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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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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.