DBM10013 ENGINEERING MATHEMATICS 1 _ TRIGONOMETRY

MUHAMMAD SALLEHUDDIN BIN ISMAIL
Trigonometry Notes & ExercFisAesZEHA NASYA BINTI ABD MALIK

DBM10013 Engineering Mathematics 1 : Trigonometry

Published by POLITEKNIK MUKAH

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Cover by Muhammad Sallehuddin Bin Ismail

Perpustakaan Negara Malaysia

Editor:
Muhammad Sallehuddin Bin Ismail
Fazeha Nasya Binti Abd Malik

e ISBN: 978-967-2097-62-4

PREFACE

The Trigonometry Notes & Exercises is an e-book and exercise book for
mathematics problem solving design for polytechnic students. The
measurement process involves the use of triangles and a branch of
mathematics known as trigonometric functions. Trigonometry is an
important tool for evaluating measurement of height and distance.

In this chapter, we discuss how to manipulate trigonometry equations
by applying various formulas and trigonometry identities. The given
worked examples serve as a step by step guide while the carefully
designed questions provide the practice. This book therefore, provides
an opportunity to discover various problem solving methods.

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

TABLE OF CONTENTS Page

SOLVE THE FUNDAMENTAL OF
TRINOGONOMETRY FUNCTIONS

1.0 1.1 Define sine, cos1ine, tangent, secant,
cosecant and cotangent .

1.2 Sketch the graph of sine, c0osine and tangent 2 - 19

1.3 Show the positive and negative values of trigonometric
function using quadrants

1.4 Calculate the values of trigonometric functions

SOLVE TR0I1G–ON1O3M|PEaTgRIeC EQUATIONS AND

IDENTITIES

2.0 2a..1triSgoonlvoemtertrigicobnaosmi2ceitdreinctietiqeusations involving: 20 - 30

b. compound angle .

c. double angle compound 0

APPLY SINE AND COSINE RULES 31 - 37

3.0 3.1 Define the sine and cosine rules

3.2 Calculate the area of triangle using the
formula ab sin c

3.3 Solve simple trigonometric problems using
sine and cosine rules

2|Page

1.0 Introduction
.
Trigonometry is one of branches, where students will learn the relationship
TOPIC 1: between angles and sides of a triangle.

The concept of trigonometry are based on a right angled triangle and
all side of a right triangle is named as figures.

DEFINE SINE, COSINE, TANGENT, SECANT,
COSECANT, AND COTANGENT

Learning Outcomes:

1.1 Define sine, cosine, L
tangent, secant, cosecant,
and cotangent.

11..22SSkkeettcchhthheeggraphh of sine, 1.1 Define sine, cosine, tangent, secant,
ccoossiinnee aanndd ttaannggeenntt GOAL cosecant, and cotangent

1.3 Show the positive and DEFINITIONS FORMULA
negative values of
trigonometric function
using quadrants.

11.4.4CCaalclcuulalate the values of
trigonometric functions.

2|Page

Based on diagram below find :

Solution
:

3|Page

Based on diagram below find :

4|Page

Based on diagram below find :

a. b.

[ Ans: 0.50] [ Ans: 1.73]
c. d.

[ Ans: 0.87] [ Ans: 0.58]

5|Page

Based on diagram below, find the :

a. i5 + i 7 b. 3i3 − 9i8

[ Ans: 0.6] [ Ans: 1.67]

c. 25 + 4i8 + 2 d. 4 i 8 − 5 i 3 − −36

[ Ans: 0.75]

6|Page

Base A right-angle triangle has an angle sin = sin Ɵ = ,



find the :

8
5

a. b.

[ Ans: 0.63] [ Ans: 0.78]
c. d.

[ Ans: 0.80] [ Ans: 1.28]

7|Page

GOAL 2 1.2 Sketch the graph of sine, cosine and tangent

Graph of Sine
The Sine Function has this beautiful up-down curve
which repeats every 360°. It starts at 0, heads up to
1 by 90° and then heads down to -1.

Graph of Cosine
The Cosine is just like Sine, but it starts at 1 and
heads down until 180° and then heads up again.

\

8|Page

Graph of Tangent
The tangent function has a completely different
shape… it goes between negative and positive
(infinity), crossing through 0 every 180°, as shown
on this plot.
At -270°, -90°, 90°, 270° the function is officially
undefined, because it could be positive infinity or
negative infinity.

9|Page

SKETCH THE GRAPH OF SINE.
Step 1 : Calculate the sine for different angles

Tittle: = for angles from 0° to ± 360° in 90° increments

Step 2 : Plot each point from the table on the graph paper
and join the dots.

