scholarly journals From ratios of right triangle to unit circle: an introduction to trigonometric functions

2019 ◽  
Vol 1157 ◽  
pp. 022124 ◽  
Author(s):  
C L Maknun ◽  
R Rosjanuardi ◽  
A Jupri
Keyword(s):  
Unit Circle ◽  
Trigonometric Functions

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.

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.

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.

The Restricted Arc-Width of a Graph

10.37236/1734 ◽  
2003 ◽  
Vol 10 (1) ◽  
Author(s):  
David Arthur
Keyword(s):  
Unit Circle ◽  
Single Point ◽  
Minimum Width ◽  
Graph Operations ◽  
Function Mapping

An arc-representation of a graph is a function mapping each vertex in the graph to an arc on the unit circle in such a way that adjacent vertices are mapped to intersecting arcs. The width of such a representation is the maximum number of arcs passing through a single point. The arc-width of a graph is defined to be the minimum width over all of its arc-representations. We extend the work of Barát and Hajnal on this subject and develop a generalization we call restricted arc-width. Our main results revolve around using this to bound arc-width from below and to examine the effect of several graph operations on arc-width. In particular, we completely describe the effect of disjoint unions and wedge sums while providing tight bounds on the effect of cones.

A New Solution for Maxterm Problem in Trigonometric Functions by Simulated Annealing Algorithm

2013 ◽  
Vol 4 (2) ◽  
pp. 20-28
Author(s):  
Farhad Soleimanian Gharehchopogh ◽  
Hadi Najafi ◽  
Kourosh Farahkhah
Keyword(s):  
Simulated Annealing ◽  
Simulated Annealing Algorithm ◽  
Trigonometric Functions ◽  
Matlab Software ◽  
Annealing Algorithm ◽  
Do So

The present paper is an attempt to get total minimum of trigonometric Functions by Simulated Annealing. To do so the researchers ran Simulated Annealing. Sample trigonometric functions and showed the results through Matlab software. According the Simulated Annealing Solves the problem of getting stuck in a local Maxterm and one can always get the best result through the Algorithm.

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