In addition to being transcendental, trigonometric functions are classified as "periodic." Mathematicians classify them as such a class of function because any value that appears once in a trigonometric function will appear an infinite number of times. I.e., there are an infinite number of ways to demonstrate that the trigonometric functions are not bijective.
The derivatives of trigonometric functions are also trigonometric functions, therefore their derivatives must also be periodic. Let us find the periodicity of the sine function to begin (defining the sine function as y = sin(x)):
Therefore, from the above we can find the nth derivative of the sine function by taking a multiplier of four away consistently until our derivative is between one and four: that is to say, in algebraic terms, n - 4x.
Let us do the same for the cosine function: