Higher Engineering Mathematics, Sixth Edition

220 Higher Engineering Mathematics

(a)

(b)

Real axis

4

0

308

Imaginary

axis

x

jy

 Real axis

7 1458

x

jy

Figure 20.7

Using trigonometric ratios,x=4cos30◦= 3. 464

andy=4sin30◦= 2 .000.

Hence 4∠ 30 ◦= 3. 464 +j 2. 000

(b) 7∠ 145 ◦is shown in Fig. 20.7(b) and lies in the

third quadrant.

Angleα= 180 ◦− 145 ◦= 35 ◦

Hence x=7cos35◦= 5. 734

and y=7sin35◦= 4. 015

Hence 7∠− 145 ◦=− 5. 734 −j 4. 015

Alternatively

7 ∠− 145 ◦=7cos(− 145 ◦)+j7sin(− 145 ◦)

=− 5. 734 −j 4. 015

Calculator

Using the ‘Pol’and‘Rec’ functions on a calculator

enables changing from Cartesian to polar and vice-versa

to be achieved more quickly.

Since complex numbers are used with vectors and

with electrical engineering a.c. theory, it is essential that

the calculator can be used quickly and accurately.

20.7 Multiplication and division in polar form

IfZ 1 =r 1 ∠θ 1 andZ 2 =r 2 ∠θ 2 then:

(i) Z 1 Z 2 =r 1 r 2 ∠(θ 1 +θ 2 )and

(ii)

Z 1

Z 2

=

r 1

r 2

∠(θ 1 −θ 2 )

Problem 12. Determine, in polar form:

(a) 8∠ 25 ◦× 4 ∠ 60 ◦

(b) 3∠ 16 ◦× 5 ∠− 44 ◦× 2 ∠ 80 ◦

(a) 8∠ 25 ◦× 4 ∠ 60 ◦=( 8 × 4 )∠( 25 ◦+ 60 ◦)= 32 ∠ 85 ◦

(b) 3∠ 16 ◦× 5 ∠− 44 ◦× 2 ∠ 80 ◦

=( 3 × 5 × 2 )∠[16◦+(− 44 ◦)+ 80 ◦]= 30 ∠ 52 ◦

Problem 13. Evaluate in polar form

(a)

16 ∠ 75 ◦

2 ∠ 15 ◦

(b)

10 ∠

π

4

× 12 ∠

π

2

6 ∠−

π

3

(a)

16 ∠ 75 ◦

2 ∠ 15 ◦

=

16

2

∠( 75 ◦− 15 ◦)= 8 ∠ 60 ◦

(b)

10 ∠

π

4

× 12 ∠

π

2

6 ∠−

π

3

=

10 × 12

6

∠

(π

4

+

π

2

−

(

−

π

3

))

= 20 ∠

13 π

12

or 20 ∠−

11 π

12

or

20 ∠ 195 ◦or 20 ∠− 165 ◦

Problem 14. Evaluate, in polar form

2 ∠ 30 ◦+ 5 ∠− 45 ◦− 4 ∠ 120 ◦.

Addition and subtraction in polar form is not possible

directly. Each complex number has to be converted into

cartesian form first.

2 ∠ 30 ◦= 2 (cos 30◦+jsin30◦)

=2cos30◦+j2sin30◦= 1. 732 +j 1. 000

5 ∠− 45 ◦= 5 (cos(− 45 ◦)+jsin(− 45 ◦))

=5cos(− 45 ◦)+j5sin(− 45 ◦)

= 3. 536 −j 3. 536

4 ∠ 120 ◦= 4 (cos 120◦+jsin120◦)

=4cos120◦+j4sin120◦

=− 2. 000 +j 3. 464

Hence 2∠ 30 ◦+ 5 ∠− 45 ◦− 4 ∠ 120 ◦