Easy To Understand Complex Number

Easy to understand

COMPLEX NUMBERS

POLITEKNIK MUKAH SARAWAK

MUHAMMAD HAZIM FAHMI BIN MOHD GHAFAR 0|Page
AFZAN BINTI OMAR

MUHAMMAD HAZIM FAHMI BIN MOHD GHAFAR
AFZAN BINTI OMAR

Easy To Understand Complex Number

Published by POLITEKNIK MUKAH

Km. 7.5, Jalan Oya
96400 Mukah
Sarawak
Tel: 084-874001
Fax: 084-874005

Website: http://www.pmu.edu.my

Copyright © 2021

All rights reserved. No part of this publication can be reproduced, stored in retrieval system or transmitted
in any forms or by any means, electronic or mechanical including photocopying, recording or otherwise
without the prior permission of the copyright owner.

Cover by Hazim Fahmi Ghafar

Perpustakaan Negara Malaysia

Editor:
Muhammad Hazim Fahmi Bin Mohd Ghafar
Afzan Binti Omar

ISBN: [Ebook ISBN number] (ebook)

PREFACE

Easy To Understand Complex Numbers is e-book and exercise book for
mathematics problem solving design for polytechnic students and all interested
in learning interesting technique to solving mathematics problem especially
complex numbers.

This highlights various methods to solve higher order thinking and challenging
question often encountered. The purpose of this e-book is to improve a student
ability to solve challenging complex number questions. The worked examples
serve as a step by step guide while the carefully design question provide the
practice. This book therefore, provide opportunity to discover problem solving
methods through video tutorial and calculator tutorial so students can try out the
methods introduce.

The questions are categorized according to the types of methods to be applied.
Challenging Examination Paper in Tutorial at the end of the book will provide with
additional practice and assessment on a student progress on mastery of the
method. Teachers and students will find the detail worked solution especially
helpful in understanding the complex numbers. Student will easy to find and
upload this e-Book interesting useful as a resource to score well in DBM10013:
Engineering Mathematics 1 as well other major examinations.

01 – 13 |Page
14 – 22 |Page

23 – 29 |Page
30 – 41 |Page

gOHD GHAFAR
5|Page

1.0 Introduction

TOPIC 1: The point of this nomplex number e-book is to supply a brief think about
and self assessment program for understudies who wish to gotten to be
THE CONCEPT OF more recognizable with complex numbers. In this topic, you should able to
COMPLEX NUMBER understand the concept of complex numbers, perform operations on
1 complex numbers such as addition and substraction of complex number,
multiplication and division of complex numbers. Understand the graphical
representation of complex numbers through an Argand diagram and
complex number in other forms such as polar form and exponential form.

GOAL2 1 GGOOAALL 1 1.1 Concept of a Complex Numbers
1122..
1.1 Concept of a Complex
Number A complex number is a number comprising real part and imaginary part.
It can be written in the form + , where a and b are real numbers,
GOAL 2 and i is the imaginary unit with the property = √−1 and 2 = −1.

1.2 Equality of Complex 1
Number

GOAL 3

1.1 Quadratic Equation With
Complex Number

GOAL 4 The simplying of

1.3 Conjugate of Complex ✓ If n is an even number, the value of is 1 or -1
Number ✓ If n is an odd number, the value of is 1 or -1

2|Page

A complex number has a combination of real part and imaginary part. So
based on table below, so every value of real part and imaginary part are
also complex number.

C

Keep In Mind..

Real part can ONLY operate (plus & minus) with real part.
Imaginary part can ONLY operate (plus & minus) with
imaginary part.

..NEVER MIX THEM WHEN DOING PLUS & MINUS..

EXAMPLE 1
Express = 2 + √−36 in the terms of + .

z = 2 + −36 Why it become √−1?To give
= 2 + 36 −1 some value of square root.
= 2 + 6i
Why it become ? √−1 =
You can check
your answer by Value of 6 come from √36 = 6
using scientific
calculator

30||PPagaeg e

EXAMPLE 2

Express the following in terms of + ∶

a. 2 − 16 b. −9 + −49

Casio 570MS: = 2 − 16 −1 = 9 −1 + 49 −1
= 2 − 4i = 3i + 7i
(-) = 10i
Press MODE 2
d. 16 + − 4
c. −81 −36 81 9

2 + √− 36 = 81 −1 49 −1 = 4 + 4 −1
= = 9i 7i 99
= 63i2
Monitor Display.. = 63(−1) = 4+2i
= −63 93
2 + √−36

2

Press SHIFT =

2 + √−36

6. EXERCISE 1
Simplify each of the following:
So, the answer is +

a. −36 + 2

[ Ans: 2 + 6 ]

b. 6 − −17

[ Ans: 6 − 4.123 ]
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c. −9 −16

d. 3 − 8 − −64 [ Ans: −12]
27
[ Ans: -23 − 8 ]

EXAMPLE 3

Simplify each of the following:

i6

= (i2 )3

= (−1)3 ➢ ( ) ÷ =
= −1 ➢ ൫ ൯ (− ).
➢ (− ) × (− ) × (− ) = −

to an odd power is i13
always equal to , just
need to determine the = (i2 )6 i ➢ ( )6 13 ÷ 2 = 4
sign. = (−1)6 i ➢ ( 2) (−1). 1
=i ➢ ℎ ∙ ? 12 ∙ 1, ℎ 13
= ±

5|Page

Simplify each of the following: ➢ ( )3 3 ÷ 2 = 3
➢ ( 2) (−1). 1
i3 ➢ ℎ ∙ ? 2 ∙ 1, ℎ 3
➢ (−1) × (−1) × (−1) = −1
= (i2 )1 i
= (−1) i
= −i

Simplify each of the following:

to an even power is i5 ➢ ( )5 5 ÷ 2 = 2.5
always equal to , just ➢ ( 2) (−1). 1
need to determine the = (i2 )2 i ➢ ℎ ∙ ? 5
sign. = (−1)2 i

= ± =i

EXERCISE 2 i75
Simplify each of the following:
b.
i88

a.

[ Ans: 1] [ Ans: − ]

i304 i405

c. d.

[ Ans: 1] [ Ans: ]

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i16 f. i7

e. [ Ans: 1]

i34 i33 [ Ans: − ]
[ Ans: ]
g. h.

Casio Classwiz 570EX [ Ans: −1]
:

EXAMPLE 4 ✓ Even power is always
equal to
Simplify each of the following:
✓ Odd power is always
i6 + 8i13 equal to
= (i2 )3 + 8(i2 )6 i
= (−1)3 + 8(−1)6 i
= −1+ 8i

So, the answer is − +

7|Page

EXERCISE 3

Simplify the power of for the following below:

a. i5 + i7 b. 3i3 − 9i8

[ Ans: 0] [ Ans: −9 − 3 ]

c. 25 + 4i8 + 2 d. 4i8 − 5i3 − −36

[ Ans: 11] [ Ans: 4 − ]

Simplify Odd Power & Even Power GGOOAALL 2 1.2 Equality of Complex Number
of Complex Number 1122..
https://www.youtube.com/watch?v 1.2
=DgVG1OJNqTI

PLEASE SCAN QR ± = ±
CODE:

Simplify of Odd
Power & Even Power

Video Tutorial

8|Page

EXAMPLE 5

(2x + 9) − (6 y − 3)i = 3 + 9i

a=c bi = di (2 + 9) − (6 − 3) = 3 + 9

is real parts the first 2x +9 = 3 −6y +3 = 9
complex number. 2x = 3−9 −6y =9−3
2x = −6 −6y = 6
is real part the second x = −6 y= 6
complex number. 2 −6
x = −3 y = −1
is imaginary part the first
complex number. EXERCISE 4

is imaginary part the Solve the following below:
second complex number.
a. x + yi = 6 + 8i
b. −2x + 4yi = (3x −1) + 5i

