As several respondents have indicated, one could choose many different trigonometric functions to serve as the basic elements in terms of which other trigonometric functions are expressed and in (ancient) history other choices were made than those standard now. What we know now that was not known to the ancients that leads us to use cosine and sine is the central place in mathematics of linear ordinary differential equations and exponential functions.
The functions $\cos{t}$ and $\sin{t}$ are a basis for the space of solutions of $\ddot{x} + x = 0$, which is probably the most important differential equation in mathematics, because it is probably the most fundamental differential equation in physics. An alternative basis is $e^{i t}$ and $e^{-i t}$, so, said another way, the choice of $\cos$ and $\sin$ is made because these are the real and imaginary parts of the complex exponential $e^{i t} = \cos{t} + i \sin {t}$. The exponential is clearly fundamental, and it is convenient to express as much as one can in terms of it.
The alternative trigonometric functions do not have such properties and this means that to make extensive use of them it turns out to be more convenient to express them in terms of cosine and sine, or, equivalently, exponentials.
Also, this point of view makes apparent the parallelism with real exponentials, or rather the hyperbolic cosine and sine, corresponding to the differential equation $\ddot{x} - x = 0$.
Moreover $\cos$ and $\sin$ admit simply geometric interpretations and sets of functions such as $\{\cos{nt}, \sin{nt}: n \in \mathbb{Z}, n > 0\}$ exhibit useful orthogonality properties (with respect to integration) that make them well adapted for use in contexts like Fourier series. There are of course many systems of orthogonal functions. What the most useful ones have in common is an origin as the solution of a homogeneous second order linear ODEs with some parameter (in this case $n$) in its coefficients.
What one teaches to children is in part dictated by what is needed later by those who will study more, and in part dictated by what has proved most tractable in diverse contexts.