Perfect material for trigonometry

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Discussion Overview

The discussion revolves around the use of illustrations and diagrams in understanding trigonometric concepts, particularly the angle-sum formulas. Participants share their experiences with different resources and methods for learning trigonometry, including graphical representations and derivations.

Discussion Character

  • Exploratory
  • Conceptual clarification
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • Some participants find the illustrations on the referenced website helpful for understanding trigonometric concepts.
  • Others express that the diagrams are complicated and suggest that recreating graphs may aid in understanding formulas.
  • A participant mentions a specific resource, a calculus book by Anton, that contains a good graph and derivation for the Law of Cosines.
  • One participant proposes that the figure could be drawn in steps to facilitate understanding, suggesting a sequence of triangles to analyze.
  • Some participants reflect on their past experiences with trigonometry, noting that they relied on memory or lookup for formulas rather than understanding through derivations.

Areas of Agreement / Disagreement

Participants express a mix of appreciation and confusion regarding the diagrams, indicating that while some find them useful, others struggle to understand them. There is no consensus on the best approach to learning these concepts, and multiple perspectives on the effectiveness of different resources and methods remain.

Contextual Notes

Some participants mention a lack of recent engagement with trigonometry, which may affect their understanding. There are also references to specific resources that may not be universally accessible or familiar to all participants.

archaic
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The illustrations on this website are perfect.
http://trigonography.com/toc/
oads%2F2015%2F09%2Ftrigonograph-sincosanglesumdiff.png

I hope it'll help!
 
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Good stuff!
 
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Doc Al said:
Good stuff!
Plus one on that.

I don't remember seeing figures like this (maybe I was sleeping in trig class that day). The angle-sum formulas and such were always a memory/lookup thing for me. Next time I can draw a little sketch and figure them out.
 
These figures, the two diagrams, are complicated for me. Maybe I have lost something in the last several years.

Some of the books I used showed derivations from graphs, which although complicated, after examining them carefully for a long time, I was able to understand. Learning to recreate some of the graphs might be a way for some people to figure how to get some of the formulas.

There is a good graph and derivation shown for Law Of Cosines, in of all places, the big thick Calculus book by Anton; a book I never used for any course but found the copy at a used-book sale.
 
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symbolipoint said:
These figures, the two diagrams, are complicated for me. Maybe I have lost something in the last several years.

Some of the books I used showed derivations from graphs, which although complicated, after examining them carefully for a long time, I was able to understand. Learning to recreate some of the graphs might be a way for some people to figure how to get some of the formulas.

There is a good graph and derivation shown for Law Of Cosines, in of all places, the big thick Calculus book by Anton; a book I never used for any course but found the copy at a used-book sale.
Look at the left graph, the side which is equal to ##\sin{(\alpha+\beta)}## is equal to the opposite side since this is a rectange, you reason the same way for the rest, for the cosine you'll be substracting what's left of the side from the opposite one.
 
archaic said:
Look at the left graph, the side which is equal to ##\sin{(\alpha+\beta)}## is equal to the opposite side since this is a rectange, you reason the same way for the rest, for the cosine you'll be substracting what's left of the side from the opposite one.
Still very tough. With time and thinking, I am able to understand the red triangle and the lower blue triangle for the figure on the left. I have not understood beyond those yet.
 
I am starting to see how the figure on the left can instead of being displayed all at one time, could be drawn IN STEPS, and then label the parts during each step. From these, begin examining the triangles in this sequence:
  1. red middle
  2. blue lower right
  3. blue upper right
  4. pink left

Now knowing these all compose the rectangle, the sine of sum of the two angles can be concluded.
 

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