[ Ans: = 6 , = 8] [ Ans: = 1 , = 5]

c. 5x + 6yi = (6 − 2x) + (3y + 4)i 54

[ Ans: = 6 , = 43]
7

9|Page

Casio Classwiz 570EX GOAL 32 1.3 Quadratic Equation With Complex Number
:
12.
Complex

The quadratic equation in the complex number system,

1 .3 2 + + = 0

Where , and are real number and ≠ 0

− ± √ 2 − 4
= 2

Then,

EXAMPLE 6

Solve the following equation using the quadratic formula.
formula:

x2 − 4x + 5 = 0

a =1 b =-4 c = 5

x = −(−4)  (−4)2 − 4(1)(5)
2(1)

= 4  16 − 20
2

= 4  −4 −1
2

= 4  2i
2

= 2 ± ∴ = 2 + = 2 −

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EXERCISE 5

Solve each the following equation using the quadratic formula

a. 2 2 − + 4 b. 2 2 − 5 + 10

So, the answer is,

[ Ans: 0.25 ± 1.392 ] [ Ans: 1.25 ± 7.416 ]

c. 2 − 6 + 12 d. 3 2 + 6 + 18

[ Ans: 3 ± 1.732 ] [ Ans: −1 + 2.236 ]

GOAL 23 1.3 Conjugate of Complex Number
12.
Complex

Complex Conjugates

+ −

1.4Same real parts and imaginary parts have
opposite signs

*
*

*

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Double Conjugates

∗ = ( + )( − )

= 2 + 2

The product of a conjugate pair is positive real
number

EXAMPLE 7

Find the conjugate of = 6 + 25 and it’s ∗

Z * = 6 − 25i CHANGE THE SIGN IN THE MIDDLE

∗
(complex number) (conjugate)

ZZ * = (6 + 25i)(6 − 25i) +
= 62 + 252
= 36 + 625
= 661

EXERCISE 6
Find the conjugate and compute ∗ of the following complex numbers.

a. = 16 b. = −8

[ Ans: = −16 , ∗ = 256] [ Ans: = 8 , ∗ = 64]

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c. = 5 + 2 d. = 3 − 11

[ Ans: = 5 − 2 , ∗ = 29] [ Ans: = 3 + 11 , ∗ = 130]
e. = −4 + 12 f. = −6 − 8

[ Ans: = −4 − 12 , ∗ = 160] [ Ans: = −6 + 8 , ∗ = 100]

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2.0 GOAL 12 2.1 Addition and Subtraction Complex Number
12.
0
Complex Must +
TOPIC 2:
( ± ) ± ( ± ) = ( ± ) + ( ± )
OPERATION OF
COMPLEX NUMBER 1.5

Real Parts Imaginary

GOAL 1 ➢
➢
2.1 Addition and Subtraction
of Complex Number EXAMPLE 1

Number
GOAL 2

2.2 Multiplication and Division Calculate the following expression.
of Complex Number
a. 2 + 6i − i
Number = 2 + (6 −1)i
= 2 + 5i

b. − 3i − 2 + 8i + 4i
= −2 + (−3 + 8 + 4)i
= −2 + 9i

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EXAMPLE 2
Solve the following expression.

Given that 1 = −1 + 3 . and 2 = 5 − 8 . Solve that:

a. z1 + z2

( ± ) ± ( ± ) = ( ± ) + ( ± )

(−1+ 3i) + (5 − 8i) = (−1+ 5) + (3 − 8)i
= 4 + (−5i)
= 4 − 5i

b. z1 − z2

( ± ) ± ( ± ) = ( ± ) + ( ± )

(−1+ 3i) − (5 − 8i)
−1+ 3i − 5 + 8i = (−1− 5) + (3 + 8)i

= −6 +11i

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EXERCISE 1 b. ( + ) − (− + )

