120 Higher Engineering Mathematics
(Note that when changing from polar to Cartesian
co-ordinates it is not quite so essential to draw
a sketch. Use of x = r cos θ and y =r sin θ automatically
produces the correct signs.)
Problem 7. Express (4.5, 5.16 rad) in Cartesian
co-ordinates.
A sketch showing the position (4.5, 5.16 rad) is shown
in Fig. 12.9.
x = r cos θ = 4.5 cos 5.16 = 1.948
y
A
B
0
x
5 5.16 rad
r 5 4.5
Figure 12.9
which corresponds to length OA in Fig. 12.9.
y = r sin θ = 4.5 sin 5.16 = −4.057
which corresponds to length AB in Fig. 12.9.
Thus (1.948, −4.057) in Cartesian co-ordinates
corresponds to (4.5, 5.16 rad) in polar co-ordinates.
Now try the following exercise
Exercise 54 Further problems on changing
polar into Cartesian co-ordinates
In Problems 1 to 8, express the given polar coordinates as Cartesian co-ordinates, correct to
3 decimal places.
1. (5, 75 ◦ )
[(1.294, 4.830)]
2. (4.4, 1.12 rad)
[(1.917, 3.960)]
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 12.10 shows 5 equally spaced holes on
an 80 mm pitch circle diameter. Calculate their
co-ordinates relative to axes 0x and 0y in (a)
polar form, (b) Cartesian form.
Calculate also the shortest distance between
the centres of two adjacent holes.
[(a) 40∠18 ◦ , 40∠90 ◦ , 40∠162 ◦ ,
40∠234 ◦ , 40∠306 ◦ ,
(b) (38.04 + j12.36), (0 + j40),
(−38.04 + j12.36), (−23.51 − j32.36),
(23.51 − j32.36)
47.02 mm]
y
x
O
Figure 12.10
12.4 Use of Pol/Rec functions on
calculators
Another name for Cartesian co-ordinates is rectangular co-ordinates. Many scientific notation calculators
possess Pol and Rec functions. ‘Rec’ is an abbreviation of ‘rectangular’ (i.e., Cartesian) and ‘Pol’ is an
abbreviation of ‘polar’. Check the operation manual for
your particular calculator to determine how to use these
(Note that when changing from polar to Cartesian
co-ordinates it is not quite so essential to draw
a sketch. Use of x = r cos θ and y =r sin θ automatically
produces the correct signs.)
Problem 7. Express (4.5, 5.16 rad) in Cartesian
co-ordinates.
A sketch showing the position (4.5, 5.16 rad) is shown
in Fig. 12.9.
x = r cos θ = 4.5 cos 5.16 = 1.948
y
A
B
0
x
5 5.16 rad
r 5 4.5
Figure 12.9
which corresponds to length OA in Fig. 12.9.
y = r sin θ = 4.5 sin 5.16 = −4.057
which corresponds to length AB in Fig. 12.9.
Thus (1.948, −4.057) in Cartesian co-ordinates
corresponds to (4.5, 5.16 rad) in polar co-ordinates.
Now try the following exercise
Exercise 54 Further problems on changing
polar into Cartesian co-ordinates
In Problems 1 to 8, express the given polar coordinates as Cartesian co-ordinates, correct to
3 decimal places.
1. (5, 75 ◦ )
[(1.294, 4.830)]
2. (4.4, 1.12 rad)
[(1.917, 3.960)]
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 12.10 shows 5 equally spaced holes on
an 80 mm pitch circle diameter. Calculate their
co-ordinates relative to axes 0x and 0y in (a)
polar form, (b) Cartesian form.
Calculate also the shortest distance between
the centres of two adjacent holes.
[(a) 40∠18 ◦ , 40∠90 ◦ , 40∠162 ◦ ,
40∠234 ◦ , 40∠306 ◦ ,
(b) (38.04 + j12.36), (0 + j40),
(−38.04 + j12.36), (−23.51 − j32.36),
(23.51 − j32.36)
47.02 mm]
y
x
O
Figure 12.10
12.4 Use of Pol/Rec functions on
calculators
Another name for Cartesian co-ordinates is rectangular co-ordinates. Many scientific notation calculators
possess Pol and Rec functions. ‘Rec’ is an abbreviation of ‘rectangular’ (i.e., Cartesian) and ‘Pol’ is an
abbreviation of ‘polar’. Check the operation manual for
your particular calculator to determine how to use these