Volume 04 Issue 01-2024
43
International Journal Of Management And Economics Fundamental
(ISSN
–
2771-2257)
VOLUME
04
ISSUE
01
P
AGES
:
43-47
SJIF
I
MPACT
FACTOR
(2021:
5.
705
)
(2022:
5.
705
)
(2023:
7.
448
)
OCLC
–
1121105677
Publisher:
Oscar Publishing Services
Servi
ABSTRACT
Recommendations on the method of using fractions, equivalent formulas, series of equivalent ratios, inverse
trigonometric functions, the largest and smallest values and finite trigonometric series in order to apply the theoretical
part of the science in practice, that is, to increase the skills of solving examples and problems, were highlighted.
KEYWORDS
Fractions, trigonometric solutions to geometric problems, radius, perimeter, face of a circle inscribed outside a
triangle.
INTRODUCTION
In general secondary schools and academic lyceums,
fractions are taught in the mathematics course,
covering topics such as comparing fractions,
operations on fractions, rational numbers, decimal
fractions, and their combinations. The didactic
methods of teaching these topics are essential in
implementing the unity of theory and practice in
education, emphasizing the significance of teaching
from the perspective of theory [4].
During the periods, educational programs were carried
out in a traditional way, that is, lectures were carried
out through large books and manuals. This, in turn, has
Research Article
BASED ON AN INTEGRATIVE APPROACH TO TRIGONOMETRY EXAMPLES
AND EXERCISES APPLICATION OF FRACTION PROPERTIES
Submission Date:
January 02, 2024,
Accepted Date:
January 07, 2024,
Published Date:
January 13, 2024
Crossref doi:
https://doi.org/10.37547/ijmef/Volume04Issue01-07
Bakirov Tolkinjon Yunusaliyevich
Associate Professor Of The Department Of Mathematics Of Fergana State University, Uzbekistan
Axmadjonova Oydinxon Soyibjon Qizi
An Instructor At The Mathematics Department Of Farg‘Ona State University, As Indicated
, Uzbekistan
Journal
Website:
https://theusajournals.
com/index.php/ijmef
Copyright:
Original
content from this work
may be used under the
terms of the creative
commons
attributes
4.0 licence.
Volume 04 Issue 01-2024
44
International Journal Of Management And Economics Fundamental
(ISSN
–
2771-2257)
VOLUME
04
ISSUE
01
P
AGES
:
43-47
SJIF
I
MPACT
FACTOR
(2021:
5.
705
)
(2022:
5.
705
)
(2023:
7.
448
)
OCLC
–
1121105677
Publisher:
Oscar Publishing Services
Servi
weakened the issue of ensuring that the quality of
education is as high as possible. In modern education,
special attention is paid to the process of digitization
of education in order to increase the quality indicators
of education. The current state of the education
system is characterized by the increasing role of non-
traditional educational technologies. Learning by the
learner with their help is much faster than with
traditional technologies. These technologies change
the nature of knowledge development, acquisition and
distribution, deepening and expanding the content of
the studied subjects, quickly updating it, using more
effective teaching methods, and also significantly
expanding the opportunity for education for everyone
will give [3].
In the current educational landscape, practical
exercises on fractions are utilized in mathematics
lessons for various topics. In addition to the basic
mathematics, algebra, and geometry courses, selected
topics in Trigonometry are introduced for 4th-year
students in higher education, where practical skills in
applying trigonometry to geometric problems are
emphasized. Specifically, equivalent formulas and
ratios are employed to address fundamental problems
in the field.
The application of the sine theorem for triangles allows expressing trigonometric elements for each side through
angles. The relationships are substantial and can be derived using the sine theorem
+ + =
, leading to
equivalent relationships. Utilizing the sine theorem, if we substitute angle
a
with the side opposite to the triangle and
the corresponding face, the formula
2
sin
sin
sin
a
b
c
R
=
=
=
(1) emerges.
Further, by substituting the left side of
2
sin
a
R
=
into the formula, we obtain the equation
2
2
sin
a
S
R
h
=
(2).
Simplifying the (2) equation by shortening both sides with the tangent of half the angle
sin
a
S
h R
=
(3),
Volume 04 Issue 01-2024
45
International Journal Of Management And Economics Fundamental
(ISSN
–
2771-2257)
VOLUME
04
ISSUE
01
P
AGES
:
43-47
SJIF
I
MPACT
FACTOR
(2021:
5.
705
)
(2022:
5.
705
)
(2023:
7.
448
)
OCLC
–
1121105677
Publisher:
Oscar Publishing Services
Servi
we can express it as 2R, as shown in formula
2
sin
sin
a
h
R
=
(4).
The equivalence ratios for 2R were derived earlier as
2
2
sin
sin
sin
sin
sin
sin
a
a
h
a
b
c
S
R
h
=
=
=
=
=
[1].
