How many ways are there to define sine and cosine? Sometimes there are many ways to define a mathematical concept, for example the natural base logarithm. How about sine and cosine?
Thanks.

An interesting construction is given by Michael Spivak in his book Calculus, chapter 15. The steps are basically the following:
$1.$ We define what a directed angle is.
$2.$ We define a unit circle by $x^2+y^2=1$, and show that every angle between the $x$-axis and a line origined from $(0,0)$ defines a point $(x,y)$ in that circle. 

$3.$ We define $x = \cos \theta$ and $y = \sin \theta$. 

$4.$ We note that the area of the circular sector is always $x/2$, so maybe we can define this functions explicitly with this fact:

$5.$ We define $\pi$ as the area of the unit circle, this is:
$$\pi = 2 \int_{-1}^1 \sqrt{1-x^2} dx$$
$6.$ We give an explicit formula for the area of the circular sector, namely:
$$A(x) = \frac{x\sqrt{1-x^2}}{2}+\int_x^1 \sqrt{1-t^2}dx$$

and show that it is continuous, and takes all values from $0$ to $\pi/2$. We may also plot it, since we can show that $2A(x)$ is actually the inverse of $\cos x$.

$7.$ We define $\cos x$ as the only number in $[-1,1]$ such that
$$A(\cos x) = \frac{x}{2}$$
and thus define
$$\sin x = \sqrt{1-\cos^2x}$$
$8.$ We show that for $0<x<\pi$ 
$$\cos(x)' = - \sin(x)$$
$$\sin(x)' =  \cos(x)$$
$9.$ We finally extend the functions to all real values by showing that for $\pi \leq x \leq 2\pi$,
$$-\sin(2\pi-x) = \sin x$$
$$\cos(2\pi-x) = \cos x$$
and then that
$$\cos(2\pi k+x) = \cos x$$
$$\sin(2\pi k+x) = \sin x$$
