INTERNATIONAL JOURNAL OF ARTIFICIAL INTELLIGENCE
ISSN: 2692-5206, Impact Factor: 12,23
American Academic publishers, volume 05, issue 05,2025
Journal:
https://www.academicpublishers.org/journals/index.php/ijai
page 1770
TRIGONOMETRIC EQUATIONS AND SOLVING METHODS
Yuldashev Jonibek Azamkulovich
head teacher of mathematics at the Academic Lyceum of
Turin Polytechnic University in Tashkent
Abstract :
This in the article trigonometric equations solution non-standard methods about
information given . Including both sides of the equation side one kind trigonometric to the
function multiplication , both
towards one odd number or one kind trigonometric function
add and subtraction , proportion , mathematics analysis of elements , vectors scalar from the
product use about information given and visols undress shown .
Key words :
Function determination domain of the function limitation property , proportion ,
vector , scalar multiplication , number inequality , equations system , equations union .
Trigonometric equations to the appearance looking at solution one how much methods
available . These instead of to put , to rationalize instead of substitutions , trigonometric
equations different ways of solving private cases , artificial form from substitutions using
trigonometric equations solution and etc. Some
in cases given we know the equations
methods with solution much complicated It will be . equations solution non-standard to the
methods stop Let's go . Artificial form requiring replacement
trigonometric equations in
solution following from methods is used .
Trigonometry algebra and geometry in sciences important importance has is , is
integrative to the feature have In this sense trigonometric equations solutions collection
generalization and them choice in students difficulty gives birth to . Because students two
variable linear equations solution methods and their application opportunities about enough for
information has not to be possible . Two variable linear equations solution methods school
mathematics in science deep It is also known that it cannot be studied . But specialized schools
and high schools mathematics study in programs two variable linear equations occurs and they
Diophantus equations ( undefined equations ) topics in the section is studied .
Two variable linear Diophantus equations solution technique trigonometric equation
roots collection one how many when general roots the set separate in receiving application
possible . Because selectively taken general roots set usually given equation solution as a result
found all roots total satisfied for they two variable linear Diophantus equations roots set that
obvious will be . Especially trigonometric equations given in the field when given roots
collection separate in receiving this obvious It seems like this . in cases trigonometric equation
roots only given to the sectors relevant those who were separate required to obtain and in this
many in cases choice through results is taken . But choice and intuitive considerations always
reliable
expected the results It doesn't give . That's why for Diophantus equations and
trigonometric of equations mutual relatedness study mathematics internal integration to
strengthen service does . Mutually relevance trigonometric of equations solutions set first of all
arithmetic progression organization to be able with explained .
INTERNATIONAL JOURNAL OF ARTIFICIAL INTELLIGENCE
ISSN: 2692-5206, Impact Factor: 12,23
American Academic publishers, volume 05, issue 05,2025
Journal:
https://www.academicpublishers.org/journals/index.php/ijai
page 1771
Trigonometric issues in solving , their general solutions find for arithmetic progressions
in the form of written roots set compare need will be , because they either of the matter
solution , or on the contrary to be possible .
Let's assume first x
k
= + 1 5 k progression k - hadi second x
m
= + 3 7 m progression m -
to the extent equal let it be . k and m parameter how in values 1 5 + = + k 3 7 m equality
appropriate that we determine . k and m parameter so value find must arithmetic progressions
found numbered values equal Let . As a result following whole with coefficient two variable
linear equation harvest becomes : 5 k − 7 m = 2. The equation whole numbers in the collection
We solve the equation . variable in front of two from the coefficient we get the most small to
value has that happened we choose ( this is 5) and on the left second the limit is 7 m = 5 m + 2
m such as we write and 5 k − − 5 m 2 m − = + 2 5 k 5 m 2 2 m . The left side of the equation is
5 submissiveness for right side is also 5 division necessary : 2 m + = 2 5 s , sZ . Thus , 2 m −
=− 5 s 2 from the equation m and s unknown whole numbers as above we will find .
Last of the equation variables in front of two from the coefficient we get the most small
to value has what happened we choose ( this is 2) and on the left second term 2 m − − =− 4 σs 2
2 m − = − 4 ss in 2 forms We write . The left side of the equation is divided by 2 . multiple ,
therefore for s − 2 is also 2 multiple to be we need : s − = 2 2 p , pZ . So as s = + 2 p
2
from the
equation unknown s and p whole numbers We will find . From the unknowns one of coefficient
1 equal was equation to be taken with solution process finally So , s = + 2 p 2 equation p of
optional whole in values will be an integer .
So as first progression k = + 7 p 6 in number present x = + = + 1 5 k 1 5(7 p + = 6) 35 p
+ 31, second progression m = + 5 p 4 in number present x = + 3 7 m = + 3 7(5 p + = 4) 35 p +
31 is equal . m = + 5 p 4 and k = + 7 p 6 , both progression general reaches the terms ( values ) .
So , the general term is x = 35 p + 31 ,
p = Z.
Conclusion as to say possibly trigonometric
equations roots collection in
generalization whole with coefficient linear equations solution to the methods separately
attention to give necessary . From the analyses this It was found that the students trigonometric
of equations roots generalization mainly choice through done This is
one how much
calculations and It takes time . Therefore for trigonometric of equations general roots collection
when choosing whole with coefficient linear equation solution from the methods use teacher
and to students We believe that this will have a positive effect . Also , such approaches
mathematics science internal mutual integration to provide service does .
Used literature list:
1. UX Khankulov .
Trigonometric issues in solution many kind of methods application .
Physics , Mathematics and Informatics.- Tashkent , 2024. -№ 4. -P.24-32.
2. Victor Shoup . A Computational Introduction to Number Theory and Algebra. Boston ,
2015.p.247.
3. Nishonov , FM, Shaev , AK, (2021). Some questions of the organization of individual
works of students in mathematics in the conditions of credit training. Theoretical &
Applied Science, (4), 1-7.
4. Nishanov , FM (2018). Some questions of design of tasks in mathematics. ISJ Theoretical
& Applied Science, 09 (65): 41-44. Doi:
INTERNATIONAL JOURNAL OF ARTIFICIAL INTELLIGENCE
ISSN: 2692-5206, Impact Factor: 12,23
American Academic publishers, volume 05, issue 05,2025
Journal:
https://www.academicpublishers.org/journals/index.php/ijai
page 1772
5. Govorova, K. F. (2018). Formation of basic competence in solving trigonometric equations
and inequalities. Scientific electronic journal Meridian, (4), 15-17.