Solve the following expression.
a. (− + ) + ( + )

[ Ans: −1 + 10 ] [ Ans: 7 + 2 ]

c. (− − ) + (− + ) d. (− + ) − (− + )

Addition Of Complex Number [ Ans: −21 − 2 ] [ Ans: 12 − 4 ]
https://youtu.be/Mw1pWQHlwVg

e. ( + ) + ( − ) f. [(− − ) − ( + )]

PLEASE SCAN QR [ Ans: 13 + 2 ] [ Ans: −27 − 18 ]
CODE:
g. (− + ) + ( − ) h. [(− + ) − (− −
Addition Of Complex
Number Video Tutorial

)]

[ Ans: 11 − 19 ] [ Ans: 64 + 224 ]
3 5 5

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GOAL 22 2.2 Multiplication and Division Complex
12. Number

Comp2lex MULTIPLICATION OF COMPLEX

( ± )( 1 ± ) = ± ± ± 2

1.6 3 4

= ± ( ± ) ± (−1)

**Be sure to replace with (-1) 2 = −1
and proceed with simplification.
= ( ± (− ) ± ( ± )
Answer should be + form.

(-) 3 ( (-) EXAMPLE 3

2+9 ) Solve each of the following expression.

= a. − 3i(−2 + 9i)
= 6i − 27i2
= 6i − 27(−1)
= 27 + 6i

b. (3 + 2i)(4 − 6i)
= 12 −18i + 8i −12i2
= 12 + (−18i + 8)i −12(−1)
= (12 +12) + (−18i + 8i)
= 24 −10i

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EXERCISE 3

−3 (−2 + 9 ) Solve the following expression.
(27 + 6 )
a. (− )(− ) b. (− )( )
So, the answer is:

27 + 6

[ Ans: −6] [ Ans: = 24]

c. ( + ) d. ( − )

[Ans: −42 + 12 ] [ Ans: 28 − 12 ]

e.(− + )( + ) f.(− − )( + )

[ Ans: −41 + ] [ Ans: −1 − 34 ]

g.( − )( − ) h. (− − )( − )

[ Ans: −11 − 78 ] [ Ans: −57 − 51 ]

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DIVISION OF COMPLEX

+ Conjugate
+

( + ) ( − ) ( + )( − )
= ( + ) × ( − ) ( + )

( − ) + ( + )
= ( 2 + 2)

2- 4 EXAMPLE 4
Solve each of the following expression.
i 3+

5i =

a. 2 − 4i Conjugate
3 + 5i

2 − 4 7 11 = 2 − 4i  3 − 5i +
3 + 5 − 17 − 17 3 + 5i 3 − 5i So, −

= (2 − 4i)(3 − 5i) +
(32 + 52 )
So, this answer is:
7 11 = (6 −10i −12i + 20i2 ) Refer Multiplication of Complex sub topic.
34
− 17 − 17
= (6 − 20) + (−10i −12i)
34

= −14 − 22i
34

= − 14 − 22 i Write the fraction
34 34 in lowest term

= − 7 − 11i
17 17

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b. 4 + 2 Make the denominators
3 + 5i 3i the same:

= 3i(4) + 2(3 + 5i) numerators
3i(3 + 5i) 3i(3 + 5i) denominators

= 12i + 6 +10i
9i +15i2

= 6 + 22i  −15 − 9i
−15 + 9i −15 − 9i

= (6 + 22i)(−15 − 9i)
(−15)2 + (9)2

= (−90 − 54i − 330i −198i2 )
306

= (−90 +198) + (−54i − 330i)
306

= 108 − 384i
306

= 108 − 384i
306 306

= 6 − 64 i Write the fraction
17 51 in lowest term

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EXERCISE 4

Calculate the equation in form + .

