It sounds like you are looking for a formula to calculate the output of a trig function for a given input. While there are methods for doing this approximately, it’s important to note that a function doesn’t need an equation in order to be a function. A function is just a special collection of inputs and outputs. How you define the function is up to you.
The key thing that allows us to define trig functions how we do is a fact from geometry. Namely, if you draw two right triangles that have the same angles the ratios of their corresponding side lengths will always be the same. In other words the ratios “opposite/hypotenuse”, “adjacent/hypotenuse”, and “opposite/adjacent” only depend on the angle, not the size of the triangle. Since these ratios only depend on the angle, we can define a function that associates one of these ratios to each angle. This defines a function. This *is* what the function is “doing.”
Actually calculating these things is interesting, but it’s a whole other story. Traditionally, these things were actually measured and tabulated in look-up tables.