Sharing Teaching Ideas: Discovering New Trigonometric Identities

2003 ◽  
Vol 96 (9) ◽  
pp. 650-653
Author(s):  
Sheldon P. Gordon
Keyword(s):  
Trigonometric Identity ◽  
Teaching Ideas ◽  
Rare Opportunity ◽  
As If ◽  
Trigonometric Identities

For most of our students, trigonometric identities appear as if by magic. A recent class activity gave me the rare opportunity to lead my students to discover what appears to be a new trigonometric identity by using a combination of graphical, numerical, and algebraic tools. In this article, I demonstrate this result and then show how it can be extended to a wider class of apparently new identities that are all extensions of the Pythagorean identity.

scholarly journals Student’s Difficulty Identification in Completing the Problem of Equation and Trigonometry Identities

2019 ◽  
Vol 2 (1) ◽  
pp. 34 ◽  
Author(s):  
Siti Maryam Rohimah ◽  
Sufyani Prabawanto
Keyword(s):  
High School Students ◽  
School Students ◽  
Quadratic Equations ◽  
Trigonometric Identity ◽  
Ability Tests ◽  
Solving Equations ◽  
Data Collection Technique ◽  
Trigonometric Equations ◽  
Identity Problems ◽  
Trigonometric Identities

This study aims to identify the types of difficulties experienced by high school students in solving equations and trigonometric identities. The method used in this research is descriptive qualitative research method because researchers want to describe or describe the facts of students' difficulties in solving equations and trigonometric identities. The data collection technique in this study is by using respondents' ability tests and interviews. Based on the results of data analysis, there are three aspects of students 'difficulties in solving trigonometric equations and also there are three aspects of students' difficulties in solving trigonometric identity problems. The difficulties of students in solving trigonometric equations, namely the difficulty of students in deciphering the form of the problem, difficulty in factoring in the form of trigonometric quadratic equations, and difficulties using the basic trigonometric equations. Whereas, the difficulties of students in solving trigonometric identity problems, namely the difficulty of students applying general trigonometry formulas, difficulty describing each of the trigonometric comparison relationships, and difficulties in performing algebraic calculations/computation.

Cosmology in the Making: Barth, the Baktaman, and Secrecy

2007 ◽  
Author(s):  
Tony Crook
Keyword(s):  
Social Relationships ◽  
Left Behind ◽  
Conceptual Work ◽  
Rare Opportunity ◽  
As If

This chapter explores Barth's work among the Baktaman, and examines how the Min got into Anthropology and how Anthropology got into the Min. It opens with an image that affords a glimpse of influences on Barth's perception of the Baktaman. The chapter also presents an in-depth example of two-way combinations between social relationships, analytical relationships, and ethnography, such that they ultimately appear inseparably a combination of the others and yet appear at moments as if they are stand-alone ‘separates’. Barth's record provides a rare opportunity to track the impressions left behind by thinking carried away from experiences at the ends of other journeys, the impressions left behind by earlier conceptual work, and to witness their development in subsequent passages. After the effects on Barth's epistemology are outlined, the discussion backtracks to focus on the Baktaman monograph. The chapter also examines the consequences of Barth having in mind a structuralist opposition of isolated symbolic elements. It then addresses Barth's discussion of creative mechanisms during storage within the minds of individual ritual specialists. Barth's methodological landscape is made concrete such that the movement of ideas can be witnessed in the passage of senior men through the forest to attend an initiation.

scholarly journals III. Observations of the Mer de Glace.—Part I

1859 ◽  
Vol 9 ◽  
pp. 245-247
Keyword(s):  
The Other ◽  
Single Source ◽  
The West ◽  
Ice Stream ◽  
The Face ◽  
Sinuous Channel ◽  
Convex Curvature ◽  
Western Side ◽  
Rare Opportunity ◽  
As If

In this paper the author communicates the first part of a series of observations upon the Mer de Glace, made during a residence of six weeks at the Montanvert last summer. He corroborates the laws regarding the swifter flow of the central portions of the ice-stream, first established by Prof. Forbes, and shows how the velo­city changes as the width of the glacier varies. The Mer de Glace moves through a valley which twice turns a convex curvature to the east, and once to the west. The points of swiftest motion at these curves are found to be not central, but thrown to that side of the valley towards which the glacier turns its convex curvature. It has hitherto been believed that the portion of the Mer de Glace derived from the Glacier du Géant moved swiftest. The author shows that the tributaries which form the Mer de Glace lose their individuality in the trunk stream, the latter flowing as if it proceeded from a single source. The point of maximum motion is sometimes on the eastern, sometimes on the western side of a line drawn along the centre of the glacier, the change from side to side depending upon the curvature of the valley. The locus of the point of swiftest motion in a glacier which moves through a sinuous valley, is exactly similar to that of a river moving through a sinuous channel; it forms a curve more deeply sinuous than the valley itself, and crosses the centre of the valley at each point of contrary flexure. A rare opportunity of determining the comparative velocities of a glacier at its surface and close to its bed, was furnished by a precipice of ice 140 feet in height, which was exposed near the Tacul. Three stakes were fixed in this, precipice, one at the top, the other near the bottom, and a third in the face of the precipice at a height of nearly 40 feet above the bottom; the velocities of the three stakes were found to be 6 inches, 4.59 inches, and 2.56 inches per day; thus furnishing additional proof of the correctness of the law first predicted by Prof. Forbes, and confirmed subsequently by his own observations and those of M. Martins.

