MAC101-Lec04 engineering foundation .pdf

mathematics foundation engineering

1. Trigonometry
MAC101 Foundation Mathematics
Lecture 04
MAC101 – 2025/2026
Foundation Level
Dr. Hossam KAMAL
[email protected]
Dr. Mohamed Hazem HASSAN
[email protected]
Contents:
4.1 The Concept of Angle
4.2 Triangle and Trigonometric Functions
4.3 Circle, wave, and trigonometry
Lancshire - EUE 2025.26 - Dr. Hossam Kamal 1

2. Assessments
Lancashire -EUE
Dr. Hossam Kamal
Pg. 2
MAC101
Lec. 04
Week 4 (18/10/2025)
Formative test 01
Auditorium
13:40 - 14:40
Week 4 (18/10/2025)
Formative test 01
Auditorium
13:40 - 14:40
Week 6 (1/11/2025)
Reflection week
No lectures or tutorials
Week 6 (1/11/2025)
Reflection week
No lectures or tutorials
Week 7 (8/11/2025)
Formative test 02
Auditorium
13:40 - 14:40
Week 7 (8/11/2025)
Formative test 02
Auditorium
13:40 - 14:40
Week 8 (15/11/2025)
Submission of Report
20% of the module marks
Week 8 (15/11/2025)
Submission of Report
20% of the module marks
Week 9 (22/11/2025)
Formative test 03
Auditorium
13:40 - 14:40
Week 9 (22/11/2025)
Formative test 03
Auditorium
13:40 - 14:40
Week 10 (29/11/2025)
Summative Written Exam
20% of the module marks
Week 10 (29/11/2025)
Summative Written Exam
20% of the module marks
Week 13
(20/12/2025)
Final Written
Exam
60% of the
module marks
Week 13
(20/12/2025)
Final Written
Exam
60% of the
module marks

3. The Report Assessment
Lancashire -EUE
Dr. Hossam Kamal
Pg. 3
MAC101
Lec. 04
The Mathematical Bases of the AI and IoTTechnologies
Your Task
1. Elaborate a scientific report exploring the subject
2. Your report should include:
Summary / Abstract
Introduction
Literature Survey
Description and history of smart systems and artificial intelligence
Description and history of internet of things
The Inter-relation between AI and IoT
The mathematical tools behind AI and IoT
The future of mathematics influenced by AI and IoT
Conclusion
References
3. Explore thoroughly the mathematical tools showing their definition, branches and basic formula
4. Organise your work in a scientific simple paragraphs
5. Support your report with necessary graphs, images, tables, and charts
6. If you use some AI tools, you have to mention it in foot notes
7. You can employ the templates of report and previous examples are provided on the module LMS Moodle page
8. Present and defend your report

4. Our lecture
Lancashire -EUE
Dr. Hossam Kamal
Pg. 4
MAC101
Lec. 04
Last LectureLast Lecture
Today’s LectureToday’s Lecture
Next LectureNext LectureGraphs:
Graphical solution of
equations, including
simultaneous equations.
Graphical solutions of
quadratic equations
Graphical solution of cubic
equations
Finding roots graphically
Slope of a curve
Graphs:
Graphical solution of
equations, including
simultaneous equations.
Graphical solutions of
quadratic equations
Graphical solution of cubic
equations
Finding roots graphically
Slope of a curve
4.1 The Concept of Angle
4.2 Triangle and Trigonometric
Functions
4.3 Circle, wave, and
trigonometry
4.1 The Concept of Angle
4.2 Triangle and Trigonometric
Functions
4.3 Circle, wave, and
trigonometry
Exponentials and Logarithms:
• Exponential function
• Logarithmic functions
• Natural logarithm.
• Graphs and properties of exponential and
logarithmic functions.
• Laws of growth and decay
Exponentials and Logarithms:
• Exponential function
• Logarithmic functions
• Natural logarithm.
• Graphs and properties of exponential and
logarithmic functions.
• Laws of growth and decay

