Ordinarily in mathematics, when you want to define a function, it is without reference to geometry. For instance the mapping f:ℝ→ℝ x→x2 And though I don't know much about mathematics I assume you somehow proof that the function is well defined for all numbers, check if the derivative exists and so forth. But for sine and cosine it seems somewhat different if you use the definition with the unit circle. It seems then that all their properties must be proven through geometric arguments. How do you for instance proove that they are defined for all numbers? How do you proove anything about their derivatives and general limits of them, when all you can resort to is a "drawing"?! Maybe my worries are for nothing, but I still wanted to ask the question and ask whether there exists a definition of them purely in terms of algebraic means.