+
a. + b. −

[Ans: 37 + 3 ] [Ans: 2 + 3 ]
13
13 13 13

− − +
c. − d. −

[Ans: 1 − 4 ] [Ans: 1 + 7 ]
10 5
5 5

( + )( − ) d. ( + ) +
d. − −

[Ans: 25 + 5 ] [Ans: 11 + 9 ]
6
13 13 2

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3.0 GOAL 12 3.1 Draw an Argand Diagram
12. Number
0
Complex nuCmombeprlecxan be represented geometrically on an Argand
TOPIC 3: Diagram.

GRAPHICAL 1.7
REPRESENTATION OF
COMPLEX NUMBER

GOAL 1
3.1 Draw an Argand Diagram
GOAL 2
3.2 Modulus and Argument

EXAMPLE 1
Sketch Argand diagram the complex number below:
−

0

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EXERCISE 1
Sketch argand diagram the complex number below:

a. +

Sketch the diagram before b. 2−
find the modulus and
argument. This is because
you need to calculate your
argument based on which
quadrant you are in.

c.− +

d.− −

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GOAL 22 3.2 Modulus and Argument
12. Number

Complex

| |, = √ +

1.8 , = −
( )



−


ii ° +
iii



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EXAMPLE 2

Calculate the modulus and the argument for the complex
number below:

. 3 + 6 So, = 3 , = 6

3 + 6 ENG , = √ 2 + 2
OPTN
= √32 + 62
Then, 1 = √45
⊳ ⦟ = 6.708
3√5⦟63.43494882
3√5 = 6.708 , = −1

So, this answer is:
6.708 ⦟ 63.44° = −1 6
3

= −1(2)

= 63.44°

Since + is in the first quadrant

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. −4 − 6 So, = −4 , = −6

, = √ 2 + 2

= √(−4)2 + (−6)2

= √52
= 7.211

, = −1


= −1 −6
−4
= −1(23)

= 56.31°

Modulus & Argument of Argand Since − − is in the third
Diagram quadrant
https://www.youtube.com/watch?v
=ODnQ3Jv4sdE = 180° + 56.31°
= 236.31°
PLEASE SCAN QR
CODE:

Argand Diagram Video
Tutorial

28 | P a g e

EXERCISE 2

Represent each of the following in an Argand Diagram:

a. = + b. = −

ANS: ANS:
= 5.099 = 10.817
= 78.69° = −56.31°@303.69°

c. = − + c. = − −

ANS: ANS:
= 15.811 = 6.708
= 124.70° = −116.57°@243.43°

29 | P a g e

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GOAL 12 4.1 Complex Number in Others Form
12.
Number

4.0 Complex

0 1.9

TOPIC 4: Cartesian Form

COMPLEX NUMBER IN Polar Form
OTHERS FORM

GOAL 1 Trigonometry
Form
4.1 Complex Number in
Other Form =

GOAL 2 Exponential
Form
4.2 Multiplication and Division
In Polar Form

GOAL 3 EXAMPLE 1

4.3 Multiplication and Division

In Trigonometry Form Express the complex number, = 2 + 6 into:

a. Trigonometric Form
b. Polar Form
c. Exponential Form

Solution: , = −1
, = √ 2 + 2
6
= √(2)2 + (6)2 = −1 2
= √40
= 6.325 = −1(3)

= 71.57°

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Plot the complex number in an Argand Diagram:


c. Trigonometric Form
= ( + )

= =

= ( + )
= 6.325( 71.57° + 71.57° )

b. Polar Form in degree
convert to
= ⦟
radian
= ⦟
= 6.325 ⦟ 71.57° =
= 6.325 1.249
a. Exponential Form
=

Argument convert to radian:


= 180 ×
71.57˚

= 180 ×
= 1.249

32 | P a g e

EXAMPLE 2

Express the complex number, = 3 < 65° into:

a. Trigonometric Form
b. Cartesian Form
c. Exponential Form

Solution:

a. Trigonometric Form
= ( + )

