Higher Engineering Mathematics

COMPLEX NUMBERS 255

E

Modulus, |Z|=r=

√

(2^2 + 32 )=

√
13 or 3. 606 ,
correct to 3 decimal places.

Argument,argZ=θ=tan−^132

= 56. 31 ◦or 56 ◦ 19 ′

In polar form, 2+j3 is written as 3. 606 ∠ 56 ◦ 19 ′.

Problem 10. Express the following complex

numbers in polar form:

(a) 3+j 4 (b)− 3 +j 4

(c)− 3 −j4 (d) 3−j 4

(a) 3+j4 is shown in Fig. 23.6 and lies in the first

quadrant.

− 2 − 1 12

−j

−j 2

−j 3

−j 4

j

j 2

j 3

(− 3 +j4) j 4 (3+j4)

(− 3 −j4) (3−j4)

− 3 3

r

r

r

Real axis

Imaginary

axis

r

θ

αα

α

Figure 23.6

Modulus, r=

√

(3^2 + 42 )=5 and argument

θ=arctan^43 = 53. 13 ◦= 53 ◦ 8 ′.

Hence 3 +j 4 = 5 ∠ 53 ◦ 8 ′

(b) − 3 +j4 is shown in Fig. 23.6 and lies in the
second quadrant.

Modulus, r=5 and angle α= 53 ◦ 8 ′, from

part (a).

Argument= 180 ◦− 53 ◦ 8 ′= 126 ◦ 52 ′ (i.e. the

argument must be measured from the positive

real axis).

Hence− 3 +j 4 = 5 ∠ 126 ◦ 52 ′

(c)− 3 −j4 is shown in Fig. 23.6 and lies in the

third quadrant.

Modulus,r=5 andα= 53 ◦ 8 ′,asabove.

Hence the argument= 180 ◦+ 53 ◦ 8 ′= 233 ◦ 8 ′,

which is the same as− 126 ◦ 52 ′.

Hence(− 3 −j 4 )= 5 ∠ 233 ◦ 8 ′or 5∠− 126 ◦ 52 ′

(By convention theprincipal valueis normally

used, i.e. the numerically least value, such that

−π<θ<π).

(d) 3−j4 is shown in Fig. 23.6 and lies in the fourth

quadrant.

Modulus,r=5 and angleα= 53 ◦ 8 ′,asabove.

Hence( 3 −j 4 )= 5 ∠− 53 ◦ 8 ′

Problem 11. Convert (a) 4∠ 30 ◦(b) 7∠− 145 ◦

intoa+jbform, correct to 4 significant figures.

(a) 4∠ 30 ◦is shown in Fig. 23.7(a) and lies in the

first quadrant.

4

30 °^

(^0) x Real axis
jy
Real axis
(b)
7
145 °
x
jy
(a)
Imaginary
axis
α
Figure 23.7
Using trigonometric ratios,x=4 cos 30◦= 3. 464
andy=4 sin 30◦= 2 .000.
Hence 4∠ 30 ◦= 3. 464 +j 2. 000
(b) 7∠ 145 ◦is shown in Fig. 23.7(b) and lies in the
third quadrant.
Angleα= 180 ◦− 145 ◦= 35 ◦