scholarly journals ANALYSIS OF STUDENTS MATHEMATICAL CONNECTION ERRORS IN TRIGONOMETRIC IDENTITY PROBLEM SOLVING

2020 ◽  
Vol 17 (35) ◽  
pp. 825-836
Author(s):  
Budi MARDIYANA USODO ◽  
. BUDIYONO ◽  
Anisa Astra JINGGA ◽  
Dwi FAHRUDIN
Keyword(s):  
Conceptual Knowledge ◽  
Procedural Knowledge ◽  
Arithmetic Operation ◽  
Critical Role ◽  
Trigonometric Identity ◽  
Mathematical Connections ◽  
Learning Mathematics ◽  
Algebraic Concepts ◽  
Algebraic Concept ◽  
Trigonometric Identities

The trigonometric identity is essential in learning Mathematics because it requires students to think critically, logically, systematically, and thoroughly. Solving trigonometric identity problems requires students to relate conceptual knowledge or procedural knowledge, which then used in questions. This study involved grade X students of senior high school, which were examined to find out the types of mathematical connections errors and causes of the errors. Before task-based interviews were conducted, 36 students were first given a test. Based on several considerations, seven students ( three males and four females) were selected to undergo a task-based interview. This research employed a qualitative research method with a case study design. The results of the analysis indicate that the errors in connecting to conceptual knowledge are most commonly the mistake of connecting the algebraic concept. On the other hand, 86.11% of students experienced errors in connecting to procedural knowledge. This error happened when the students worked on problems with trigonometric identities, which they had rarely encountered in exercises. Errors in mathematical connections in trigonometric identity are caused by the lack of understanding of the algebraic arithmetic operation, emphasis on the concept, and strategic knowledge. It shows that students need a variety of problems to be able to master various forms of trigonometric identities. This research's result also reinforces the critical role of algebraic concepts as prior knowledge in studying trigonometric identity.

On q-analogues for trigonometric identities

Analysis ◽  
2020 ◽  
Vol 40 (2) ◽  
pp. 105-112
Author(s):  
Sarah Abo Touk ◽  
Zina Al Houchan ◽  
Mohamed El Bachraoui
Keyword(s):  
Short Proof ◽  
Elliptic Functions ◽  
Main Argument ◽  
Residue Theorem ◽  
Trigonometric Identity ◽  
Trigonometric Identities

AbstractIn this paper we will give q-analogues for the Pythagorean trigonometric identity {\sin^{2}z+\cos^{2}z=1} in terms of Gosper’s q-trigonometry. We shall also give new q-analogues for the duplicate trigonometric identity {\sin(x-y)\sin(x+y)=\sin^{2}x-\sin^{2}y}. Moreover, we shall give a short proof for an identity of Gosper, which was also established by Mező. The main argument of our proofs is the residue theorem applied to elliptic functions.

Improvement of an artificial test substrate technique for evaluation of em immunocytochemical labeling

1987 ◽  
Vol 45 ◽  
pp. 762-763
Author(s):  
G. D. Gagne ◽  
M. F. Miller
Keyword(s):  
Bovine Serum Albumin ◽  
Serum Albumin ◽  
Bovine Serum ◽  
Artificial Substrate ◽  
Chemical Fixation ◽  
As If

We recently described an artificial substrate system which could be used to optimize labeling parameters in EM immunocytochemistry (ICC). The system utilizes blocks of glutaraldehyde polymerized bovine serum albumin (BSA) into which an antigen is incorporated by a soaking procedure. The resulting antigen impregnated blocks can then be fixed and embedded as if they are pieces of tissue and the effects of fixation, embedding and other parameters on the ability of incorporated antigen to be immunocyto-chemically labeled can then be assessed. In developing this system further, we discovered that the BSA substrate can also be dried and then sectioned for immunolabeling with or without prior chemical fixation and without exposing the antigen to embedding reagents. The effects of fixation and embedding protocols can thus be evaluated separately.

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