scholarly journals ABOUT DEVELOPMENT OF STUDENTS ' THINKING WHEN SOLVING TRIGONOMETRIC EQUATIONS AND INEQUALITIES IN THE SCHOOL ALGEBRA COURSE

2020 ◽  
Vol 69 (1) ◽  
pp. 138-143
Author(s):  
D.M. Nurbayeva ◽  
◽  
Zh.M. Nurmukhamedova ◽  
S. Yeraliyev ◽  
B.M. Kossanov ◽  
...  
Keyword(s):  
Unit Circle ◽  
Circle Method ◽  
Trigonometric Functions ◽  
Detailed Explanation ◽  
Trigonometric Inequalities ◽  
Trigonometric Equations ◽  
Development Of Students

The article deals with solutions of trigonometric inequalities using the unit circle. Specific examples show its application for all trigonometric functions, namely sinus, cosine, tangent and cotangent. An explanation of how to correctly define the period for solving inequalities is also provided. Before analyzing the solution to trigonometric inequalities, the authors present the solution of trigonometric equations according to the formula, but his roots are depicted on the unit circle, where detailed explanation of the record of solutions of this equation. The pictures in the article demonstrate the images that should be presented by the teacher on the blackboard when solving trigonometric inequalities. The article is written in an accessible language, when reading which the unit circle method will be understandable not only to current teachers, but also to students of Junior courses of pedagogical universities.

scholarly journals A Novel Comprehensive Evaluation Method of Forest State Based on Unit Circle

Forests ◽  
2018 ◽  
Vol 10 (1) ◽  
pp. 5
Author(s):  
Ganggang Zhang ◽  
Gangying Hui ◽  
Gongqiao Zhang ◽  
Yanbo Hu ◽  
Zhonghua Zhao
Keyword(s):  
Forest Management ◽  
Unit Circle ◽  
Comprehensive Evaluation ◽  
Mixed Forest ◽  
Circle Method ◽  
Indicator System ◽  
Gansu Province ◽  
Pinus Koraiensis ◽  
Arc Length ◽  
Closed Graph

Comprehensive evaluation of forest state is the precondition and critical step for forest management. To solve the problem that the radar plot and unit circle only focus on the value of each the evaluation index, this paper proposes a novel method for comprehensively and simultaneously evaluating the functionality and inhomogeneity of forest state based on the modified unit circle method. We evaluated the forest state of the Quercus aliena BL. var. acuteserrata Maxim. ex Wenz. broad-leaved mixed forest in the Xiaolong Mountains Forest Area of Gansu Province and the Pinus koraiensis Sieb. et Zucc. broad-leaved mixed forest in Jilin Province in China. According to the principle of comprehensive, scientific and operability, 10 evaluation indices on forest structure and vitality were selected to construct the evaluation indicator system. Each index was normalized based on the assignment method and ensured to be strictly positive based on reciprocal transformation method. The areas and arc length of the closed graph, formed by connecting every two adjacent indicators, in the radar plot and unit circle were extracted. Based on the isoperimetric theorem (isoperimetric inequality), a comprehensive evaluation model was constructed. Compared with radar chart and unit circle method, each index in the newly proposed unit circle method is represented by an independent sector region, reflecting the contribution of the index to the overall evaluation result. Each index has the same relative importance weight, contributing to the estimation the relative sizes of each aspect of forest state. The unique area and arc length of the closed graph help summarize the overall performance with a global score. The expression effect of improved unit circle has been enhanced, and as an English proverb put it, “A picture is worth a thousand words.” The new proposed method simultaneously evaluates the functionality and inhomogeneity of the forest state and it is a powerful tool for the diagnosis of forest state problems and the decision-making of forest management.

scholarly journals Gaussian Qualitative Trigonometric Functions in a Fuzzy Circle

2018 ◽  
Vol 2018 ◽  
pp. 1-9
Author(s):  
M. Clement Joe Anand ◽  
Janani Bharatraj
Keyword(s):  
Particle Swarm Optimization ◽  
Unit Circle ◽  
Fuzzy Theory ◽  
Particle Swarm ◽  
Membership Functions ◽  
Trigonometric Functions ◽  
Swarm Optimization ◽  
Qualitative Representation ◽  
Trigonometric Identities

We build a bridge between qualitative representation and quantitative representation using fuzzy qualitative trigonometry. A unit circle obtained from fuzzy qualitative representation replaces the quantitative unit circle. Namely, we have developed the concept of a qualitative unit circle from the view of fuzzy theory using Gaussian membership functions, which play a key role in shaping the fuzzy circle and help in obtaining sharper boundaries. We have also developed the trigonometric identities based on qualitative representation by defining trigonometric functions qualitatively and applied the concept to fuzzy particle swarm optimization using α-cuts.