Press, 6 = 3 = 65°
3 ENG
5= = ( + )
= 3( 65° + 65° )
OPTN = 3(0.423 + 0.906 ) Make sure your
⊳ < = 1.269 + 2.719 check answer by
using calculator

Then, 2 b. Cartesian Form
⊳ + = +

1.267854785+2.71892i = 1.269 + 2.719

So, this answer is: c. Exponential Form
1.269 + 2.719 =

Argument convert to radian: =
= 3 1.135

= 180 ×

65˚
= 180 ×

= 1.135

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EXAMPLE 3

Express the complex number, = 2.132 1.4573 into:

a. Polar Form =
b. Trigonometric Form
c. Cartesian Form

Solution: = .
( ) = .
a. Polar Form
= <

= ⦟ ( )
= 2.132 ⦟ 83.50° ( ) = 180 ×

b. Trigonometric Form 1.457
( ) = × 180

= 83.50°

= ( + )

= ( + )
= 2.132( 83.50° + 83.50° )
= 2.132(0.113 + 0.994 )
= 0.241 + 2.119

c. Cartesian Form
= +

= 0.241 + 2.119

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EXERCISE 1

a. Express of the complex number, = 2 + 4 in the forms of
trigonometric, polar and exponential.

ANS:
Trigonometric form: 4.4721 ( 63.44°) + ( 63.44°)
Polar form: 4.4721 ⦟ 63.44°
Exponential form: 4.4721 1.1072

b. Express of the complex number, 6 ⦟ 75° in the forms of
exponential, trigonometric and cartesian.

ANS:
Exponential form: 6 1.309
Trigonometric form: 6 ( 75°) + ( 75°)
Cartesian form: 1.553 + 5.796

c. Express of the complex number, 3.606 1.5141 in the forms
polar, trigonometric and cartesian

ANS:
Polar form: 3.606 ⦟ 86.75°
Trigonometric form: 3.606( 86.75°) + ( 86.75°)
Cartesian form: 0.204 + 3.600

35 | P a g e

GOAL 22 4.2 Multiplication and Division In Polar Form

12. Number

Complex

Multiply value, add the angle

( ⦟ )( ⦟ ) = ⦟( + )

1.10

Divide value, subtract the angle

⦟ = ⦟ ( − )
⦟
Then, press like screen display:
(3⦟ 45°)(6 ⦟ 80°)

−19.32437585 + 14 ⊳ EXAMPLE 4

OPTN Given 1 =3 ⦟ 45° and 2 = 6( 80° + 80° ). :

. 1 2

. 2
1

Then, 1 = . 1 2 Convert into Polar Form
⊳ + 1 =3 ⦟ 45° 2 =6 ⦟ 80°
2 = 6( 80° + 80° ).
18⦟125
( ⦟ )( ⦟ ) = ⦟( + )
So, this answer is: ( ⦟ °)( ⦟ °) = ( )⦟ ( ° + °)
18⦟125°
= ⦟ °

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Formula:

⦟ = ⦟ ( − )
⦟

⦟ °
⦟ ° = ⦟ ( ° − °)

= ⦟ ( °)

EXERCISE 2

Multiplication & Division In Polar Given 1 = 6 ⦟ 70° and 2 = 12 ⦟ 85°. Calculate each of the
https://www.youtube.com/watch?v
=q3LSSavqiqc following in polar form:

PLEASE SCAN QR a.
CODE:
b.
Multiplication &
Division In Polar Form
ANS:
Video Tutorial
a. 72 ⦟ 155°
b. 2 ⦟ 15°

Given 1 = 8 ⦟ 35° , 2 = 10(cos 20° + 20°) and
3 = 4 ⦟ 5°. Calculate each of the following in polar form:

a.

b.


ANS:

c. a. 80⦟ 55°

b. 4 ⦟ 15°

5

c. 2 ⦟ 30°

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Given 1 = 2 ⦟ 40° , 2 = 10(cos 80° + 80°) and
3 = 6.94 2.314 . Calculate each of the following in polar form:

a.

b.


c.

d.


