ENG1090
School of Mathematics
Monash University
Semester 1 - 2022
Applied Class 4 - Week 5
1 Define the modulus and the argument of a complex number z = x + yi.
2 State De Moivre’s theorem and give an example of it.
3 Convert the following complex
numbers from polar form to Cartesian form
√
√ or vice versa:
−3π
5π
(a) 2 cis( 6 )
(b) 2 cis( 4 )
(c) −1 + i
(d) − 3 − 3i
4 Carry out the following operations in polar form and then simplify your answers to Cartesian form if
possible:
√
π
32 cis( 10
) × 45 cis( −π
2
−π
11π
3 )
(a) 2 cis
× 3 cis
(b)
(c)
3π
8π
10
10
cis( 4 )
16 cis( 9 )
c) (2 + 3i)/(-1 + i) = ((2 + 3i)(-1 - i))/
5 Find |z|, the principal argument Arg(z), and z for the following complex numbers:
2 + 3i
1
(b) z = (−1 − i)(−2 + 3i)
(c) z =
(a) z =
−2 − i
−1 + i
((-1 + i)(-1 - i)) = (1 - i), Arg() =
atan2(-1, 1)
6 Three electrical circuits having complex impedance z1 = 10 + 30i ohms, z2 = 20 + 0i ohms, and z3 =
1 − 20i ohms are connected in parallel across a 200 volt supply (i.e., the complex voltage is V = 200 + 0i
volts). If the total complex current I amps in the network is given by
I=
V
1
1
1
1
, where =
+
+ .
z
z
z1 z2 z3
Find the complex current I and then find the actual current flowing in the network (i.e., |I| amps) and its
phase angle (i.e. Arg(I) radians).
7 Convert the following complex numbers from exponential form to Cartesian form or vice versa:
(a) 3eπi/2
(b) (2 − i)e(3+4i)x
(c) −3 − 4i
(d) cos(2t) + i sin(2t)
8 Express each of the following as a sum of terms, each involving a complex (or real) exponential:
(a) sin(5x)
(b) cos(2xi)
(c) sin(i)
9 Apply De Moivre’s theorem to find each of the following in Cartesian form:
3
(8 cis( 4π
1
18 ))
(a)
(b)
π 4
(2 cis( 3 ))
(1 + i)3
1+i
10 Use De Moivre’s theorem to verify that √ 21/6 is a root of the equation z 6 + 2z 3 + 2 = 0.
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Additional Questions
In addition to your applied class problem set, you can practice and prepare for the unit, at your own
time, using the following questions.
1.1
Polar form of a complex number
11 Express the following polar complex numbers in Cartesian form:
(a) cis(0)
(b) 2cis(π)
(c) cis( 2π
(d) cis( 4π
3 )
3 )
−π
−3π
−π
(e) cis( 2 )
(f) cis( 2 )
(g) 2cis( 6 )
12 With the aid of an Argand diagram, express the following Cartesian complex numbers in polar form
(in √
terms of
√ their principal
√argument):
(a) 2 + 2i√ (b) 1 − 3i (c) −i
(d) −4
(e) 5 (f) − 3 − i
(g) 2i
13 Carry out the following operations in polar form (in terms of their principal argument) and then convert
your answers to Cartesian form (in terms of their principal argument):
π π cis( −π
3 )
(a) cis
× cis
(b)
6
3
cis( π6 )
14 By expressing i in polar form, show that the effect of multiplying any complex number z = r cis(θ) by
i is to leave the modulus unchanged and to rotate the radius vector through π2 (90◦ ) in the anticlockwise
sense.
