Analysis of Pythagoras Theorem and Trigonometry

The provided document analyzes the concepts of Pythagoras theorem and trigonometry, including a discussion on the importance of these mathematical concepts in solving problems. It reviews various resources, such as books and journals, to provide insights into how these concepts are taught and applied in different contexts.

Finding the missing sides using Pythagoras
theorem and trigonometry

ABSTRACT
Pythagoras theorem or trigonometric functions can be used to calculate the missing side or the
size of a right-angled triangle. Pythagoras Theorem and Trigonometric functions play a key role
in GCSE mathematics both in foundation and higher papers. It is important that the students
understand the concept of the theorem to build further in Geometry. The basic aim of this paper
to analyse that how Pythagoras and trigonometric theorem usually beneficial for real world
because there concept applied for the purpose of construction of building and also measure the
distance of mountain and rope. This paper is discussed about the issue related to the
contemporary pedagogy in this topic nowadays. In current scenario, Modern approach help for
teach the meaning full concept and understand the idea that applied in the real world.

Table of Contents
ABSTRACT.....................................................................................................................................2
INTRODUCTION...........................................................................................................................4
DISCUSSION..................................................................................................................................5
LITERATURE REVIEW................................................................................................................9
MISCONCEPTIONS.....................................................................................................................11
CONCLUSION..............................................................................................................................17
REFERENCES..............................................................................................................................18

INTRODUCTION
The Pythagoras theorem is basically the fundamental relation with the Euclidean geometry that
consists of three side of right triangle. The Pythagoras theorem plays an important role in the
mathematics and geometry. This theorem is beneficial for the two dimensional navigation and
also easily identify the shortest path and distance. This will be used in different areas and field
such as mathematics, science, music etc. There is a long 4000 years of history behind this
theorem. Many researchers and great mathematicians had proved this theorem and embraced its
importance in their work (Maor, 2007). According to Pythagoras, It can be easily calculate the
actual solution with the help of number and values. Mathematic is the basic foundation for
everything. Geometry is the highest form of mathematical studies to apply the new approach for
the measurement. The physical world could be understood through mathematics. He connected
the patterns and contributed theories to understand the relations of angles, triangles, areas,
proportion and polygons.

DISCUSSION
“To this day, the Pythagoras’ theorem remains the most important single theorem in the whole of
mathematics” – Jacob Bronowski, (2017). The Ascent of Man, P160
The Pythagorean Theorem is a cornerstone of mathematics and continues to be so
interesting to mathematicians that there are more than 400 different proofs of the theorem,
including an original proof by President Garfield. In this study and research carried out the
national curriculum and pedagogy of mathematics in schools all over the country and
internationally. The Pythagorean theorem is consists of various proofs:
According to the right triangle, the length square of hypotenuse always equal to the sum of
another side of triangle.
a² + b² = c²
Method- 1
According to this triangle, It will be prove that a² + b² = c²
First of all, it can be constructed the perpendicular from C to segment AB.
Illustration 1
Illustration 2

In this right triangle, The altitude is divided into the right angle to the hypotenuse in the
different two segments e and f. According to geometric, the length of each side of right triangle
is the geometric means that the length of hypotenuse. On the other hand, the hypotenuse
segments is the adjacent to the side.
Therefore,
c / a = a / e and c/ b = b / f
Step 1: it is cross multiply in the both the sides and generate the ratio.
ce = a^2
cf= b^2
Step 2: Add a and b
a^2 + b^2 = ce+cf
Step 3: Factoring out the c on the right side that given:
a^2 + b^2 = c(e+f)
Step: Substitution
c= e+f
a² + b² = c²

Method-2
This is the another method to proof the a² + b² = c² by using this diagram.
It is notice that the square root can be expressed in two different ways.
= a² + b²
= (a+b)..............equation-1
There are different areas of four triangles ½ ^ab and areas of the smaller square of c^2
= 4(1/2^a b)+c^2..............equation-2
Both the side is representing the large square and combine to equation one and two-dimensional
(a+b) = 4(1/2^a b)+c^2
Afterwards, it simplifies the equation
a^2+2ab+b^2 = 2ab +c^2
Subtracted the 2ab in the both side of equation
a^2 +b^2 = c^2
Illustration 3