By substituting this information into the equation
(
)
(
)
(
)
(
)
2
sin
sin
sin
2sin
cos
sin
2
2
2
2
2sin
cos
sin
2sin
cos
2sin
cos
2
2
2
2
2
2
2 sin
cos
cos
4 sin
cos
cos
4 cos
cos
cos
2
2
2
2
2
2
2
2
2
R
a b c
p
R
R
R
R
R
R
+
+
+ +
+
−
=
=
=
+
−
+
=
+
−
+
−
+
+
=
+
+
=
+
=
+
−
+
−
=
+
=
=
we find an
equivalent ratio for the perimeter of the triangle in terms of 2R
2
2 cos
cos
cos
2
2
2
p
R
=
(5)
with the given lengths of the sides and the external circumradius, the formula for calculating the area
2 sin
2 sin
2 sin
2
2
4
2
2
2sin sin
sin
abc
abc
R
R
R
S
S
R
R
R
S
S
=
=
=
=
(6)
is obtained.
We get the equivalent ratios obtained above:
Volume 04 Issue 01-2024
46
International Journal Of Management And Economics Fundamental
(ISSN
–
2771-2257)
VOLUME
04
ISSUE
01
P
AGES
:
43-47
SJIF
I
MPACT
FACTOR
(2021:
5.
705
)
(2022:
5.
705
)
(2023:
7.
448
)
OCLC
–
1121105677
Publisher:
Oscar Publishing Services
Servi
2
2
sin
sin
sin
sin
sin
sin
2sin sin
sin
2 cos
cos
cos
2
2
2
a
a
h
a
b
c
S
p
S
R
h
=
=
=
=
=
=
=
In geometric problems, there is a connection between the area and perimeter of a triangle through its sides, but
there is no direct connection between the angles. This problem can be addressed through the application of
trigonometric equivalent ratios. In other words, we utilize the fundamental property of fractions for this equation,
2sin
sin
sin
2 cos
cos
cos
2
2
2
p
S
=
,
applying the product of the tangent of half-angles and using the connection between the acute angles to determine
2
2
2
cos
cos
cos
sin
sin
sin
2
2
2
2
2
2
S
p
S
p tg
tg
tg
=
=
Problem 1
: For any given triangle with the area , and two angles and provided, determine its perimetr.
Solution: In this problem, the area and perimeter of the triangle are related through its sides and the internal
circumradius. Although there are no formulas directly connecting the area and perimeter to the angles between the
sides, we can employ equivalent ratios established for the external circumradius of a triangle, as explained before.
2
2sin
sin
sin
2 cos
cos
cos
2
2
2
p
S
R
=
=
S
Volume 04 Issue 01-2024
47
International Journal Of Management And Economics Fundamental
(ISSN
–
2771-2257)
VOLUME
04
ISSUE
01
P
AGES
:
43-47
SJIF
I
MPACT
FACTOR
(2021:
5.
705
)
(2022:
5.
705
)
(2023:
7.
448
)
OCLC
–
1121105677
Publisher:
Oscar Publishing Services
Servi
We will determine the relationship between the product of the acute angles' tangents in this triangle and the
perimeter by utilizing the product of tangents, aiming to define this connection accurately:
2
2
2
2
2
2
2
2
2
2
2
2sin
sin
sin
4 cos
cos
cos
2
2
2
2 cos
cos
cos
2 cos
cos
cos
2
2
2
2
2
2
sin
sin
sin
8sin
cos
sin
cos
sin
cos
2
2
2
2
2
2
cos
cos
cos
1
1
2
2
2
4
2
2
2
2
2
2
2
4sin
sin
sin
2
2
2
2
p
S
S
S
p
S
Sctg
ctg
ctg
p
Sctg
ctg
ctg
P
p
=
=
=
=
=
=
=
=
2
2
2
P
Sctg
ctg
ctg
=
It is possible to find the third angle from the equation stated here
+ + =
[2].
REFERENCES
1.
Обид
Каримий
.
Тригонометриядан
мисол
ва
масалалар
ечиш
.
Тошкент.1972
-
й.
2.
С.О.Новосёлов.Специальный
курс
тригонометрии.Советская наукаю1954 й
3.
Bakirov
T.Y,
Axmadjonova
O.S.
Maktab
matematikasida kasrlar mavzusini o’qitishda
zamonaviy raqamli texnologik online xizmatlaridan
foydalanish samaradorligi.
“Texnika va raqamli
texnologiyalarning amaliyotda qo’llanilishi va
ularning innovatsijn yechimlari”
// Xalqaro ilmiy-
texnik konferensiya materiallari.
Farg’ona,
4-may
2023-yil III qism 200-202 betlar
4.
Bakirov T.Y, Axmadjonova O.S.
Umumiy o‘rta ta’lim
maktablarida kasr tushunchalarini o‘rganish haqida
//
Qo’qon DPI Ilmiy xabarlar jurnal
i.
Qo’qon,
2023-yil
№ 9
-son. 286- 289 betlar
