θ), where r is a Cartesian to polar co-ordinates a diagram should
‘radius’ from a fixed point and θ is an angle from always be sketched.
a fixed point.

Problem 1. Change the Cartesian co-ordinates

13.2 Changing from Cartesian into (3, 4) into polar co-ordinates.
polar co-ordinates
In Fig. 13.1, if lengths x and y are known, then the A diagram representing the point (3, 4) is shown in
length of r can be obtained from Pythagoras’ theo- Fig. 13.2.
rem (see Chapter 12) since OPQ is a right-angled
triangle. Hence r 2 = (x 2 + y2 )

*
from which, r = x2 + y2

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−1 43 = 53.13◦ or 0.927 rad.
[note that 53.13◦ = 53.13 × (π/180) rad = 0.927 rad]
Figure 13.1
Hence (3, 4) in Cartesian co-ordinates corres-
ponds to (5, 53.13◦ ) or (5, 0.927 rad) in polar
From trigonometric ratios (see Chapter 12), co-ordinates.
y
tan θ =
x Problem 2. Express in polar co-ordinates the
position (−4, 3).

y
from which θ = tan−1 A diagram representing the point using the Cartesian
x
co-ordinates (−4, 3) is shown in Fig. 13.3.
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  √
r = 2 2 + 52 = 29 = 5.385 correct to
√
From Pythagoras’ theorem, r = 42 + 32 = 5. 3 decimal places
By trigonometric ratios, α = tan−1 43 = 36.87◦ or 5
0.644 rad. α = tan−1 = 68.20◦ or 1.190 rad
2
Hence θ = 180◦ − 36.87◦ = 143.13◦ or
Hence θ = 360◦ − 68.20◦ = 291.80◦ or
θ = π − 0.644 = 2.498 rad.
Hence the position of point P in polar co-ordinate θ = 2π − 1.190 = 5.093 rad
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 Fig. 13.4.

r = 52 + 122 = 13
12
and α = tan−1
5
= 67.38◦ or 1.176 rad Figure 13.5

Hence θ = 180◦ + 67.38◦ = 247.38◦ or

Thus (2, −5) in Cartesian co-ordinates corres-
θ = π + 1.176 = 4.318 rad
ponds to (5.385, 291.80◦ ) or (5.385, 5.093 rad) in
polar co-ordinates.

Now try the following exercise.

Exercise 61 Further problems on changing

from Cartesian into polar co-ordinates
In Problems 1 to 8, express the given Carte-
sian 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)]
Figure 13.4  
(6.61, 20.82◦ ) or
2. (6.18, 2.35) (6.61, 0.36 rad)
Thus (−5, −12) in Cartesian co-ordinates corres-
ponds to (13, 247.38◦ ) or (13, 4.318 rad) in polar  
co-ordinates. (4.47, 116.57◦ ) or
3. (−2, 4) (4.47, 2.03 rad)
 CARTESIAN AND POLAR CO-ORDINATES 135
 
(6.55, 145.58◦ ) or
4. (−5.4, 3.7) (6.55, 2.54 rad)
 
(7.62, 203.20◦ ) or
5. (−7, −3) (7.62, 3.55 rad)
 
(4.33, 236.31◦ ) or
6. (−2.4, −3.6) (4.33, 4.12 rad) B
  Figure 13.7
(5.83, 329.04◦ ) or
7. (5, −3) (5.83, 5.74 rad) Hence (4, 32◦ ) in polar co-ordinates corresponds
  to (3.39, 2.12) in Cartesian co-ordinates.
(15.68, 307.75◦ ) or
8. (9.6, −12.4) (15.68, 5.37 rad)
Problem 6. Express (6, 137◦ ) in Cartesian
co-ordinates.

13.3 Changing from polar into A sketch showing the position (6, 137◦ ) is shown in
Cartesian co-ordinates Fig. 13.8.
From the right-angled triangle OPQ in Fig. 13.6. x = r cos θ = 6 cos 137◦ = −4.388
x y which corresponds to length OA in Fig. 13.8.
cos θ = and sin θ = , from
r r
trigonometric ratios y = r sin θ = 6 sin 137◦ = 4.092

Hence x = r cos θ and y = r sin θ which corresponds to length AB in Fig. 13.8.

Figure 13.8

Thus (6, 137◦ ) in polar co-ordinates corresponds

Figure 13.6 to (−4.388, 4.092) in Cartesian co-ordinates.
If lengths r and angle θ are known then x = r cos θ (Note that when changing from polar to Cartesian
and y = r sin θ are the two formulae we need to co-ordinates it is not quite so essential to draw
change from polar to Cartesian co-ordinates. a sketch. Use of x = r cos θ and y = r sin θ auto-
matically produces the correct signs.)
Problem 5. Change (4, 32◦ ) into Cartesian
co-ordinates. Problem 7. Express (4.5, 5.16 rad) in Cartesian
co-ordinates.
A sketch showing the position (4, 32◦ ) is shown in
Fig. 13.7. A sketch showing the position (4.5, 5.16 rad) is
shown in Fig. 13.9.
Now x = r cos θ = 4 cos 32◦
= 3.39
and ◦
y = r sin θ = 4 sin 32 = 2.12 x = r cos θ = 4.5 cos 5.16 = 1.948
136 GEOMETRY AND TRIGONOMETRY

3. (7, 140◦ ) [(−5.362, 4.500)]

4. (3.6, 2.5 rad) [(−2.884, 2.154)]
5. (10.8, 210◦ ) [(−9.353, −5.400)]
6. (4, 4 rad) [(−2.615, −3.207)]
7. (1.5, 300◦ ) [(0.750, −1.299)]
8. (6, 5.5 rad) [(4.252, −4.233)]
9. Figure 13.10 shows 5 equally spaced holes
Figure 13.9 on an 80 mm pitch circle diameter. Calculate
their co-ordinates relative to axes 0x and 0y
which corresponds to length OA in Fig. 13.9. in (a) polar form, (b) Cartesian form.
y = r sin θ = 4.5 sin 5.16 = −4.057 Calculate also the shortest distance between
the centres of two adjacent holes.
which corresponds to length AB in Fig. 13.9.
Thus (1.948, −4.057) in Cartesian co-ordinates y
corresponds to (4.5, 5.16 rad) in polar
co-ordinates.

13.4 Use of R → P and P → R

functions on calculators
Another name for Cartesian co-ordinates is rect-
angular co-ordinates. Many scientific notation cal- O x
culators possess R → P and P → R functions. The
R is the first letter of the word rectangular and the P is
the first letter of the word polar. Check the operation
manual for your particular calculator to determine
how to use these two functions. They make changing
from Cartesian to polar co-ordinates, and vice-versa,
so much quicker and easier.

Now try the following exercise.

Figure 13.10
Exercise 62 Further problems on changing
polar into Cartesian co-ordinates [(a) 40∠18◦ , 40∠90◦ , 40∠162◦ ,
In Problems 1 to 8, express the given polar co- 40∠234◦ , 40∠306◦ ,
ordinates as Cartesian co-ordinates, correct to (b) (38.04 + j12.36), (0 + j40),
3 decimal places. (−38.04 + j12.36),
1. (5, 75◦ ) [(1.294, 4.830)] (−23.51 − j32.36), (23.51 − j32.36)
2. (4.4, 1.12 rad) [(1.917, 3.960)] 47.02 mm]