Higher Engineering Mathematics, Sixth Edition

Cartesian and polar co-ordinates 119

Now try the following exercise

Exercise 53 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 dec-

imal places, in both degrees and in radians.

1. (3, 5) [(5.83, 59.04◦) or (5.83, 1.03rad)]


2. (6.18, 2.35)

[

( 6. 61 , 20. 82 ◦)or

( 6. 61 , 0 .36rad)

]

3. (−2, 4)

[

( 4. 47 , 116. 57 ◦)or

( 4. 47 , 2 .03rad)

]

4. (−5.4, 3.7)

[

( 6. 55 , 145. 58 ◦)or

( 6. 55 , 2 .54rad)

]

5. (−7,−3)

[

( 7. 62 , 203. 20 ◦)or

( 7. 62 , 3 .55rad)

]

6. (−2.4,−3.6)

[

( 4. 33 , 236. 31 ◦)or

( 4. 33 , 4 .12rad)

]

7. (5,−3)

[

( 5. 83 , 329. 04 ◦)or

( 5. 83 , 5 .74rad)

]

8. (9.6,−12.4)

[

( 15. 68 , 307. 75 ◦)or

( 15. 68 , 5 .37rad)

]

12.3 Changing from polar into

Cartesian co-ordinates

From the right-angled triangleOPQin Fig. 12.6.

cosθ=

x

r

and sinθ=

y

r

,from

trigonometric ratios

Hence x=rcosθ and y=rsinθ

y

y

(^0) x Q x
P
r

Figure 12.6
If lengthsrand angleθare known thenx=rcosθand
y=rsinθare the two formulae we need to change from
polar to Cartesian co-ordinates.
Problem 5. Change (4, 32◦) into Cartesian
co-ordinates.
A sketch showing the position (4, 32◦)isshownin
Fig. 12.7.
Now x=rcosθ=4cos32◦= 3. 39
and y=rsinθ=4sin32◦= 2. 12
y
y
0 x
x
r 54
 5328
Figure 12.7
Hence (4, 32◦) in polar co-ordinates corresponds to
(3.39, 2.12) in Cartesian co-ordinates.
Problem 6. Express (6, 137◦) in Cartesian
co-ordinates.
A sketch showing the position (6, 137◦)isshownin
Fig. 12.8.
x=rcosθ=6cos137◦=− 4. 388
which corresponds to lengthOAin Fig. 12.8.
y=rsinθ=6sin137◦= 4. 092
which corresponds to lengthABin Fig. 12.8.
B
A 0
y
x
r 56
 51378
Figure 12.8
Thus (6, 137◦) in polar co-ordinates corresponds to
(−4.388, 4.092) in Cartesian co-ordinates.