Higher Engineering Mathematics, Sixth Edition

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 ofx=rcosθandy=rsinθautomatically

produces the correct signs.)

Problem 7. Express (4.5, 5.16rad) in Cartesian

co-ordinates.

A sketch showing the position (4.5, 5.16rad) is shown

in Fig. 12.9.

x=rcosθ= 4 .5cos5. 16 = 1. 948

y

A

B

0 x

 5 5.16 rad

r 5 4.5

Figure 12.9

which corresponds to lengthOAin Fig. 12.9.

y=rsinθ= 4 .5sin5. 16 =− 4. 057

which corresponds to lengthABin 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 co-

ordinates as Cartesian co-ordinates, correct to

3 decimal places.

1. (5, 75◦) [(1.294, 4.830)]


2. (4.4, 1.12rad) [(1.917, 3.960)]


3. (7, 140◦)[(−5.362, 4.500)]


4. (3.6, 2.5rad) [(−2.884, 2.154)]


5. (10.8, 210◦)[(−9.353,−5.400)]
   6. (4, 4rad) [(−2.615,−3.207)]
   7. (1.5, 300◦) [(0.750,−1.299)]
   8. (6, 5.5rad) [(4.252,−4.233)]
   9. Figure 12.10 shows 5 equally spaced holes on
   an 80mm pitchcircle diameter. Calculatetheir
   co-ordinates relative to axes 0xand 0yin (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 .02mm]

y

O x

Figure 12.10

12.4 Use of Pol/Rec functions on calculators

Another name for Cartesian co-ordinates isrectangu-

larco-ordinates. Many scientific notation calculators

possessPolandRecfunctions. ‘Rec’ is an abbrevi-

ation 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