A Strategy For Avoiding A Hidden Trap In Trigonometry

1982 ◽  
Vol 75 (6) ◽  
pp. 502
Author(s):  
James P. Herrington
Keyword(s):  
Included Angle ◽  
Two Sides

The solution of an oblique triangle, given two sides and the included angle, can contain a hidden trap often overlooked by both students and textbooks. If one of the unknown angles is obtuse, it is possible to obtain an erroneous solution for the triangle.

scholarly journals Precision Cutting of the Molds of an Optical Functional Texture Film with a Triangular Pyramid Texture

Micromachines ◽  
2020 ◽  
Vol 11 (3) ◽  
pp. 248
Author(s):  
Huang Li ◽  
Zhilong Xu ◽  
Jun Pi ◽  
Fei Zhou
Keyword(s):  
Area Ratio ◽  
Shape Error ◽  
Average Roughness ◽  
Optical Texture ◽  
Included Angle ◽  
Triangular Pyramid ◽  
Technical Requirements ◽  
Exit Burr ◽  
Five Axis ◽  
Two Sides

Based on an analysis of the precision and preparation technology of an optical texture film with a triangular pyramid texture, the technical requirements of the original mold were determined, and precision shaping planning technology was adopted to process the original mold. The shape error of the optical texture mold of the triangular pyramid was assessed by defining the area ratio of the retro-reflection. The influence of the tool nose radius and exit burr on the area ratio of the retro-reflection were analyzed. By optimizing the cutting tools, cutting materials and cutting boundaries, a five-axis ultra-precision machining system was used to plan the triangular pyramid structure with a base length of 115 µm and an included angle between two sides of 70.5°. The experimental results indicate that the dimension error of the triangular pyramid element is less than 1 µm, the angle error of the included angle between two sides is less than 0.05°, and the average roughness of the side of the triangular pyramid can reach 9.2 nm, which satisfies the processing quality requirements of the triangular pyramid texture mold.

Cosine Triples

1976 ◽  
Vol 69 (2) ◽  
pp. 119-124
Author(s):  
Charles G. Moore
Keyword(s):  
The Other ◽  
Included Angle ◽  
The Law ◽  
Law Of Cosines ◽  
Two Sides

This investigation of cosine triples was instigated by a chance observation. For years I have used triangles with integral sides for the purpose of quizzing students on the law of cosines. The law of cosines states that the square of the length of any side of a triangle is equal to the sum of the squares of the lengths of the other two sides minus twice the product of the lengths of those two sides and the cosine of the angle between them. For example, given the triangle with sides 6 and 7, and included angle 20 degrees, the student is expected to find the remaining side using the law of cosines (fig. 1).

scholarly journals XI. A proposed Improvement in the Solution of a Case in Plane Trigonometry

1826 ◽  
Vol 10 (1) ◽  
pp. 168-170
Author(s):  
William Wallace
Keyword(s):  
Present State ◽  
Considerable Improvement ◽  
Mathematical Science ◽  
Included Angle ◽  
The Third ◽  
Two Sides ◽  
Made In ◽  
Elementary Theories

In the present state of mathematical science, cultivated as it has been, with assiduity, during the two preceding centuries, it can hardly be expected that any considerable improvement remains to be made in Plane Trigonometry, one of its most elementary theories. There is, however, one case in the resolution of oblique-angled triangles, which appears to me to admit of a solution somewhat more simple and convenient than those which are commonly known; it is that in which two sides and the included angle are given to find the third side.

scholarly journals Peter Chew Triangle Diagram and Application

2021 ◽  
Author(s):  
Peter Chew
Keyword(s):  
Triangle Diagram ◽  
Included Angle ◽  
The Third ◽  
The Future ◽  
Online Calculator ◽  
Teaching Of Mathematics ◽  
Two Sides

Abstract: The objective of Peter Chew Triangle Diagram is to clearly illustrate the topic solution of triangle and provide a complete design for the knowledge of AI age. Peter Chew's triangle diagram will suggest a better single rule that allows us to solve any problem of topic solution of triangle problems directly, more easily and more accurately. There are two important rules for solving the topic solution of triangle today [1,2], namely the sine rule and the cosine rule. The sine rule is used to find a non-included angle when are given two sides and a non-included angle or the opposite side angle given when are given two angles and one side. The cosine rule normally is used to find the included angle when are given three sides or the third side when are given two sides and the included angle. Generally, we only think that when given two sides and an included angle, the cosine rule is used to find the third side. In fact, when two sides and one non included angle are given, the cosine rule is also more easier for finding the third side. For problem given 2 sides and an included angle, directly find the non included angle. We need to use Peter Chew rule [1] to solve this problem. Peter Chew Rule allows us to find the non included angles directly, easier and more accurately. The application of Peter Chew's triangle diagram in the PCET calculator allows the PCET calculator to directly solve any problem in the topic solution of triangle, which is easier and more accurate. The Peter Chew diagram provides a complete design of the topic solution of triangle, which can help students solve any problems in the topic solution of triangle directly, more easily, and more accurately. Apply Peter Chew diagram to the new calculator (PCET calculator) , allows the PCET calculator to solve any problems in the topic solution of triangle and solve some problem that can not solve by current online calculator such as Math Portal and Symbolab. Which can make PCET calculator effectively help the teaching of mathematics, especially when similar covid-19 problems arise in the future.

When is a Proof Not a Proof.

1926 ◽  
Vol 19 (8) ◽  
pp. 499-505
Author(s):  
P. Stroup
Keyword(s):  
Formal Proof ◽  
The Other ◽  
Geometric Proof ◽  
Included Angle ◽  
Two Sides

We have all heard of pupils who “got by” in geometry by merely memorizing the proofs given in the book and of teachers who permitted them to “get by.” To such persons those proofs were not proofs. Assuming that such methods are passe if they have not passed all together, what limits are we to approach in the other direction? How long should a student be asked to hold in his memory any completed proof? Is it sufficient to remember the fact and that it has been proved? Are the proofs presented really designed to convince the student of the truth of the fact being proved? Are they the kind of proof that is really convincing to him? Is the logical proof convincing to him? Why prove to him facts that he is sure are true? We ask why he thinks that two triangles must be alike if they have two sides and the included angle the same and he replies that anybody can see that with the implication that somebody around is a “boob.” If he can see no exceptions then is it not a proof to him? Is it not more likely to prejudice him against the geometric proof than to raise any respect for it to drag him through a formal proof under those conditions?

Pitfalls in the Trigonometric Solution of an Oblique Triangle

1944 ◽  
Vol 37 (7) ◽  
pp. 311-313
Author(s):  
Irwin M. Rothman
Keyword(s):  
Complete Understanding ◽  
Included Angle ◽  
Trigonometric Solution ◽  
The Law ◽  
Law Of Cosines ◽  
The One ◽  
Ambiguous Case ◽  
Two Sides

Textbooks in plane trigonometry generally treat the solution of oblique triangles by the Law of Sines, Law of Cosines, etc. However, if one uses some of these laws without a complete understanding of their limitations, incorrect results are often obtained. The only case which most textbooks treat adequately in this respect is the solution of a triangle in which two sides and an angle opposite one of them are given, usually referred to as the “Ambiguous Case.” However, in other cases, such as the one in which two sides and the included angle are given, or the one in which the three sides are given, care must be taken if incorrect solutions are to be avoided.

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