OK, I know the SDMB frowns upon homework-related questions (or is that disallows it?) but anyways, I just need a point in the right direction. It isn’t a homework question, but I will probably need it for my math class, so here goes:
What is the geometry behind the proof of the identity sin(a - b) = sin(a)cos(b) - sin(b)cos(a)?
I’ve found/done the proofs for the other main three identities (addition and the cosine difference ones), but I can’t get this one.
Before anyone says “sin(x) = cos(pi/2 - x),” I want the main geometry that leads to the proof; i.e., I do not want to use another identity.
BTW, for the cosine proofs I used the unit circle and for the sin(a + b) one I found it on the Internet. However, I can’t find the geometric part of the proof to the sine difference identity on the Internet or in my textbook.
And while we’re at it, can you prove the sin(a + b) identity from the unit circle without using one of the cosine identities?
I have a good hunch that this proof will be on my test.
If you are asked to prove that sin(A-B) = sinAcosB - CosAsinB, from ‘first principles’, using geometry, it is perfectly good maths to first prove that [list=1][li]sin(A+B) = sinAcosB + cosAsinB, then prove that[/li][li]sin(-B) = -sin(B), and then prove that[/li][li]cos(-B) = cosB[/list=1][/li]Then you can replace B in (1) by -B to prove what you wanted.
It might not produce something that is intuitively obvious (although later it will seem so), but IMHO the best proofs are the ones that go beyond the obvious, either in the result or the method, i.e. they make you say Wow!
I found both of these by using the search string “proof of angle addition formula” (without the quotes) in google. This search string, or a similar one, will probably bring you other proofs if neither of the above is to you liking.
The thing is, the teacher has already said that one of the proofs for the main for identities will be on the next test, and this proof will need to contain the geometry part.
Also, as I said, I don’t want to use another identity to prove it (other than the obvious ones like cos^2(x) + sin^2(x) = 1 and cos(-x) = cos(x) for example), so using the sin(a + b) identity is out.
The geometric proof of the sine of the difference of angles isn’t in any of my texts and I can only find the addition one on the Internet.
The proof we used in class was to use the unit circle and distance formula, but using that always yields:
cos^2(a) + 2cos(a - b) + sin^2(b) = …
Therefore, there is no sin(a - b) I can work with. On a similar note, if I’m working with theta = a + b, there’s no sin(a + b) that I can work with.
So basically, my question is: can the sine identities be proven using the unit circle without using the cosine identities?
But more specifically, what’s the geometry behind sin(a - b) = …?
Any proof that works for the addition identity will also work for the difference identity, since subtraction is just another form of addition. It’s the same as adding a negative angle.
Thanks Cabbage, but I guess I am still left wondering about the second part of my question.
Can you derive the sine sum and difference of angle identities using the unit circle and distance formula? Once again, without proving the cosine ones first.
I am a bit thoughtful about why the first two replies were “you don’t need to know it”, when the motto of this site is “to fight ignorance”. Sure, the first reply was probably in jest, but the second seemed serious to me.
I use trig all the time in my “real” life, as do a lot of people. Why not give the OP the benefit of the doubt?
Thanks Tyrrell, but by unit circle and distance formula I meant using, well, the unit circle and distance formula. In other words, to have all the angles originating from the centre, then geometrically proving the congruence of the triangles formed by the angles and using that to derive the equation. The problem with this however is, because I can only work with point (1, 0), it doesn’t give me a sin(a +/- b) to work with.
The reason I ask this is because the teacher showed only the proof using the unit circle, but expects us to be able to prove all the identities. She also said that we are not allowed to use other identities; whether that includes not being allowed to prove one identity first to derive another, I am not sure, hence the question if it can be done.
Bear in mind that every step of a proof of a trigonometric formula is an identity. Therefore, if you can’t prove one identity and use it to prove another, then all proofs are unacceptable.
