#trigonometry

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old hawk
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So im in 11th grade, and we just studied the formula cos(a+b) = cosacosb -sinasinb
now our teacher didnt really show us the demonstration of this formula, however he did say it involved using triangles and the Thales theorem. Whenever I try to construct both Triangles on the trigonometric circle, I end up with 2 right angled triangles whom do not have 3 aligned points and I am therefore unable to use the Thales theorem. Could someone help with the demonstration?

foggy ridgeBOT
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foggy ridgeBOT
obsidian mulch
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Try to read it, and if you have questions about a step, ask about that step.

But the general idea is to take 2 points on the unit circle and measure the length of their chord. Rotate so one of the points moves to (1,0) and hence the other is at angle a+b. Since rotation preserves lengths, the new chord is the same length, and the algebra spits out the identity

tribal mesa
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if u want i can show u my notes

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(even if we only showed how to calcualte sin(a+b), but for cos(a+b) its basically the same thing)

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anyways its an easy demonstraion, its very geometrical

old hawk
old hawk
tribal mesa
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didnt remmeber we did the cos formula too

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thats good

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bassically, u start by drawing the angle alfa and the angle beta

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then take a point P on the 3rd line (b' in the drawing)

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do its projection on "b" (u get point Q)

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and on "a" (u get point H)

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then project Q on "a" and on the line PH

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you know that cosine (alfa + beta) is equal to HP/VP

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and from there...

tribal mesa
old hawk
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ahhh

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interestinggg

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just small question why did we do the projections

tribal mesa
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breaking down the alfa+beta angle allows you to calculate it ig

obsidian mulch
old hawk
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ohh okay thanks

foggy ridgeBOT
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@old hawk

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