Method-3
This is the last method to simplify the equation and prove that a² + b² = c².
This method is expressed the trapezoid formula to simplify the equation of Pythagoras theorem.
= 1 / 2 (b1+b2).h
Let uses the diagram to fill the formula for the areas of trapezoid.
= 1 / 2 (a+b)(a+b)
Multiplying the 1 / 2 in both the brackets
= 1 /2 ab + 1/ 2 ab + 1/ 2 c^2
Since, it expresses the two areas in the trapezoid and also set that equal to each other. It will be
use the algebra to easily simplify the equation in proper manner.
=> 1 /2 ab + 1/ 2 ab + 1/ 2 c^2 = 1 / 2 (a+b)(a+b)
=> ab + 1 / 2 c^2= 1 / 2 (a^2 +2ab+ b^2)
=> 2ab + c^2 =a^2 + 2ab+ b^2
c² = a² + b²
a² + b² = c²
Illustration 4

LITERATURE REVIEW
As per Yang, (2009) in UK the secondary school students would be introduced to
Pythagoras theorem in Key stage 3 and develop more in key stage 4. GCSE foundation and
higher tier exam papers contains at least one question based on Pythagoras theorem. The main
concepts of Trigonometry is calculated the missing the sides or angles using Trigonometric
functions such as Sine, Cos and Tan. Pythagoras theorem can be used only in the right-angled
triangle. Identifying the angles of the triangle and classifying into right category such as right-
angled triangles contains 90 degree angles, Isosceles triangle contains two equal sides and
angles, etc.
According to Mirna, (2018) Trigonometric is based on the real world because it helps for
calculating the values, length of triangle. It is generally implements the new concept to show the
calculation, reasoning and study about the properties of triangle. It must be the simplest figure
that applicable in the real time application to measure the distance in effective manner. These
concepts are applicable in different field and also choose the potential career path according to
the interest. On the basis of Singh and Bajaj, (2018) Math is used in the music world where the
people listen the songs on the basis of their choices but the concept used behind the music is
sound wave. This will help for transmitting from one place to another by using graphical
representation. The people cannot see the wave but only listen the particle that move. This
graphical representation can be formed by the cos and sine function that used in the
trigonometric. The engineer and music producer are able to manipulate the sound wave with the
help of trigonometric function. The pattern of wave attributes such as volume, base and trebles
which suitable for the listener.
On the other hand, Trigonometric concept is also useful in the video game development
because it helps for buildings obstacles in the video game services. It also calculated the actual
trajectory of projectile for the purpose of accuracy if it leaves the gun. This will be new concept
that useful for deploying the space instrument and equipment such as rocket and satellite. This
will help for calculating the accurate speed of long distances. Even, the miscalculation sends to
the satellite in the wrong side so that it is the best concept to determine the proper direction and
distance.

According to Williams, (2018). Pythagoras theorem is also used in the real time application for
constructing the building to measure the accurate angle in proper manner. This theorem is also
beneficial in the lift to set the proper measurement and also identify the accurate distance.
The Pythagorean Theorem is also known as Pythagoras' theorem. It explains the basic
relationship between the three sides of the right-angled triangle. According to the theorem, the
square of the longest side, called Hypotenuse, which is opposite to the right angle, is always
equal to the sum of the squares of the other two sides. This theorem is based on the well-known
mathematical statement that describes the right angle triangle.
For Example-
Illustration 5
a2 + b2 = c2
However, There are various number of facets in the Pythagoras theorem and this facet is
also known as Euclid's formulas that allow generating the results by using theorem. It also
satisfied the set of integers such as
a = p2 -q2
b = 2pq
c = p2+q2
For Example. It might that p= 2 and q= 1
In case, p = 22-12 = 3, q= 2 x 2 x 1 = 4
c = 22 +12 =5

In this way, it is calculated the three number 3, 4 and 5 which is the smallest Pythagoras
numbers. This formula is basically work on the integer values for different range of values.
MISCONCEPTIONS
To find the missing side in the right angle triangle the students use Pythagoras Theorem. The
common misconception in Pythagoras theorem are as follows:
 Fail to identify the right-angled triangle
 Fail to identify the hypotenuse
 Misrepresentation of the equation,
for example, + = instead of  =
 Miscalculations such as eliminating the squares on both the sides instead of calculating
the square roots on equations.
 In a higher exam paper, it is hard to identify the triangle in a complex diagram or a word
problem.
According to Williams, (2018) these common errors can be rectified by deriving the
equations or formulae, exploring the theorem and proving it. It always helps pupils to see the big
picture by visualising the angles in both two-dimensional and three-dimensional shapes. This
would help the students identify, and give a clear image of the given shapes. Also, it will be very
interesting to connect the theory to the real life examples. For instance, calculate the distance
from ground to the mountain, ladder etc.
Researcher also investigating that teachers from three different countries and cultures
namely Czech Republic, Hong Kong and Shanghai teach Pythagoras’ theorem to year 8 in
schools. The analysis showed that the teachers in Hong Kong and Shanghai emphasized
exploring the theorem visually and proved mathematically while the Czech teacher verified the
theorem by demonstrating. It is very important to involve the students actively in the process of
learning by providing various problems within both mathematical and daily life contexts. It is
complicated and more challenging implementing in the normal classroom settings. Moreover, it
can be achieved by following Jo Boaler’s suggestions such as grouping students with mixed
ability and using technology. For example, group the students in each table with mixed ability
(two higher and three lower) and provide different tasks which involve all the pupils and bring
out their potential like making a project using Pythagoras theorem. In this way, it can be said that

this theorem is beneficial for student to apply the concept in their life. It explores their
knowledge and skill towards the growth and development. Mathematically concept is the best
learning to understand the challenging issues and problem. Many researchers always focus
towards their new idea and though develop through the Pythagoras theorem because it can be
implemented to the daily life.
Misconception:
According to author, There are some misconception generated when the pupils are
confused over the side the triangle. It is difficult for identifying that particular hypotenuse and
triangle is orientated. For Example-
Pupil must understand the hypotenuse by applying the Pythagoras theorem which applicable only
on the right angle triangle.
According to Brezov, (2018) The Japanese teaching approach is to group the students. It
also motivates towards the learning of theorem and also implements the concept in real world.
Learning from each other and get along and grow together. Research tells that this approach in
teaching at school maintains equality between them. On the other hand, in UK, the teaching
approach is towards attaining the best GCSE results rather than the achievements of the students
in their later life. It is important to achieve higher marks but the standard of education should not
be comprised. This can be achieved in schools by providing the students enough time to explore
and learn the topic by practical implementation of the theory. So the pupils can relate the topic to
the real life examples and develop their subject knowledge further.
As per Cullen and Martin, (2018) in many countries the schools still follow the traditional
pedagogy to teach the subjects especially Mathematics. However, the technology inventions help
the teachers and students to understand and solve the problems effortlessly. Also, it embeds the
applications of other knowledge such as ICT skills, logical thinking and more. It connects to the
real time and provides an experience to achieve higher results. According to Mansfield, 2018
Pythagoras theorem is successfully discovered by the author. He also offered to sacrifice of oxen
and also contribute in the term of mathematical to expand the need and requirement of
development. This theorem is introduced for student during the school years to understand the

concept. It is important theorem in school days where student see the formula and applicable on
the calculation. It also connected with the geometric. The learning and teaching of this theorem
can help for enhance the understanding and use for the purpose of paper folding, dot paper,
computer technology geo-boards and various type of materials. It is the beneficial for used in the
other field and areas to manipulate the information in proper manner. In this way, student easily
calculated the calculation by using Pythagoras formulas. On the basis of Fleurant and Bodin-
Fleurant, (2019) mathematics learning is mart and adoptable technique to solve the problem and
generate the solution in tricky ways, it is brain teaser that builds more complex crises and basic
law applications etc. It is needed for student to build the strong concept that help for easily
resolve all the problems and issues on every day, undoubtedly, it will help for handle and
manage the situation by using the positive skill and approach. Pythagoras and trigonometric are
art of science that applicable in the real world to solve the problem in easy ways. The generation
of computer is also imaginary concept that possible by using the pattern of geometry because it
can be defined the proper locations, colour of image and each points. Image is also made by
modern and accurate technique to refer to the triangulations. It is designed to connect one point
to another so that it will create a proper image. Pythagoras is slightly helps for people to create
own ideas and maintain the every integer number in proper manner. The mathematicians always
use the new concept and implemented idea in the different method that applied in the theorems.
According to Güner, (2018) Pythagoras theorem is famous that help for generating the
accurate results for the specific triangles. It is originally demonstrated the proof of this theorem
and method used the different Euclid's elements for calculating the accurate outcome in effective
ways. It is expressed the ratio of integer number in multiple ways. Student understands this
theorem to learn about the proof and their formulas that generating the values. The Egyptians
used the theorem for constructing the building by using right angle concepts. It uses the rope
which formed into the 3, 4 and 5 triangles that giving the angle exactly look like 90 degree. In
this way, this concept is applied in the school to use many examples related to the rope. In this
way, student checks the right angle at home and at school by using Pythagoras theorem concepts.
According to Hashmi, Hussain and Shoaib, (2018) Pythagoras and trigonometric are the
best partner in the term of modern architecture that help for curved the surfaces in stone, steel
and glass. On the other hand, it is applied in the straight planes in the buildings but at the angle to
another. It is the advanced technology to easily measure the distances and their space. In hill

