CHHOREPATAN SECONDARY SCHOOL

Chhorepatan, Pokhara

Project report of Mathematics

Submitted by: Submitted to:

Anup Ranabhat Arjun Aryal
12 “G”
804
 Acknowledge

I would like to take this opportunity to express my sincere gratitude and appreciation to
everyone who has contributed to the successful completion of this mathematics project on
trigonometry. First and foremost, I extend my heartfelt thanks to my respected mathematics
teacher, Arjun Aryal & Upendra Pokhrel, whose guidance, encouragement, and constructive
feedback have been invaluable throughout the process. Their support and expertise have helped
me gain a deeper understanding of trigonometry, its fundamental concepts, and its real-life
applications. I am also extremely grateful to my school, Chhorepatan Secondary School, for
providing me with the necessary resources and an environment conducive to learning and
exploration. This project would not have been possible without the continuous support and
motivation from my family, who have always encouraged me to work hard and strive for
excellence. Their patience, understanding, and belief in my abilities have played a significant role
in keeping me focused and determined. I would also like to extend my appreciation to my friends
and classmates, who have been a source of motivation and inspiration, making the learning
process more engaging and enjoyable. In addition, I would like to acknowledge the various books,
online materials, and educational resources that have provided me with valuable insights and
information, helping me to enhance the quality of this project. Throughout this journey, I have
not only expanded my knowledge of trigonometry but have also developed critical thinking,
problem-solving, and research skills, which will be beneficial in my academic journey ahead. This
project has been a truly enriching experience, and I am thankful to everyone who has been a
part of this learning process in one way or another.
 Certificate

This is to certify that Anup Ranabhat, a student of Class 12, Chhorepatan Secondary School,
has successfully completed the mathematics project on "Trigonometry: Concepts and
Applications" as per the guidelines provided by the school and under the supervision of Arjun
Aryal. This project has been submitted as part of the academic requirements for the
mathematics curriculum for the session 2081/82.

The project has been thoroughly reviewed and is found to be an original and sincere effort
demonstrating a deep understanding of the topic. It effectively covers various aspects of
trigonometry, including its fundamental concepts, important formulas, practical applications in
real life, and problem-solving techniques. The student has shown dedication, research skills, and
analytical thinking in compiling and presenting this project in a structured and comprehensive
manner.

I appreciate the hard work and enthusiasm put into this project and commend Anup Ranabhat
for the effort in making this project informative, well-researched, and neatly presented. I hope
this learning experience will enhance the student’s understanding of trigonometry and inspire
further exploration of mathematical concepts.

Student Signature Teacher Signature

 Preface

Mathematics has always been an essential part of our lives, helping us understand the world
through numbers, patterns, and logical reasoning. Among its various branches, trigonometry
holds a significant place, as it plays a crucial role in geometry, physics, engineering, astronomy,
and various other fields. This project, titled "Trigonometry: Concepts and Applications," is an
attempt to explore the fundamental principles of trigonometry, its historical background, key
formulas, and real-life applications. Through this project, I have delved into the origins of
trigonometry, dating back to ancient civilizations, and studied how it has evolved over time to
become one of the most widely used mathematical disciplines. The project also covers
trigonometric functions, identities, equations, and their importance in solving practical problems.
Furthermore, I have included various solved examples and illustrations to make the concepts
clearer and more engaging. Working on this project has been an enlightening experience,
allowing me to develop a deeper understanding of how trigonometry is applied in different fields,
from measuring heights and distances to analyzing wave motions and navigation. This project
would not have been possible without the continuous guidance of my mathematics teacher,
ARJUN ARYAL, whose support and encouragement have helped me immensely in understanding
the subject. I am also grateful to my school,Chhorepatan Secondary School, for providing the
necessary resources and an environment conducive to learning. Additionally, I extend my
gratitude to my family and friends, who have motivated me throughout this journey. This project
has not only enriched my mathematical knowledge but has also helped me enhance my research,
analytical, and problem-solving skills. I hope that this project serves as an informative and
valuable resource for those who wish to learn more about trigonometry and its vast applications.
 Contents

