Defining trigonometric functions via complex sequences

In the literature we find several different ways of introducing elementary functions. For the exponential function, we mention the following ways of characterising the exponential function: (a) (b) , also for complex values of x; (c) x → exp (x) is the unique solution to the initial value problem [4] (d) x → exp (x) is the inverse of (e)x → exp (x) is the unique continuous function satisfying the functional equation f (x + y) = f (x) f (y) and f(0) = 1 [6]; the corresponding definition is done for the logarithm in [7]; (f) Define dr for rational r, and then use a continuity/density argument [8]. All of them have their advantages and disadvantages. We like (a) and (c), mostly because they have natural interpretations, (a) in the setting of compound interest and (c) being a simple model of many processes in physics and other sciences, but also because they are related to methods and ideas that are (usually) introduced rather early to the students.