ANS:

a. 20 ⦟ 120°
b. 5 ⦟ 40°
c. 138.8 ⦟ 252.58°
d. 34.7⦟ 172.58°

GOAL 32 4.3 Multiplication and Division In Trigonometric Form

12.

1.11 [( ( + )( ( + )]
= ( [ ( + ) + ( ( + ) ]

( + )
( + )

= [ ( − ) + ( − ) ]


38 | P a g e

EXAMPLE 5

Given 1 =4(cos60° + 60° ) and 2 = 8 1.5707 . Calculate:

. 1 2

. 2
1

. Convert into Trigonometric Form

1 =4(cos60° + 60° ) = .
.
2 = 8 1.5707
×
= °
= ( ° + ° )

( [ ( + ) + ( ( + ) ]
( )[ ° + °) + ( ( ° + °) ]

= [ ° + ° ]

. ( +

( ° + ° )
( ° + ° )

[ (+ − ) + ( − ) ]


= [ ( ° − ° ) + ( ° − °) ]


= [ ° + ° ] ]



39 | P a g e

EXERCISE 3

Given 1 = 3( 20° + 20°) and 2 = 9( 80° +
80°). Calculate each of the following in trigonometric form:

a.

b.


ANS:

a. 27( 100° + sin 100°)
b. 3( 60° + sin 60°)
c. °

Given 1 = 4 + 8 , 2 = 10(cos 20° + 20°) and
3 = 4 ⦟ 5°. Calculate each of the following in trigonometric form:

a.

b.


c.


ANS:
a. 89.443( 83.43° + sin 83.43°)
b. 0.894( 43.43° + sin 43.43°)
c. 2.236( 58.43° + sin 58.43°)

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Given 1 = 2 ⦟ 40° , 2 = 10(cos 80° + 80°) and
3 = 3 0.349 . Calculate each of the following in trigonometric form:

a.

b.


c.

d.


ANS:

a. 20( 120° + sin 120°)
b. 5( 40° + sin 40°)
c. 15( 60° + sin 60°)

41 | P a g e

FINAL EXAMINATION PAPERS

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Given = 4 + 8 and = −1 − 3 , determine each of the following expression and write the answer
in + form: [ Final exam Paper June 2018 Session]

i. +
ii.
iii. − 2

ANS:
i. 3 + 5
ii. 20 − 20

iii. 12 + 2

Given that = −4 + 6 , = 7 − 5 and = −6 − 2 . Calculate each of the following in the form of
+ . [ Final exam Paper June 2019 Session]

i. −
ii. 2( + )
iii.



ANS:

i. 2 + 8

ii. −20 + 8

iii. 29 11
− 26 − 26

43 | P a g e

Given 1 = 3 + 2 and 2 = 3( 30° + 30°)[ Final exam Paper June 2018 Session]
i. Calculate the modulus and argument for 1
ii. Sketch the Argand Diagram for 1

ANS:
i. = 3.606 , = 33.69°
ii. = 33.69°

Given that = 5 − 10 and = −8 + 2 . Calculate the modulus and argument. Then, sketch the Argand
diagram for + .[ Final exam Paper June 2019 Session]

ANS:

+ = −3 − 8
= 8.544 = −110.55° =249.45°

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Given equation = 6 + and = −1 + 5 , and = 4 − 8 determine each of the following
expression and write the answer in + form: [ Final exam Paper June 2017 Session]

i. 2 + 4
ii. ×

iii.


ANS:

i. 28 + 30

ii. 36 + 28

iii. 1 31
− 26 − 26

Given that = 6 − 8 and , = −4 + . Sketch the Argand Diagram. Then, determine the modulus
and the argument for the complex number below: [ Final exam Paper June 2017 Session]

i.


ii.

ANS:

i. = 10, z = - 53.13°, = 306.87°

ii. − 32 + 26 , r = 2.425, = 140.91°
17 17

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