Numeric Representation of Trig Functions I'm taking a first semester calculus course and we are learning about the hyperbolic functions. For example, $\sinh(x)$, $\cosh(x)$, etc. The thing that most interests me is that there are actual numeric functions being represented by the hyperbolic functions. 
For example $\sinh(x)=\frac{e^x-e^{-x}}{2}$ and $\cosh(x)=\frac{e^x+e^{-x}}{2}$. And of course all of the other hyperbolic functions are build from either of these two, or combinations of them. So the way that I'm looking at $\sinh(x)$ and $\cosh(x)$ is as aliases for $\frac{e^x-e^{-x}}{2}$ and $\frac{e^x+e^{-x}}{2}$.
Are the trig functions similar in this way? What I'm asking is do $\sin(x)$ and $\cos(x)$ have some actual numeric function which they are aliases for? I should probably already know the answer to this, but I've struggled in the past with understanding trig functions. My understanding of the hyperbolic functions seems to be better because there are actual concrete numeric functions tied to them. Is there anything analogous to this with the trig functions? 
 A: I'm not sure 100% what you mean by "numeric" but in terms of "analogous":
$$\sin(x) = \frac{e^{ix} - e^{-ix}}{2i}$$
$$\cos(x) = \frac{e^{ix} + e^{-ix}}{2}$$
Although, funny enough, when I have $e^{i\theta}$ I usually think of it in terms of $\cos$ and $\sin$ in order to compute the values...
A: If you're not satisfied with Dair's answer because it uses complex numbers, here are a couple of ways you can define $\sin x$. Neither is very elementary.
First, you can define $\sin x$ as the sum of an infinite series,
$$\sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \dots .$$
No matter what number $x$ is, the partial sums of this series eventually start getting closer and closer to $\sin x$. They converge faster when $x$ is small. For example, 
$$\sin 0.1 \approx 0.1 - 0.1^3/3! + 0.1^5/5! = 0.009983341666\dots$$ is an excellent approximation of $\sin 0.1 = 0.009983341664\dots$.
Second, when you learn about integrals, you'll see that $\arcsin x$ can be defined as the integral $\int_0^x \frac{dt}{\sqrt{1-t^2}}$ whenever $-1 \leq x \leq 1$. Then $\sin$ can be defined as the inverse function of $\arcsin$, at least for arguments between $-\pi/2$ and $\pi/2$.
Definitions similar to those above can be given for $\cos$. You may be interested to know that
$$\sinh x = x + \frac{x^3}{3!} + \frac{x^5}{5!} + \frac{x^7}{7!} + \dots ,$$
where the only difference with $\sin x$ is the sign of the terms.
I should add, lastly, that it is possible to give a rigorous definition of the sine function that follows the typical geometric definition, but then the difficulty lies in giving a mathematically precise definition of at least one of the following closely related concepts: angle measure, the length of a circular arc, or the area of a circular sector. In high school math, these concepts are always taken for granted as being intuitively obvious, but their correct formulation takes a good deal of work.