Cover Page
2. Certificate
3. Acknowledgment
4. Preface
5. Introduction to Trigonometry
6. History and Development of Trigonometry
7. Basic Trigonometric Ratios
8. Trigonometric Formula Sheet
9. Trigonometric Functions and Their Graphs
10. Inverse Trigonometric Functions Graphs
11. Applications of Trigonometry in Real Life
12. Solved Problems and Examples
13. Fun Facts and Interesting Insights
14. Conclusion and Summary
15. Reference
Introduction to Trigonometry
Trigonometry is one of the most fundamental and widely used branches of mathematics that deals
with the study of angles, triangles, and the relationships between their sides and angles. The word
"trigonometry" is derived from the Greek words "trigonon" (triangle) and "metron" (measure),
signifying its primary focus on the measurement and analysis of triangles. Initially developed to
solve problems in astronomy and land surveying, trigonometry has now expanded its applications
into various fields, including physics, engineering, computer graphics, architecture, navigation, and
even medicine. The core of trigonometry lies in the study of trigonometric ratios—sine, cosine,
tangent, and their reciprocals—which help determine unknown sides and angles in a triangle.
These ratios form the basis of trigonometric functions, which are extensively used in periodic
phenomena like sound waves, light waves, and alternating current electricity. The origins of
trigonometry can be traced back to ancient civilizations, particularly the Egyptians and Babylonians,
who used basic trigonometric principles for construction and astronomical calculations. However,
it was the Greek mathematician Hipparchus, often regarded as the "Father of Trigonometry," who
systematically developed the first trigonometric tables. Later, Indian and Islamic scholars made
significant advancements, introducing new concepts and functions, which were further refined by
European mathematicians during the Renaissance. Over time, trigonometry has evolved from a
tool for measuring distances and angles to a sophisticated mathematical discipline with
applications in calculus, algebra, and complex number theory. Today, trigonometry continues to
play a crucial role in solving real-world problems, from calculating the height of mountains and the
depth of oceans to designing intricate structures and predicting planetary movements.
Understanding trigonometry not only enhances mathematical problem-solving skills but also
provides a deeper insight into the fundamental principles governing the natural world.

History
The history of trigonometry dates back thousands of years, with its origins rooted in the ancient
civilizations of Egypt, Babylon, and Greece. Early applications of trigonometric principles can be
found in Egyptian and Babylonian mathematics, where they used basic geometric relationships to
construct pyramids and study celestial movements. However, it was the Greek mathematician
Hipparchus (c. 190–120 BCE) who is often credited as the "Father of Trigonometry" for developing
the first known trigonometric table and systematically studying the relationships between angles
and chord lengths in a circle. Later, the Indian mathematician Aryabhata (476–550 CE) made
significant contributions by introducing the modern sine function and formulating trigonometric
concepts in a more structured way. During the Islamic Golden Age, scholars like Al-Battani and
Omar Khayyam expanded upon Greek and Indian knowledge, refining trigonometric tables and
developing the six fundamental trigonometric functions: sine, cosine, tangent, cosecant, secant, and
cotangent. These advancements were later transmitted to Europe during the Renaissance, where
mathematicians such as Johannes Müller (Regiomontanus) and Leonhard Euler further formalized
trigonometry, integrating it into algebra and calculus. With the advent of modern mathematics,
trigonometry became an essential tool in fields like astronomy, physics, engineering, and navigation.
Today, trigonometry is not only confined to theoretical mathematics but also plays a crucial role in
scientific discoveries, technological innovations, and real-world problem-solving. Its historical
evolution demonstrates how mathematical knowledge has been built upon over centuries, shaping
the way we understand and interact with the world around us.
Basic Trigonometry Ratio
Trigonometry is based on six fundamental ratios that define the relationship between the angles and
sides of a right-angled triangle. These ratios—sine, cosine, tangent, cosecant, secant, and cotangent—
are used to calculate unknown angles or side lengths in triangles. They play a crucial role in solving
problems related to heights, distances, and periodic functions.

Trigonometric Formula Sheet

Definition of the Trig Functions
Right Triangle Definition Unit Circle Definition
Assume that: Assume θ can be any angle.
0 < θ <2 or 0◦ < θ < 90◦
π

(x, y)
1
y
θ
hypotenuse
opposite

θ
adjacent

opp hyp
sin θ = csc θ = y 1
hyp opp sin θ = csc θ =
1 y
adj hyp x 1
cos θ = sec θ = cos θ = sec θ =
hyp adj 1 x
opp adj y x
tan θ = adj cot θ = opp tan θ = x cot θ = y

Domains of the Trig Functions

sin θ, θ ∈ (−∞, ∞) csc θ, θ /= nπ, where n ∈ Z

cos θ, θ ∈ (−∞, ∞) 1
sec θ, θ /= n + π, where n ∈ Z
2
1
tan θ, θ /= n + π, where n ∈ Z cot θ, θ /= nπ, where n ∈ Z
2

Ranges of the Trig Functions

−1 ≤ sin θ ≤ 1 csc θ ≥ 1 and csc θ ≤ −1
−1 ≤ cos θ ≤ 1 sec θ ≥ 1 and sec θ ≤ −1
−∞ ≤ tan θ ≤ ∞ −∞ ≤ cot θ ≤ ∞
Identities and Formulas
Tangent and Cotangent Identities Half Angle Formulas
sin θ cos θ
tan θ = cot θ =
cos θ sin θ
Reciprocal Identities

1 1
sin θ = csc θ =
csc θ sin θ
1 1
cos θ = sec θ =
sec θ cos θ
Sum and Difference Formulas
1 1
tan θ = cot θ =
cot θ tan θ sin(α ± β) = sin α cos β ± cos α sin β

Pythagorean Identities cos(α ± β) = cos α cos β ∓ sin α sin β

sin2 θ + cos2 θ = 1
tan(α ± β) tan α ± tan β
tan2 θ + 1 = sec2 θ =
1 ∓ tan α tan β
1 + cot2 θ = csc2 θ
Product to Sum Formulas
Even and Odd Formulas 1
sin α sin β = [cos(α − β) − cos(α + β)]
sin(−θ) = − sin θ csc(−θ) = − csc θ 2
1
cos(−θ) = cos θ sec(−θ) = sec θ cos α cos β = [cos(α − β) + cos(α + β)]
2
tan(−θ) = − tan θ cot(−θ) = − cot θ
1
Periodic Formulas sin α cos β = [sin(α + β) + sin(α − β)]
2
If n is an integer 1
cos α sin β = [sin(α + β) − sin(α − β)]
sin(θ + 2πn) = sin θ csc(θ + 2πn) = csc θ 2
cos(θ + 2πn) = cos θ sec(θ + 2πn) = sec θ Sum to Product Formulas
tan(θ + πn) = tan θ cot(θ + πn) = cot θ
α+β α−β
sin α + sin β = 2 sin cos
Double Angle Formulas 2 2
α+β α−β
sin(2θ) = 2 sin θ cos θ sin α − sin β = 2 cos sin
2 2
α+β α−β
cos(2θ) = cos2 θ − sin2 θ cos α + cos β = 2 cos cos
= 2 cos2 θ − 1 2 2
= 1 − 2 sin2 θ α+β α−β
cos α − cos β = −2 sin sin
2 tan θ 2 2
tan(2θ) = Cofunction Formulas
1 − tan2 θ
Degrees to Radians Formulas π π
sin − θ = cos θ cos − θ = sin θ
If x is an angle in degrees and t is an angle in 2 2
radians then: π π
csc − θ = sec θ sec − θ = csc θ
2 2
π t πx 180◦t π π
= ⇒ t= and x = tan − θ = cot θ cot − θ = tan θ
180◦ x 180◦ π 2 2
 Unit Circle
(0, 1)

1
√
3 90◦, π2 √
(− 2, 2 )
( 21 , 3
2 )

√ √ √ √
2 2 2 2
(− 2 , 2 ) ( 2 , 2 )
120◦, 2π3 60◦, π3
√ √
3 1 135◦, 3π4 45◦, π4 ( 23 , 12 )
(− 2 , 2)

150◦, 5π6 30◦, π6

180◦, π
(−1, 0) (1, 0)
0◦, 2π

210◦, 7π6 330◦, 11π

6
√ √
3 1 3 1
(− 2 , − 2) 225◦, 5π4 315◦, 7π4 ( 2 , − 2)

√ √
240◦, 4π3 300◦, 5π3 √ √
(− 22 , − 22 ) ( 2
2 , − 2
2 )

√ √
3 3
(− 12 , − 2 ) ( 12 , − 2 )
270◦, 3π2

(0, −1)

For any ordered pair on the unit circle (x, y) : cos θ = x and sin θ = y
 Inverse Trig Functions
Definition Inverse Properties
These properties hold for x in the domain and θ in
θ = sin−1(x) is equivalent to x = sin θ the range

θ = cos−1(x) is equivalent to x = cos θ θ

sin(sin−1(x)) = x sin−1(sin(θ)) = θ
= tan−1(x) is equivalent to x = tan θ
cos(cos−1(x)) = x cos−1(cos(θ)) = θ

