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.

If sin functions are long polynomials, why do they require trig identities? Most of the problems come from the fact that the inputting values are in radians or degrees. The function converts an angle to a ratio, while most functions convert one quantity to the next and rarely change the unit. Mr.Tolly explained how “sin functions are a headache because we need to restrict their domains to make them fit algebra.” The concept of domain refers to the idea that some functions just can’t give a certain range of results. For example, a car tank can’t have 200% of fuel or -29%; it can only go from zero to 100. The same concept applies to Trig Functions. In theory, they can only go between 0 and 360 degrees before repeating themselves.

If you haven’t taken physics yet, then there might not be a reason to use sin and cos. Those concepts appear in multiple engineering jobs and trigonometry is one of the fundamental tools for those jobs. For obscure angles and bends trig function might be the easiest way to go, if not the only way of solving problems. Math was never invented to be hard, but rather to help us understand the world we live in. Although trigonometric functions might be a headache, it would be orders of magnitude more challenging to do the same computation with Algebra.

Perhaps you will never use those math tools again. If this is the case, you will be spared the dreads of trigonometry, but if you don’t then there is no way around them. Some enjoy them, some don’t, and that doesn’t mean the end for anyone; it is beneficial to specialize and get help from others. We, humans, thrive on others’ work and help. With that, thank you to the people who invented it so we didn’t have to.

Image courtesy by Desmos Graphic Calculators