What are some rigorous definitions for sine and cosine? Here are some of my ideas:
1. Addition Formula: $\sin{x}$ and $\cos{x}$ are the unique functions satisfying:

*

*$\sin(x + y) = \sin x \cos y + \cos x \sin y $


*$\cos(x + y) = \cos x \cos y - \sin x \sin y$


*$\sin 0 = 0\quad$ and $\quad\displaystyle{\lim_{x \rightarrow 0} \frac{\sin x }{x} = 1}$


*$\cos 0 = 1\quad$ and $\quad\displaystyle{\lim_{x \rightarrow 0} \frac{1-\cos x}{x} = 0}$
2. Taylor Series:

*

*$\displaystyle{\sin x = \sum_{n = 0}^{\infty} \frac{(-1)^n}{(2n+1)!}\;x^{2n+1}}$


*$\displaystyle{\cos x = \sum_{n=0}^{\infty} \frac{(-1)^n}{(2n)!}\;x^{2n}}$
3. Differential Equations: $\sin(x)$ and $\cos(x)$ are the unique solutions to $y'' = -y$, where $\sin(0) = \cos^\prime(0) = 0$ and $\sin^\prime(0) = \cos(0) = 1$.
4. Inverse Formula: We have:
$$\begin{align}
\arcsin x &= \phantom{\frac{\pi}{2} + } \int_0^x \frac{1}{\sqrt{1 - t^2}}\, dt \\[6pt]
\arccos x &= \frac{\pi}{2} - \int_0^x \frac{1}{\sqrt{1 - t^2}}\, dt
\end{align}$$
Then $\sin x$ is the inverse of $\arcsin x$, extended appropriately to the real line, and $\cos x$ is similar.
Question: Are there any others that you like? In particular, are there any good rigorous ones coming from the original geometric definition?

My preferred definition is:

$\cos x$ and $\sin x$ are the real and imaginary parts of the
  exponential function $\exp(ix)$.

Since we have:
$$
\begin{split}
e^{ix}=   \sum_{k=0}^\infty\dfrac{(ix)^k}{k!}&=1+ix+\dfrac{(ix)^2}{2!}+\dfrac{(ix)^3}{3!}+\cdots+\dfrac{(ix)^n}{n!}+\cdots\\
&=1-\dfrac{(x)^2}{2!}+\dfrac{(x)^4}{4!}-\dfrac{(x)^6}{6!}+ \cdots +i\left[ x-\dfrac{(x)^3}{3!}+\dfrac{(x)^5}{5!}+\cdots  \right]\\
&=
\sum_{k=0}^\infty\dfrac{(-1)^k\,x^{2k}}{(2k)!}+i \sum_{k=0}^\infty\dfrac{(-1)^k\,x^{2k+1}}{(2k+1)!}\end{split}
$$
we find the usual series definitions.