Domain and Range tan(tan−1(x)) = x tan−1(tan(θ)) = θ

Function Domain Range

π π Other Notations
θ = sin (x)
−1
−1 ≤ x ≤ 1 — ≤θ≤
2 2
sin−1(x) = arcsin(x)
θ = cos−1(x) −1 ≤ x ≤ 1 0≤θ≤π
cos−1(x) = arccos(x)
π π
θ = tan (x)
−1 −∞ ≤ x ≤ ∞ — ≤θ≤
1 2 tan−1(x) = arctan(x)

Law of Sines, Cosines, and Tangents

a c

C A
b

Law of Sines Law of Tangents

sin A sin B sin C a − b = tan 2 (A − B)
1
= =
a b c a + b tan 21 (A + B)
Law of Cosines
b−c tan 1 (B − C)
= 2
a2 = b2 + c2 − 2bc cos A b+c tan 21 (B + C)

b2 = a2 + c2 − 2ac cos B a−c

= tan 2 (A − C)
1

a + c tan 1 (A + C)
c2 = a2 + b2 − 2ab cos C
2
 Graphs of Trigonometry Functions

Function Parent Graph of Function Characteristics

Name Function
Domain: (−∞, ∞)
Range: [−1, 1]
Sine f(x) = sin(x) Odd/Even: Odd
Period: 2π

Domain: (−∞, ∞)
Range: [−1, 1]
Cosine f(x) = cos(x) Odd/Even: Even
Period: 2π

Domain: (−∞, ∞) except for

π
x = ± nπ
f(x) = tan(x) 2
Range: (−∞, ∞)
Tangent sin(x) Odd/Even: Odd
= Period: π
cos(x) π
Asymptotes at x = ± nπ
2

Domain: (−∞, ∞) except for

x = ±nπ
f(x) = csc(x) Range: (−∞, −1] ∪ [1, ∞)
Odd/Even: Odd
Cosecant 1
= Period: 2π
sin(x)

Domain: (−∞, ∞) except for

π
x = ± nπ
2
f(x) = sec(x)
Range: (−∞, −1] ∪ [1, ∞)
Secant 1 Odd/Even: Even
= Period: 2π
cos(x)

f(x) = cot(x) Domain: (−∞, ∞) except for

x = ±nπ Range:
1 (−∞, ∞)
= Odd/Even: Odd
Cotangent tan(x)
Period: π
cos(x) Asymptotes at x = ±nπ
=
sin(x)
 Graphs of Inverse Trigonometry Functions

Function Parent Graph of Function Characteristics

Name Function

Domain: [−1,1]
f(x) = sin−1(x)
Inverse Sine Range:
−π
,
π
= arcsin(x) 2 2

Domain: [−1,1]
f(x) = cos−1(x)
Inverse
Cosine Range: [0, π]
= arccos(x)

Domain: (−∞, ∞)
f(x) = tan−1(x)
Inverse −π π
Tangent Range: ( , )
= arctan(x) 2 2

Domain: (−∞, −1] ∪ [1, ∞)

f(x) = csc−1(x)
Inverse −π π
Cosecant Range: , , y≠ 0
= arccsc(x) 2 2

Domain: (−∞, −1] ∪ [1, ∞)

f(x) = sec−1(x)
Inverse π
Secant Range: [0, π], y≠
= arcsec(x) 2

Domain: (−∞, ∞)
f(x) = cot−1(x)
Inverse
Cotangent Range: (0, π)
= arccot(x)
Application of Trigonometric Function
Trigonometry has a wide range of applications in various fields of science, engineering, and daily
life. Some of its most important real-world uses include:

 Astronomy and Space Science – Trigonometry is used to measure the distance between
planets, stars, and celestial bodies. Astronomers use it to calculate the positions of planets
and the trajectory of satellites.
 Navigation and GPS – Ships, airplanes, and even mobile GPS systems use trigonometry to
determine locations, distances, and directions accurately.
 Architecture and Construction – Engineers and architects use trigonometric functions to
design buildings, bridges, and tunnels by calculating angles, heights, and slopes.
 Surveying and Cartography – Land surveyors use trigonometry to measure distances and
map terrains, helping in urban planning and construction projects.
 Physics and Engineering – Trigonometry plays a crucial role in mechanics, waves, optics,
and electrical circuits, helping engineers design machines, vehicles, and electronic systems.
 Medical Imaging and Ultrasound – Trigonometric calculations help in CT scans, MRI scans,
and ultrasounds by reconstructing images based on wave reflections.
 Sound and Light Waves – Trigonometry is used in analyzing wave patterns in music,
acoustics, and light reflections to improve sound quality and optical instruments.
 Criminology and Forensic Science – Crime scene investigators use trigonometry to analyze
bullet trajectories, blood spatter angles, and accident reconstructions.
 Sports and Game Development – Trigonometry is used in sports like golf, basketball, and
baseball to calculate angles and trajectories. It is also widely used in video game
development to create realistic movements and physics.
 Aviation and Rocket Science – Pilots use trigonometry to navigate airplanes, calculate
altitude, and determine optimal flight paths, while space agencies use it to launch rockets
into orbit.