station, this concept will use for the purpose of measure the length of mountains. In this way,
researcher easily identified the information of value for the aircraft in navigation and designing.
The Pythagoras theorem is the most important part of mathematics whereas it is the
generalisations and consequences of wide range of implications. After 6th BC The philosopher
and great mathematician develop the ancient civilization theorem. It has strong evidence before
1000 years ago.

Question- 10
Ans: In ABC is a right triangle.
AB= 6 cm
BC= 9 cm
(BC)² - (AC)² = (AB)²
81- 36 = (AB)²
45 = (AB)²
6.70 = AB
Question- 11
Ans: ABC is a right triangle
AB = 10 cm
AC = 20 cm

BAC=90 degree
(BC)² = (AB)² + (AC)²
(BC)² = 100 + 400
(BC)² = 500
BC= 22.3
Question- 13
Ans:
AD = 10 cm
DC = 9 cm
DC = 3 cm
angle ABC =angle BCD = 90 degree
By joining the DE
AE = 6 cm and BE = 3 cm
BC = 8 cm
(AC)² = (AB)² + (BC)²
(AC)² = 81+ 64
(AC)² = 145
AC = 12.04

Question- 14
Ans:
ABC is a right triangle.
AB= ?
AC= 6 m
BC= 1.5 m
(AB)² = (AC)² - (BC)²
(AB)² = 36 – 2.25
AB = 5.809
CB
A

CONCLUSION
As per discussion, It concluded that Pythagoras theorem and trigonometric are the
important concept that help for solving the mathematical problem in proper manner. There are
different misconception and errors generated in daily life which help for solving the issues. It
analyses and reviews the concept of Pythagoras theorem. Researchers always manipulate the
problem that manage and handle by the logical concept of mathematics in effective ways.

REFERENCES
Books and journals:
Brezov, D.S., 2018. INTRODUCTION TO LINEAR ALGEBRA AND ANALYTIC
GEOMETRY.
Cullen, C.J. and Martin, T.S., 2018. Discovering Trigonometric Identities in Geometric
Representations. Mathematics Teacher, 112(3). pp.240-240.
Fleurant, C. and Bodin-Fleurant, S., 2019. Trigonometry, Geometry of Plane and Space.
In Mathematics for Earth Science and Geography (pp. 67-96). Springer, Cham.
Güner, N., 2018. How to Teach the Pythagorean Theorem: An Analysis of Lesson Plans. Ankara
University Journal of Faculty of Educational Sciences, 51(1). pp.119-141.
Hashmi, A., Hussain, T. and Shoaib, A., 2018. Alignment between Mathematics Curriculum and
Textbook of Grade VIII in Punjab. Bulletin of Education and Research, 40(1).pp.57-76.
Huang, R. and Leung, F.K., 2002. How Pythagoras' theorem is taught in Czech Republic, Hong
Kong and Shanghai: A case study. Zentralblatt für Didaktik der Mathematik, 34(6).
pp.268-277.
Mansfield, D., 2018. The world's oldest and most accurate trigonometry table. Australasian
Science, 39(2).p.25.
Maor, E., 2007. The Pythagorean theorem: a 4,000-year history.
Mirna, M., 2018, April. Errors Analysis of Students in Mathematics Department to Learn Plane
Geometry. In IOP Conference Series: Materials Science and Engineering (Vol. 335, No.
1.p. 012116). IOP Publishing.
Singh, J. and Bajaj, R., 2018. Pythagoras, Binomial, and de Moivre Revisited Through
Differential Equations. arXiv preprint arXiv:1803.11458.

Williams, G., 2018. 102.09 Measurement of polygons and circles. The Mathematical
Gazette, 102(553).pp.121-127.
Yang, Y., 2009. How a Chinese teacher improved classroom teaching in Teaching Research
Group: A case study on Pythagoras theorem teaching in Shanghai. ZDM, 41(3). pp.279-
296.