5. Trigonometry
Lancashire -EUE
Dr. Hossam Kamal
Pg. 5
MAC101
Lec. 04
The term "trigonometry" was derived
from Greek τρίγωνονtrigōnon, "triangle"
and μέτρονmetron, "measure"
The term "trigonometry" was derived
from Greek τρίγωνονtrigōnon, "triangle"
and μέτρονmetron, "measure"
3.0
Motivated by wonder and curiosity early
peoples linked events in the heavens to the
seasons and their explanations provided
the elements for primitive cosmologies and
religious beliefs.
By 2300 BCE Egyptian priests had divided
the ecliptic into 36 sections of 10 degrees
each. (The sun’s apparent daily movement
across the sky is about 1/360 of a circle.)
By 500 BCE the Babylonians had
established twelve 30o divisions of the sky
that became the 12 houses of the zodiac.
Motivated by wonder and curiosity early
peoples linked events in the heavens to the
seasons and their explanations provided
the elements for primitive cosmologies and
religious beliefs.
By 2300 BCE Egyptian priests had divided
the ecliptic into 36 sections of 10 degrees
each. (The sun’s apparent daily movement
across the sky is about 1/360 of a circle.)
By 500 BCE the Babylonians had
established twelve 30o divisions of the sky
that became the 12 houses of the zodiac.
From 2000 BCE the Vedic peoples of India
were constructing altars accurately oriented
in an E-W direction, and working with the
ratios of the shadow-stick. Accurate Lunar
and astronomical calendars were essential
for religious and social organisation and the
use of horoscopes. Similar activities have
also been recorded in China from about
2000 BCE.
From 2000 BCE the Vedic peoples of India
were constructing altars accurately oriented
in an E-W direction, and working with the
ratios of the shadow-stick. Accurate Lunar
and astronomical calendars were essential
for religious and social organisation and the
use of horoscopes. Similar activities have
also been recorded in China from about
2000 BCE.
Greekscholars were the first to develop and
elaborate a mathematical theory of the
cosmos, (from about 500 BCE) and were the
first to begin systematic mapmaking of their
world. Using Babylonian data and
observational techniques they began to
estimate the distances and sizes of the Sun
and Moon
Greekscholars were the first to develop and
elaborate a mathematical theory of the
cosmos, (from about 500 BCE) and were the
first to begin systematic mapmaking of their
world. Using Babylonian data and
observational techniques they began to
estimate the distances and sizes of the Sun
and Moon
Ptolemy’sAlmagest (150 CE) contained all
the then known astronomical knowledge:
geometrical and numerical procedures, the
longitude and latitude of heavenly bodies,
information about parallax, the distance
and relative sizes of the Sun and the Moon,
lunar theory, solar motion, and the
occurrence of eclipses, transits, and
occultations.
Ptolemy’sAlmagest (150 CE) contained all
the then known astronomical knowledge:
geometrical and numerical procedures, the
longitude and latitude of heavenly bodies,
information about parallax, the distance
and relative sizes of the Sun and the Moon,
lunar theory, solar motion, and the
occurrence of eclipses, transits, and
occultations.
For the Arabs, the key importance of trigonometry was religious. The first sight of the
crescent moon to establish the new month, finding the direction of Mecca, and the correct
times for daily prayers were essential.
For the Arabs, the key importance of trigonometry was religious. The first sight of the
crescent moon to establish the new month, finding the direction of Mecca, and the correct
times for daily prayers were essential.