15 Find |z|, the principal argument Arg z, and z for the following complex numbers and in each case
indicate z, |z|, Arg z, and z on an Argand diagram:
√
1 1
1−i
(a) z = 3 + i
(b) z = − i
(c) z =
2 3
i
1.2
Exponential form of a complex number
16 Express each of the following in Cartesian form:
(a) exi
(b) 4eπi
(c) 2e5πi/6
(d) e2+4πi
3−(2πi)/3
(h) e
17 Find:
(a) Im{e2−xi }
(b) Re{e(−2+3i)t }
18 Express the following in exponential form:
(a) −1
(b) 4i
(c) 1 + i
(d) 2 − 2i
(e) i exi
(f) (2−3i)e−πi/2
(c) Re{(3+i)e2xi }
(e) −1 +
√
3i
(d) Im
(g) e−4+3xi
1
e(1+2i)t
1 + 2i
(f) −2 + i
19 Express each of the following as a sum of terms, each involving a complex (or real) exponential:
(a) cos(2θ)
(b) sin(−3y)
(c) cos(3t/4)
(d) sin(3ti)
(e) cos(i)
1.3
Logarithm of a complex number
Using Euler’s formula eiθ = cis(θ) = cos(θ) + i sin(θ) we can define all complex exponentials using
properties of exponentials:
ex+iy = ex eiy = ex cis(y) = ex (cos(θ) + i sin(θ)) = ex cos(θ) + iex sin(θ)
If we write z = ex+iy we see that ex is simply the modulus of z, and y is the argument of z. Therefore we
can define the complex logarithm function as follows:
ln(z) = ln(|z|) + iArg(z)
2
Here we use the principal argument Arg(z) ∈ (−π, π], and ln(0) is not defined.
20 Express each of the following
in√exact
form x + iy:
3
1
(a) ln(5 + 12i)
(b) ln − 2 − 2 i
1.4
De Moivre’s theorem and roots of complex numbers
21 Apply De Moivre’s theorem
to find each of the following in polar form and then convert to Cartesian:
√
7
6
3
−
i)
(a) (2cis( 5π
))
(b)
(
18
√
3−i
is a root of the equation z 4 − 2z 2 + 4 = 0.
22 Use De Moivre’s theorem to verify that √
2
Hint
z 4 − 2z 2 + 4 = (z 2 − 1)2 + 3 = 0.
23 Find all roots of the equation z 5 = 1 + i in polar form (with principal arguments in radians) and draw
them on an Argand diagram.
24 Find the following in polar form and then convert them to Cartesian form:
(a) the cube roots of 8i
(b) the square roots of 15 + 8i*
(c) all roots of the equation z 6 + 64 = 0
*Hint: evaluate r and θ with a calculator then convert to Cartesian form and check whether the apparent
answer is correct.
√
25 Find the the square roots of 2 3 − 2i in polar form (with principal arguments in radians).
26 Instead of using De Moivre’s theorerm, use the Cartesian form w = x√+ yi to solve for the appropriate
real and imaginary parts x & y for which w2 = (x2 − y 2 ) + i(2xy) = 2 3 − 2i.
27 Find cos(π/12) in exact form. (Hint: use the previous two questions.)
√
3 + i and z2 = 3 cis(−π/3).
28 Consider the two
complex
numbers
z
=
−
1
√
(a) Write z1 = − 3 + i in polar form, in terms of its principal argument.
(b) Use your answer to evaluate z1 /|z1 | in polar form, and then convert that into Cartesian form.
(c) Using their poler forms, determine z1 z2 and z2 /z1 in terms of their principal argument.
(d) Determine (z2 )−2 in polar form, in terms of its principal argument.
(e) Determine all distinct values of (z2 )1/3 in polar form, in terms of their principal argument.
Need more questions to practice? Please check the following questions from the suggested textbook.
From James G, (2015) Modern Engineering Mathematics (5th ed.)
Complex algebra: §3.2.5, p196, questions 1, 6, 7, 8(a), 10, 12, 15, 17.
Polar form of a complex number: §3.2.8, p204, questions 24, 26, 27.
§3.2.11, p209, questions 28, 29(a)-(b).
De Moivre’s theorem and roots of complex numbers: §3.3.3, p217, questions 34, 37 - 39
The book can be accessed for free via the Monash Library reading list.
https://ebookcentral-proquest-com.ezproxy.lib.monash.edu.au/lib/monash/
reader.action?docID=5176272
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