Trigonometry is an essential mathematical tool that influences many aspects of our lives, helping
us understand and solve complex problems in practical scenarios.
Problem:
Two boats leave a dock at the same time. One travels north at 20 km/h, and the other travels
east at 15 km/h. Find the distance between them after 2 hours.

Solution:
After 2 hours,

 The northbound boat has traveled 20×2=4020 \times 2 = 4020×2=40 km.

 The eastbound boat has traveled 15×2=3015 \times 2 = 3015×2=30 km.

The boats form a right-angled triangle. Using the Pythagorean theorem:

d2 =402 + 302 =2500

d = 50

Answer: The distance between the boats is 50 km.

Fun Facts and Interesting Insights

Ancient Origins – The word "trigonometry" comes from the Greek words trigonon (triangle) and
metron (measure), meaning "measuring triangles." The earliest known trigonometric table was
created by the Greek mathematician Hipparchus around 140 BCE.
Inspired by the Stars – Trigonometry was first developed for astronomical studies, helping
ancient civilizations calculate the positions of stars and planets. Even today, NASA uses
trigonometry to navigate space missions.
Hidden in Music – Trigonometry plays a major role in music and sound waves. The sine and
cosine functions describe how sound waves move, helping in tuning musical instruments and
designing audio systems.
The Mysterious 90° Angle – Right-angled triangles are the foundation of trigonometry. Without
a 90° angle, trigonometric ratios like sine, cosine, and tangent wouldn’t exist in the way we know
them!
The Sine Wave Everywhere – The sine function appears in nature, physics, and technology. It
describes ocean waves, light waves, radio signals, and even heartbeats on an ECG machine.
Used in Video Games – Trigonometry is widely used in game development to create realistic
motion, lighting, and 3D animations. Popular games like Minecraft and Call of Duty rely on trig
functions for movement and physics.
Trigonometry in Crime Solving – Forensic experts use trigonometry to analyze blood spatter
angles in crime scenes, helping to determine how incidents occurred.
Essential for GPS and Navigation – Without trigonometry, Google Maps and GPS systems
wouldn’t work! Trig helps in calculating distances and locations based on satellite data.
Used in Roller Coaster Design – Engineers use trigonometry to design roller coasters, ensuring
the right angles and slopes for a thrilling but safe ride.
Trigonometry and Shadows – The length of shadows changes based on the position of the sun,
and trigonometry helps calculate the height of objects just by measuring their shadows.
Trigonometry is all around us, from the smallest sound waves to the vast universe! It continues
to shape modern science, engineering, and technology in ways we often don’t even realize.

Conclusion
Trigonometry is a fundamental branch of mathematics that plays a vital role in various aspects of
life, from scientific advancements to everyday applications. It primarily deals with the
relationships between the angles and sides of triangles, with functions such as sine, cosine, and
tangent forming the basis of its calculations. The subject has evolved over centuries, starting
from ancient civilizations that used it for astronomy and navigation to modern-day applications
in engineering, physics, medicine, and technology.

Through this project, we explored key trigonometric concepts, including identities, ratios,
formulas, and real-world applications. We also examined how trigonometry is used in fields like
architecture, aviation, space exploration, and even digital technology such as GPS and video
game design. The problem-solving section demonstrated how trigonometry can be applied in
practical situations, reinforcing its significance in real life.

In conclusion, trigonometry is an essential mathematical tool that continues to shape various

disciplines. Its impact is seen in everything from the construction of buildings to the study of
waves and the movement of celestial bodies. Understanding trigonometry not only enhances
mathematical knowledge but also opens the door to countless career opportunities in science
and engineering. As technology advances, trigonometry will remain a crucial element in solving
complex problems and shaping the world around us.
 Reference
Books:
Michael J. Peterson, Basic Trigonometry for Beginners, Starwood Publications, 2010.
Lisa R. Hoffman, Understanding Trigonometry: A Step-by-Step Guide, BrightLeaf Press, 2013.
Daniel K. Simmons, Trigonometry Made Easy, Maplewood Educational, 2009.

Websites:
Desmos (https://www.desmos.com/calculator)
Cymath (https://www.cymath.com)