6. Trigonometry
Lancashire -EUE
Dr. Hossam Kamal
Pg. 6
MAC101
Lec. 04
Previous works from India and Greece were later translated and expanded in
the medieval Islamic world by Muslim mathematicians of mostly Persian and
Arab descent, who enunciated a large number of theorems which freed the
subject of trigonometry from dependence upon the complete quadrilateral,
as was the case in Hellenistic mathematics due to the application of
Menelaus' theorem. According to E. S. Kennedy, it was after this
development in Islamic mathematics that "the first real trigonometry
emerged, in the sense that only then did the object of study become the
spherical or plane triangle, its sides and angles."
Previous works from India and Greece were later translated and expanded in
the medieval Islamic world by Muslim mathematicians of mostly Persian and
Arab descent, who enunciated a large number of theorems which freed the
subject of trigonometry from dependence upon the complete quadrilateral,
as was the case in Hellenistic mathematics due to the application of
Menelaus' theorem. According to E. S. Kennedy, it was after this
development in Islamic mathematics that "the first real trigonometry
emerged, in the sense that only then did the object of study become the
spherical or plane triangle, its sides and angles."
Following Abul Wafa, the great scholar al-Biruni(973-1048)
showed how many functions could be related, and he
introduced the idea of a parametric representation as
shown in this modern diagram.
Following Abul Wafa, the great scholar al-Biruni(973-1048)
showed how many functions could be related, and he
introduced the idea of a parametric representation as
shown in this modern diagram.
In this Astrolabe the celestial sphere is
projected onto the plane of the equator.
SurveyingSurveying
RoboticsRobotics
NavigationNavigation
Fluid
Mechanics
Fluid
Mechanics
Signal
processing
Signal
processing
Kinematics
and
dynamics
Kinematics
and
dynamicsVibrations
and
oscillations
Vibrations
and
oscillations
Computer
Graphics
Computer
Graphics
AC circuitsAC circuits
Structural
design
Structural
design
Flight
dynamics
Flight
dynamics
Geotechnical
Engineering
Geotechnical
Engineering
ConstructionConstruction

7. The Concept of Angle
Lancashire -EUE
Dr. Hossam Kamal
Pg. 7
MAC101
Lec. 04
3.1
The rays AB and AC are called sides of the
angle BAC; point A is the vertex. Ray AC is
the right side of the angle.
The rays AB and AC are called sides of the
angle BAC; point A is the vertex. Ray AC is
the right side of the angle.
An angle is a plane geometrical figure
formed by two rays extending from a
common point of the plane.
An angle is a plane geometrical figure
formed by two rays extending from a
common point of the plane.
L
e
c
t
t u rt u r 04 u eLc

8. The measure of A = ∠1 , B = ∠2, and C = ∠3
Given, ⃡// 1–
We know that thealternate anglesare equal in parallel lines.
∴∠2 = ∠4 -- (1) and ∠3 = ∠5 --- (2)
Adding (1) and (2), ∠2 + ∠3 = ∠4 + ∠5
By adding ∠1 on both sides, ∠1 + ∠2 + ∠3 = ∠4 + ∠5 + ∠1
The sum of the angles at a point on a line is equal to 180 degrees.
∠4 + ∠1 + ∠5 = 180° So, ∠1 + ∠2 + ∠3 = 180°
Therefore, ∠A + ∠B + ∠C = 180°
The Relationships of Angles
Lancashire -EUE
Dr. Hossam Kamal
Pg. 8
MAC101
Lec. 04
3.1
L
e
c
1
5
/
6
2
3
4
56
7
8
Lelsslc5and /6 intersects the two lines, then
∠3 = ∠6and ∠4 = ∠5Alternate interior
∠1 = ∠8and ∠2 = ∠7Alternate exterior
∠4,∠6,∠3,and ∠5Interior
∠1,∠7,∠2,and ∠8Exterior
∠1 = ∠5 , ∠2 = ∠6 , ∠3 = ∠7 , ∠4 = ∠8Corresponding
∠1 = ∠4 , ∠2 = ∠3 , ∠5 = ∠8 , ∠6 = ∠7Vertical
Consider the arbitrary triangle ABC. Prove that the sum of all
interior angles of the triangle is equal to 180º
Example
L
e
c
Action: Draw a line through A,
parallel to BC
1
2
3
4
5Required: We have to prove that
∠A + ∠B + ∠C is equal to 180 º.

9. Measurement of Angles
Lancashire -EUE
Dr. Hossam Kamal
Pg. 9
MAC101
Lec. 04
3.1
The main systems for measuring angles are the sexagesimal system (using degrees, minutes, and seconds), the centesimal
system (using grades), and the circular system(using radians).
The sexagesimal system is the most common for everyday use, dividing a circle into 360 degrees. The centesimal system,
also known as the French system, divides a circle into 400 grades. The circular system, which is standard in mathematics and
physics, defines a radian as the angle subtended by an arc equal to the radius of the circle, with a full circle being 2π radians.
The main systems for measuring angles are the sexagesimal system (using degrees, minutes, and seconds), the centesimal
system (using grades), and the circular system(using radians).
The sexagesimal system is the most common for everyday use, dividing a circle into 360 degrees. The centesimal system,
also known as the French system, divides a circle into 400 grades. The circular system, which is standard in mathematics and
physics, defines a radian as the angle subtended by an arc equal to the radius of the circle, with a full circle being 2π radians.

