Graphing Trigonometric Functions and computing their derivative

For an upcoming midterm, I need to be able to do the following: graph, indicate direction & speed, compute the derivative, and find critical points of trigonometric and hyperbolic functions.

For example:

$(sin\theta, cos\theta)$

$(csc\theta, cot\theta) \theta \neq k\pi, k\in Z$

$(cosh t, sinh t)$

What perplexes me is the form: (a, x) with respect to theta. Are these parametric plots or the two functions mapped onto one plot? My interpretation is that they are parametric. So, for example,

$(sin\theta, cos\theta)$

Should be a unit circle except $sin\theta = x$ and $cos\theta = y$

If this is true, do I take the derivative of $sin\theta = x => cos\theta$ and $cos\theta = y => -sin\theta$ ?

But then the plot of its derivative is also graphically the same as the original function. Any help would be greatly appreciated, thanks for your time.