Get some graph paper and prepare it by scaling off 0° to 360° in 90°
increments along the x-axis, and scaling off -1 to +1 on the y-axis. Now
plot each point from the table on the graph. Then join the dots as neatly
as you can.

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Sketch the graph of cosine.

Tittle: = for angles from 0° to ± 360° in 90° increments

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Sketch the graph of tangent.

Tittle: = for angles from 0° to ± 360° in 45° increments

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1.3 Show The Positive And Negative Values Of
Trigonometric Function Using Quadrants
Determine the positive and negative values of
trigonometric functionusing quadrant.

Reference Angles (0° < θ < 360° )

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Find Reference Angle for the following:

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Find Reference Angle for the following:

DON’T FORGET TO ° ° °
CALCULATE THE
QUADRANT I , II , III & IV

[ Ans: 45° ] [ Ans: 20°] [ Ans: 40°]
° ° °

[ Ans: 89°] [ Ans: 60°] [ Ans: 1°]
° ° °

[ Ans: 27°] [ Ans: 50°] [ Ans: 25°]

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Determine the positive and negative values of the
trigonometric function using quadrant :

[ Ans: - 0.5] [ Ans: - 0.707]
[ Ans: 1] [ Ans: - 0.577]

[ Ans: 0.951] [ Ans: 0.649]

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4 1.4 Calculate the values of trigonometric functions

Determine the values of trigonometric function.

Trigonometric Quadrant

REMARKS

SIN

Inverse of Trigonometry:

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Find the values for θ where 0° ≤ θ ≤ 360°

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Find the values for θ where 0° ≤ θ ≤ 360°

[ Ans: 72°3’ & 289°57’] [ Ans: 40° & 140°]

[ Ans: 26°34’ & 206°34’] [ Ans: 133°39’ & 313°39’]

[ Ans: 225° & 315°] [ Ans: 159°21’ & 200°31’]

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TOPIC 2: Introduction

Trigonometry The point of this Trigonometry E-Book is to supply a brief think and self-
assessment program for understudies who wish to gotten to be more
recognizable with trigonometry. In this topic, you should able to
understand the concept of trigonometry, perform operations on
trigonometry such as trigonometric equation and identities and Sine and
Cosine rules of trigonometric.

GOAL 1 GOAL 1 2.1 Trigonometric Basic Identities
2.1 Trigonometric Basic
Identities Trigonometric basic identities was divided into three different identities
which is basic identities, compound angle and double angle.
GOAL 2
2.2 Compound Angle Basic identities Basic identities Formula :
GOAL 2
GOAL 3 2 + 2 = 1
2 + 2 = 1
2.3 Double Angle 2 + 1 = 2
1 + 2 = 2

Compound Angle Compound Angle Formula :
Double Angle
sin( ± ) = sin cos ± cos sin
cos( ± ) = cos cos ∓ sin sin

tan ± tan
tan( ± ) = 1 ∓ tan tan

Double Angle Formula :
sin(2 ) = 2 sin cos
cos(2 ) = cos2 − sin2
cos(2 ) = 1 − 2sin2
cos(2 ) = 2cos2 − 1

2 tan
tan(2 ) = 1 − tan2

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EXAMPLE 1

Pythagorean Identities

+ = Using identities and factorization solve 2 + sin − 1 = 0.
Where 0° ≤ ≤ 360°
or
sin2θ = 1 − cos2θ Solution :
cos2θ = 1 − sin2θ

2 + 2 = 1 Refer to Trigonometry
Identities
+ = 2 = 1 − 2
or = −
tan2θ − sec2θ = −1 You can(c1h−ec k a 2n s w) +ersin − 1 = 0 + =

tan2θ = sec2θ − 1 by use s−cie n t 2if i c+ sin − 1 + 1 = 0 *Reference for quadrant
Sin (+ve)
calculat−or 2 + sin = 0

+ = sin (− sin + 1) = 0

or sin = 0 −sin + 1 = 0
cot2θ = csc2θ − 1 [= −1 0
1 = csc2θ − cot2θ = 0° −sin = −1

sin [== 1
−1
1

= 90°

180° 0°

Basic identities Formula : Reference angle ß = 0°
In quadrant I : = ß = 0°
2 + 2 = 1 In quadrant II : 180° − 0°=180°
2 + 2 = 1 And,
2 + 1 = 2
1 + 2 = 2 90°

0°

Reference angle ß = 90°
In quadrant I : = ß = 90°
In quadrant II : 180° − 90°=90°

