Euler's Forumla, Trig Addition, and Equating Coefficients

[elarson89]

One can look into any precalculus book and find a proof of the addition formulas of sine and cosine. Though as most are aware there is a quick way to get the formulas by using Euler's Formula. But to get the formulas by eulers formula, you must equate coefficients with respect to the imaginary part i.

My question is this, equating coefficients was taught to be used for polynomials, because a set of coefficients uniquely determines a polynomial. How can you show the same is true with respect to i? Yes it looks very intuitive, but I'm wondering if there's something a little more powerful than that.

 

[Tac-Tics]

What do you mean by "equate coefficients"? Maybe you could show us what theorem you're trying to prove to make it clearer.

 

[elarson89]

Theres, only one addition formula for sine... and then there is the euler's formula e^ix=... and only one way to equate coefficients... I would write them out but I don't know how to embed tex.

 

[matticus]

if you have a + bi = c + di, then by the definition of equality of complex numbers you must have a = c and b = d. so you can equate the real and imaginary parts, if that's what you're asking.