12
$\begingroup$

Back at the university we have proven (lot of work) that if $S(X)C(Y)+C(X)S(Y) = S(X+Y)$ and $C(X)C(Y)-S(X)S(Y) = C(X+Y)$ then $S(X)$ is $\sin(x)$ and $C(X)$ is $\cos(x)$ (or constant $0$, meh). What is this theorem called..?

Later note by someone other than the original poster:

An amazingly large number of people, in posted answers and comments (some now deleted) have MISSED THE POINT. These are not the angle addition formulas for the sine and the cosine. In those formulas, one assumes the function are the sine and cosine and shows that these equations hold. In this problem, it's the other way around: One assumes these equations hold and then proves, rather than assuming from the outset, that the functions are the sine and cosine. I even rejected an edit to the original posting that would have written $\sin$ and $\cos$ in place of $S$ and $C$. That would have made the question incomprehensible!

Please: stop doing this. --- Michael Hardy

  • 1
    BTW, $C(x)$ would be $\cos(cx)$ for some constant $c$, and $S(x)$ would be $\sin(cs)$ with that _same_ constant.2012-03-28

3 Answers 3

10

Write $E(x) = C(x)+i S(x)$. Then $E(x+y)=E(x)E(y)$. This is a multiplicative variant of Cauchy's functional equation. Without further hypotheses on $S$ and $C$ it is likely that there are many solutions.

If $S$ and $C$ are assumed differentiable, then $E$ satisfies E'=E. If moreover $E$ is not identically zero, then $E(0)=1$ and so $E=\exp$. By Euler's formula, $S=\sin$ and $C=\cos$.

So, one answer to your question is uniqueness of the exponential function from its differential equation.

  • 0
    $E'$ would be some constant multiple of $E$. The constant need not be $1$.2012-03-28
3

Hint $\:$ Said addition laws are true iff $\rm\: E(X) = C(X) + {\it i}\: S(X)\:$ satisfies $\rm\:E(X+Y)\: =\: E(X)\:E(Y)\:.\:$

  • 0
    That's what I expected - I was just curious...2012-03-27
1

I found an article that seems directly relevant to the solution of the equations at: Dr. W Harold Wilson 'On Certain Related Functional Equations' AMS (read Dec 27,1917) Hope this is useful.