Trigonometric functions are the dread of PreCalc and Algebra 2, but why are they so awful? This hate for trig has nothing to do with being bad at math or a general disinterest in them. Trig functions give headaches to even the very best Calculus students. What are they? And why do they not follow the regular rules of math?

First, we need to know what Trig functions do. In simple terms, Sinusoids (sin) follow an equation that takes a simple exponential function: x11! , where 1! is a factorial. Factorials are functions that spit out simple numbers similar to squares or cubes. They work by multiplying all numbers up to the input together. For example 4! is 1*2*3*4 and 5! is 1*2*3*4*5. The next step is adding another polynomial only that the “1” gets replaced with a three: x33!. This process is repeated to infinity, increasing the exponent by two so that 3 becomes 5, then 7, and so forth. If we would graph this, we can see that it is not entirely on a sin function but gets unbelievably close, at least for numbers close to zero.

The Red graph is Sin(x) and the green function is the approximate equation. We can, of course, not add infinite polynomials, but it is visible that an increase in terms increases the accuracy.