∴ The solution is = °, °, °

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EXAMPLE 2

Using identities and factoring where 0° ≤ x ≤ 360°
3 + = +

Solution :

3 2 + = 2 + (1 + 2 )

3 2 + = 3 + 2

3 2 − 2 + − 3 = 0

2 2 + − 3 = 0

( 2 tan + 3)(tan − 1) = 0

2 tan + 3 = 0 tan − 1 = 0

2 tan = 3 tan = −1 *Reference for quadrant
= −11 Sin (+ve)
tan = −1.5 =45°

*Reference for = −1(−)1.5
quadrant “tan (-ve)”
=(-)56.31°

56.31° 45°

56.31° 45°

Reference angle ß = 56.31° Reference angle ß = 45°
In quadrant II : = 180° − 56.31° In quadrant I : = ß = 45°
In quadrant III = 180° + 45°
= 123.69°
In quadrant IV : 360° − 56.31° =225°

=303.69°

Important !! ∴ The solution is = °, . °, °, . °

Ignore the negative when
get the value of x.

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EXERCISE 1
Calculate x where 0° ≤ x ≤ 360°

a. ( − ͦ) = ( + )ͦ b. − − =

2

SHIFT

Ans : 130.9°, 310.91° Ans : 71.57°, 146.31°, 251.57°, 326.31°

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EXERCISE 2
Calculate x where 0° ≤ x ≤ 360°

a. 5 2 + 26 = 0 b. 2 2 − 5 + 2 = 0

MODE

SHIFT

Ans : 101.36°, 168.69°, 281.36°, 348.69° Ans : 26.57°, 63.43°, 206.57°, 243.43°

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GOAL 2 2.2 Trigonometric Compound Angle

Compound Angle
What are compound angle formulas?
A compound angle formula or addition formula is basically a trigonometric
identity that expresses a trigonometric function of (A+B) or (A−B) in
expressions of trigonometric functions of A and B.

They are of interest because their trigonometric functions are derived
from the trigonometric functions of their constituent angles as follows:

=

EXAMPLE 1
a) Solve the following
̊ ̊ + ̊ ̊
cos 70° cos 30°+sin 70° sin 30°= cos (70°−30°)

= cos 40°
= 0.766
b) Solve the following
̊ ̊ + ̊ ̊
sin 30° cos 45° + cos 30° sin 45° = sin (30° + 45°)
= in 75°
= 0.966

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EXAMPLE 2

a. Calculate the following equation:
cos( + 60) = sin

AB
cos cos 60 − sin sin 60 = sin

cos (0.5) − sin (0.8660) = sin

cos (0.5) = sin (1 + 0.8660)
0.5 = sin

1+0.866 cos

0.268 = tan *Reference for quadrant
= −1 0.268 tan (+ve)

= 15°

Tan x : 0.564 15°
15°
Means = Tan +
(Refer Q1 & Q3) Reference angle ß = 15° Ans : 15° & 195°
In quadrant I : = 15°
Sin x : - 0.457
In quadrant II : 180° + 15°
Means = Sin - =195°
(Refer Q3 & Q4)

Cos x : 0.235

Means = Cos +
(Refer Q1 & Q4)

b. By using compound angle identities solve for (− °)

tan( ± ) = tan ±tan

1 ∓ tan tan

tan(−60 ± 45) = tan(−60)±tan 45

1 ∓ tan(−60) tan 45

= −1.732+1

1−(−1.732)

= −0.732

2.732

= −0.268

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EXERCISE 1 b. 2 tan + 3 tan( + 45°) = 0

Calculate x where 0° ≤ x ≤ 360°
a. cos = sin( + 60°)

Ans : = ° & ° Ans : = . ° , . °, . ° & . °

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GOAL 3 2.3 Trigonometric Double Angle

Double Angle
What are double angle formulas?

A double angle is mathematically exactly what it sounds like. A double angle is
when an angle is doubled or multiplied by 2. for example, 30° doubled is 60°.
When doubling an angle, make sure that the degree measure is actually
multiplied by 2. For trigonometric functions such as sine, cosine or tangent,
doubling the angle is not the same as doubling the function.