10. Triangle Applications
Lancashire -EUE
Dr. Hossam Kamal
Pg. 10
MAC101
Lec. 04
3.2
The theorem of PythagorasThe theorem of Pythagoras
In any right-angled triangle, the square
on the hypotenuse is equal to the sum
of the squares on the other two sides
In any right-angled triangle, the square
on the hypotenuse is equal to the sum
of the squares on the other two sides
ec
!
: Lc
!
. Le
!

11. Trigonometric Functions
Lancashire -EUE
Dr. Hossam Kamal
Pg. 11
MAC101
Lec. 04
3.2
Angle of elevation
Angle of depression

12. Trigonometric Functions
Lancashire -EUE
Dr. Hossam Kamal
Pg. 12
MAC101
Lec. 04
3.2
A surveyor measures the angle of elevation of the top of a
perpendicular building as 19◦. He moves 120m nearer the building
and finds the angle of elevation is now 47◦. Determine the height
of the building.
Example
height of building, h=1.0724 x = 1.0724(56.74) = 60.85m
The angle of depression of a ship viewed at a particular instant
from the top of a 75m vertical cliff is 30◦.
Find the distance of the ship from the base of the cliff at this
instant. The ship is sailing away from the cliff at constant speed
and 1 minute later its angle of depression from the top of the cliff
is 20◦. Determine the speed of the ship in km/h.
Example

13. Trigonometric Identities
Lancashire -EUE
Dr. Hossam Kamal
Pg. 13
MAC101
Lec. 04
3.2

14. Trigonometric Identities
Lancashire -EUE
Dr. Hossam Kamal
Pg. 14
MAC101
Lec. 04
3.2
Prove that
#
tan %
" tan% =
#
sin % cos%
Example
Prove that
sin % − sin% cos%
+
=sin%
,
Example
Evaluate the following expression, correct
to 4 significant figures:
-.+ /01-2° +4'6− ,. 7 89144° #'
7. # :;8+2° ,-'
Example

15. Graphs of Trigonometric Functions
Lancashire -EUE
Dr. Hossam Kamal
Pg. 15
MAC101
Lec. 04
3.2
The wave functionThe wave function

16. Circle Applications
Lancashire -EUE
Dr. Hossam Kamal
MAC101
3.4
Equation of Ellipse
Circle is a special case of Ellipse when a=b=r
Pg. 16Lec. 04

17. Circle Applications
Lancashire -EUE
Dr. Hossam Kamal
Pg. 17
MAC101
Lec. 04
3.4
When the circle has its
center at the origin trigonometric relations
at a circle
trigonometric relations
at a circle

18. Circle, wave, and trigonometry
Lancashire -EUE
Dr. Hossam Kamal
Pg. 18
MAC101
Lec. 04
3.4
Let OR be a vector 1 unit long and free to rotate anticlockwise
about O. In one revolution a circle is produced and is shown
with 15◦sectors. Each radius arm has a vertical and a
horizontal component. For example, at 30◦, the vertical
component is TS and the horizontal component is OS. From
trigonometric ratios,
The vertical component TS may be projected across to
TS, which is the corresponding value of 30◦on the graph
of y against angle x◦. If all such vertical components as
TS are projected on to the graph, then a sine wave is
produced.
If all horizontal components such as OS are projected on
to a graph of y against angle x◦, then a cosine wave is
produced.

19. Periodic Functions
Lancashire -EUE
Dr. Hossam Kamal
Pg. 19
MAC101
Lec. 04
3.4
The wave functionThe wave function
< = =LS : ylPgE,LS
< = =LS : ylWkP,LS
amplitude period

20. Wave Functions
Lancashire -EUE
Dr. Hossam Kamal
Pg. 20
MAC101
Lec. 04
3.2
Example

21. The End
Thank you
Lancshire - EUE 2025.26 - Dr. Hossam Kamal 21