Math Is Fun Forum

bossk171

Member

Registered: 2007-07-16

Posts: 305

Proof of the sum angle identity

I'm looking for an easy to understand proof of the sum angle formulas. The best I can find is Wikipedia's (http://en.wikipedia.org/wiki/Proofs_of_trigonometric_identities#Angle_sum_identities)

Also, no fair using Euler's Formula, I know that, but that's not really what I'm looking for.

Any help (links, explanations, proofs, books, etc...) would be very much appreciated.

Thanks!

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There are 10 types of people in the world, those who understand binary, those who don't, and those who can use induction.

Kurre

Member

Registered: 2006-07-18

Posts: 280

Re: Proof of the sum angle identity

A nice geometric construction I found:
http://www.geocities.com/kurre999/Trigaddition.bmp
not a complete proof though since it doesnt work for all angles, but atleast a good explanation for it

edit: actually I think its easy to prove the formula completely using this. the picture works for angles pi/2<x+y<pi. If x+y<pi/2 the geometric construction just changes a bit. We can easily(?) derive the formula for cos(x+y) with the same set of angles x,y. Now if x+y>pi (or x+y<0) we just scale off all multiples of 2pi until we are in 0<x+y<2pi. If we are in 0<x+y<pi we are done, otherwise we use sin(a+pi)=cos(a) and use the addition formula for cos.
edit2: but I realize we get problems since we can only scale of x and y seperately, so it doesnt really work...

edit3: But, using sin(x+y)=-sin(-x-y)=-sin((pi-x)+(pi-y)) if x+y>pi but 0<x,y<pi will do the job

Last edited by Kurre (2009-03-01 07:07:26)

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: Proof of the sum angle identity

Also, no fair using Euler's Formula, I know that, but that's not really what I'm looking for.

So what are you looking for?

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"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

bossk171

Member

Registered: 2007-07-16

Posts: 305

Re: Proof of the sum angle identity

So what are you looking for?

Something I can show to a Trig student (who hasn't taken calculus). e^ix won't really cut it anyways because you need calc to prove Taylor Series (at least that's the only way I know how to do it) and in order to take the derivative of the trig functions you need to know the sum angle formulas. It's circular reasoning, right?

I like Kurre's picture, thanks! That's definitely something I can use. Does anyone else have anything else to offer?

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There are 10 types of people in the world, those who understand binary, those who don't, and those who can use induction.