Math Is Fun Forum

  Discussion about math, puzzles, games and fun.   Useful symbols: ÷ × ½ √ ∞ ≠ ≤ ≥ ≈ ⇒ ± ∈ Δ θ ∴ ∑ ∫ • π ƒ -¹ ² ³ °

You are not logged in.

#1 2009-02-28 12:14:51

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!


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

Offline

#2 2009-03-01 06:49:12

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 tongue

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

Offline

#3 2009-03-01 11:51:36

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?


"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..."

Offline

#4 2009-03-02 18:44:16

bossk171
Member
Registered: 2007-07-16
Posts: 305

Re: Proof of the sum angle identity

Ricky wrote:

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?


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

Offline

Board footer

Powered by FluxBB