Unit Circles and Inverse Trigonometric Functions

2014 ◽  
Vol 108 (2) ◽  
pp. 114-119
Author(s):  
Azael Barrera
Keyword(s):  
Unit Circle ◽  
Trigonometric Functions ◽  
Unit Circles ◽  
Inverse Trigonometric Functions

A method to determine all the inverse trigonometric functions directly from the unit circle.

scholarly journals ESSENTIAL TRIGONOMETRY WITHOUT GEOMETRY

2019 ◽  
Vol 71 (1) ◽  
Author(s):  
John Gresham ◽  
Bryant Wyatt ◽  
Jesse Crawford
Keyword(s):  
Unit Circle ◽  
Elementary Level ◽  
Comprehensive Treatment ◽  
Advanced Mathematics ◽  
Trigonometric Functions ◽  
Geometric Constructions ◽  
Rigorous Approach ◽  
Sine And Cosine Functions ◽  
Cosine Functions

Abstract The development of the trigonometric functions in introductory texts usually follows geometric constructions using right triangles or the unit circle. While these methods are satisfactory at the elementary level, advanced mathematics demands a more rigorous approach. Our purpose here is to revisit elementary trigonometry from an entirely analytic perspective. We will give a comprehensive treatment of the sine and cosine functions and will show how to derive the familiar theorems of trigonometry without reference to geometric definitions or constructions. Supplemental material is available for this article online.

Algorithm for rational determination of the area of a triangle for the development of students' creative thinking and practical application. Properties of the 90 Degrees angle tangent

2021 ◽  
pp. 46-55
Author(s):  
V. P. Zudin
Keyword(s):  
Creative Thinking ◽  
Visual Basic ◽  
Educational Process ◽  
Practical Application ◽  
Decimal System ◽  
Microsoft Word ◽  
Trigonometric Equations ◽  
Visual Basic For Application ◽  
Development Of Students

The article presents the formulas derived by the author for determining the area of a triangle by a side and two adjacent angles. These for - mulas are used to activate the educational process — students in practice figure out how to rationally determine the area of land plots, buildings, draw up and solve trigonometric equations, prove the properties of the tangent of an angle equal to 90 degrees, correct the graph of the function y = tg x. To carry out calculations of the areas of triangles, to prove the properties of 90 degree angle tangent, to solve trigonometric equations, programs are written in the Visual Basic For Application language in Microsoft Word. Using the binary-decimal system and VBA programs in Word, the value of the tangent of an angle is calculated so close to an angle of 90 degrees that this value can be roughly taken as the 90 degrees angle tangent. The study of this material at informatics lessons contributes to the development of creative thinking of students, increasing their motivation to study informatics and information technology.

scholarly journals New Refinements and Improvements of Some Trigonometric Inequalities Based on Padé Approximant

2020 ◽  
Vol 2020 ◽  
pp. 1-7
Author(s):  
Lina Zhang ◽  
Xuesi Ma
Keyword(s):  
Padé Approximant ◽  
Multiple Point ◽  
Trigonometric Functions ◽  
Pade Approximant ◽  
Trigonometric Inequalities ◽  
Jordan’S Inequality ◽  
Better Than ◽  
Approximant Method

A multiple-point Padé approximant method is presented for approximating and bounding some trigonometric functions in this paper. We give new refinements and improvements of some trigonometric inequalities including Jordan’s inequality, Kober’s inequality, and Becker-Stark’s inequality. The analysis results show that our conclusions are better than the previous conclusions.

Students' Understanding of Trigonometric Functions : A Case Study

2021 ◽  
Vol 2 (2) ◽  
Author(s):  
Gregory A. Downing ◽  
Keyword(s):  
Active Learning ◽  
Unit Circle ◽  
Student Outcomes ◽  
Approaches To Learning ◽  
Learning Approach ◽  
Trigonometric Functions ◽  
Mnemonic Devices

In trigonometry, students are often pushed toward the memorization mnemonic devices or acronyms. Instead, students should be able to use procedures and explain why they are appropriate and justify why concepts in mathematics have the properties they do (Weber, 2005). Motivated by the dichotomous approaches to learning trigonometry by the work of Weber (2005) and Kendal and Tall (1998), this study aims to explore how students in a college trigonometry course understand trigonometric functions in a unit circle learning approach course and if students in a unit-circle-first approach in a college trigonometry course can justify why trigonometric functions have the properties they do? The context of the course studied was designed to introduce active learning components to students to study how these new practices are implemented and how they affect student outcomes.

Sign in / Sign up

Export Citation Format

Share Document