DE MOIVRE’S THEOREM 263

E

but the arguments,θ, are different. It is shown in
Problem 3 that arguments are symmetrically spaced
on an Argand diagram and are (360/n)◦apart, where
nis the number of the roots required. Thus if one of
the solutions to the cube root of a complex number
is, say, 5∠ 20 ◦, the other two roots are symmetrically
spaced (360/3)◦, i.e. 120◦from this root and the
three roots are 5∠ 20 ◦,5∠ 140 ◦and 5∠ 260 ◦.

Problem 4. Find the roots of [(5+j3)]

1

(^2) in
rectangular form, correct to 4 significant figures.
(5+j3)=
√
34 ∠ 30. 96 ◦
Applying De Moivre’s theorem:
(5+j3)
1
(^2) =
√
34
1
(^2) ∠^12 × 30. 96 ◦
= 2. 415 ∠ 15. 48 ◦or 2. 415 ∠ 15 ◦ 29 ′
The second root may be obtained as shown above,
i.e. having the same modulus but displaced (360/2)◦
from the first root.
Thus, (5+j3)
1
(^2) = 2. 415 ∠(15. 48 ◦+ 180 ◦)
= 2. 415 ∠ 195. 48 ◦
In rectangular form:
2. 415 ∠ 15. 48 ◦= 2 .415 cos 15. 48 ◦
+j 2 .415 sin 15. 48 ◦
= 2. 327 +j 0. 6446
and 2. 415 ∠ 195. 48 ◦= 2 .415 cos 195. 48 ◦
+j 2 .415 sin 195. 48 ◦
=− 2. 327 −j 0. 6446
Hence [(5+j3)]
1
(^2) = 2. 415 ∠ 15. 48 ◦and
2. 415 ∠ 195. 48 ◦or
±( 2. 327 + j 0 .6446).
Problem 5. Express the roots of
(− 14 +j3)
− 2
(^5) in polar form.
(− 14 +j3)=
√
205 ∠ 167. 905 ◦
(− 14 +j3)
− 2
(^5) =
√
205
− 2
(^5) ∠
[(
−
2
5
)
× 167. 905 ◦
]
= 0. 3449 ∠− 67. 164 ◦
or 0. 3449 ∠− 67 ◦ 10 ′
There are five roots to this complex number,
(
x
− 2
(^5) =
1
x
2
5

1
√ (^5) x 2
)
The roots are symmetrically displaced from one
another (360/5)◦, i.e. 72◦apart round an Argand
diagram.
Thus the required roots are 0. 3449 ∠− 67 ◦ 10 ′,
0. 3449 ∠ 4 ◦ 50 ′, 0. 3449 ∠ 76 ◦ 50 ′, 0. 3449 ∠ 148 ◦ 50 ′
and 0. 3449 ∠ 220 ◦ 50 ′.
Now try the following exercise.
Exercise 106 Further problems on the
roots of complex numbers
In Problems 1 to 3 determine the two square roots
of the given complex numbers in cartesian form
and show the results on an Argand diagram.

1. (a) 1+j(b)j [
   (a)±(1. 099 +j 0 .455)
   (b)±(0. 707 +j 0 .707)

]

2. (a) 3−j4 (b)− 1 −[j 2
   (a)±(2−j)
   (b)±(0. 786 −j 1 .272)

]

3. (a) 7∠ 60 ◦(b) 12∠

3 π

[^2

(a)±(2. 291 +j 1 .323)

(b)±(− 2. 449 +j 2 .449)

]

In Problems 4 to 7, determine the moduli and

arguments of the complex roots.

4. (3+j4)

1

(^3) [
Moduli 1.710, arguments 17◦ 43 ′,
137 ◦ 43 ′ and 257◦ 43 ′
]