How can algebra and calculus be used to derive the identity for sin(a+b)?

[FulhamFan3]

How do you find the identity for sin(a+b) using algebra or calculus? I already know how to do it with geometry and by using imaginary numbers.

 

[Ryoukomaru]

Here is how to do it by using rotation matrices:
https://www.physicsforums.com/showthread.php?t=61109&page=2
(scroll down a bit)

 

[Hurkyl]

Pure algebra won't help -- sin is a so-called transcendental function. At least, algebra won't help unless you provide some initial algebraic relations to get started. (Though, this particular identity would usually be such a thing that you'd use to start, not something you'd derive)


As for calculus, of the most rigorous proofs, it is the easiest to follow. sin is simply a power series:

$$\sin z := \sum_{n=0}^{\infty} (-1)^n \frac{z^{2n + 1}}{(2n+1)!}$$

And similarly for cos. If you plug in z = p + q, then you can algebraically derive the identity from this.

 

[cepheid]

I seem to remember a more intuitive "geometric" proof (if you could call it that) in high school. Just wondering, because that was back in the day before Maclaurin series, and it would be nice if there were an easier derivation (easier to remember) for the sum and difference formulas. The reason why I care is that I know how to derive the double angle formulas from the sum/difference ones in like two seconds, and I know how to get the half angle formulas from the double angle ones, so if I just knew how to get at the sum/difference ones quickly without memorizing them (a bit tedious, although they're sort of half-memorized already), then I'd have all these trig identities at my fingertips... I have a pretty good memory, but memorization is not the best way, IMO.

Edit: didn't read Fullham's post. he mentioned the geometric method. Gotta link? And how do you do it using imaginary numbers? The Euler identity?

 

[dextercioby]

There's a thread on this specific matter right in this subforum.It's called "2 proofs".You'll find the proof given by HallsofIvy really charming.

Daniel.

 

[Hurkyl]

Frankly, I'd consider the sine and cosine angle addition formulas sufficiently fundamental that they're worth memorizing on their own merit, rather than trying to justify in terms of other concepts.

 

[dextercioby]

Well,only one of them is worth it.The other can be proven immediately,once one is "given"...

Let's chose the SINE...

Daniel.