Double Angle Formula

( ) =
( ) = −

= −
= −


( ) = −

EXAMPLE 1

Given = 45, where 0° ≤ α ≤ 180°

a) b)

2 tan = 2 - 2
2 = 1 − tan2 = (4)2 − (3)2

= 2 (34) 55
1 − (34)2
=7

25

24
=

7

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EXERCISE 1

a) Prove the following identity b) Solve cos(2 ) + cos = 0

2 − 2 = ( 2 )( 2 )

MODE 2

SHIFT =

c) Simplify the expression Ans: 60°, 180° & 300°
2 2(12°) − 1
d) Simplify the expression
8 (3 ) (3 )

Ans: cos 24° Ans:
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Introduction

TOPIC 2: The point of this Trigonometry E-Book is to supply a brief think and self-
assessment program for understudies who wish to gotten to be more
Apply Sine & Cosine Rules recognizable with trigonometry. In this topic, you should able to
understand the concept of trigonometry, perform operations on
trigonometry such as trigonometric equation and identities and Sine and
Cosine rules of trigonometric.

GOAL 1 GOAL 1 3.1 Sine and Cosine Rules

3.1 D e f i n e t h e s i n e The Sine Rule can be used in any triangle (not just right-angled triangles)
and cosine rules where a side and its opposite angle are known.

GOAL 2

3.2 C al c u l a t e a r e a o f
tGrOiAaLn g2l e u s i n g t h e
formula
GOAL 3

3G.O3ALS o l3v e s i m p l e
trigonometric
problems using sine
and cosine rules.

`

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The cosine rule is a commonly

used rule in trigonometry. It can

be used to study the properties

of non-rectangular triangles,

allowing missing information

such as side lengths and angle

measures to be determined. The

fPoyGrtmOhAaugLlaor1eaisn similar to the
theorem and is

relatively easy to memorize. Further explanation as per below for & :

GOAL 2

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EXAMPLE 1
Solve the triangle below :



26.3 17.5

105°






a) Angle A


= =

17.5 26.3
= =

17.5 26.3
=

26.3 = 17.5 105°
= 0.64
= −1 0.6427
= 39.99° @ 40°

b) Angles C
= 180° − (105° + 40°)
= 180° − (145°)
= 35°

c) Length C
2 = 2 + 2 − 2
2 = (17.5)2 + (26.3)2 − 2(17.5)(26.3) 35°
2 = 997.94 − 754.03
2 = 243.91
= √243.91
= 15.62

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EXAMPLE 2



175

Area Of
Triangle
95°
86



a) Find the area of triangle above
= 175
= 95°
= 86

= 1 ac sin B

2

= 1 (175)(86) sin 95°

2

= 7496.37 2



15

45°


37

b) Find the area of triangle above
= 37
= 45°
= 15

= 1 bc sin A

2

= 1 (37)(15) sin 45°

2

= 196.22 2

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EXERCISE 1
1. Find area the following below:

a. b. 7

73 9 5
35°
156


Ans: 3265.94 2 Ans: 17.41 2
2. Find the angle C :-

8
56°

5

Ans: 31.21°

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+

EXERCISE 2

1. Find all unknown angle and length of side AC below:


7

47°


8

Ans: = 56°, = 76°, = 9.2996
2. Given a triangle ABC with Angle A =28.5°, angle B =102.2° and
a = 45.5cm.
Find Angle C, length b and length c

Ans: = 49.3°, = 72.3 , = 93.2

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1. Based on Diagram 1 (a),

9



4
Diagram 1(a)

Solve the values of:

i. sin
ii. tan
iii. sec

ANS : 0.406
i. 0.444
ii. 1.094
iii.

2. Calculate the angles between 0    360 for the following equations:

i. cos = 0.358

ANS : ∴   ℎ               69.02°&, 290.98°

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3. Calculate the angles between 0    360 for the following equations:

i. sec2 + 10 tan = −4

ANS : ∴   ℎ               96.03°, 152.17°, 276.03°     332.17°
4. Based on Diagram 1 (c),

55°

A 12cm
17cm B

Diagram 1(c)

Calculate:
i. Angle A
ii. The length of AC

ANS : A = 35.51°
i. AC = 20.75cm
ii.

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Noorhayani Yahya & Shahrol Faizal Abdullah (2014). Trigonometry Notes With
Exercises (Second Edition). Perak: Politeknik Sultan Azlan Shah

Intanku Salwa Binti Shamsudin, Asmarizan Binti Mat Esa, Noor Hidayah Binti
Awang, Norzaliza Binti Mohamed Nor & Rasyidah Binti Abd Rahman (2015).
Engineering Mathematics (Level 1). Melaka: Politeknik Merlimau

Wan Azliza Binti Wan Zakaria, Rahimah Bt Mohd Zain @ Ab. Razak & Nik Noor
Salisah Binti Nik Ismail (2021). Engineering Mathematics 1. Pahang:
Politeknik Sultan Haji Ahmad Shah



Editor :

Muhammad Sallehuddin Bin Ismail
Fazeha Nasya Binti Abd Malik

e ISBN: 978-967-2097-62-4