The first time someone encounters trigonometry in schools is just 3 definitions of 3 relationships (\(\sin x, \cos x, \tan x\)) on a triangle with a 90 degree angle, that's how it is presented and then the poor child is presented with the multplicative inverses of those functions (\(\csc x, \sec x, \cot x\)), the poor child is probably going to ask why do we put names to the multiplicative inverses of those functions and the answer is legacy...

Notation is fine for inverse functions as we are using the -1 in the function so it's not going to get confuded with the multiplicative inverse but then the teacher comes and writes \(\sin^{2} x\), the teacher asks what does this mean, and the child answers, it means \(\sin(\sin(x))\) because we are squaring the sine function, to which the teacher responds, no this is the way we write \(\sin x \cdot \sin x\).

In the final encounter of the children with trigonometry in school, they are presented with "The fundamental identity of trigonometry", which it is \(\sin^{2} x + \cos^{2} x = 1\), after parsing the stupid notation of the teacher for trigonometric function they see that the identity is saying that the squares of sin and cos of x is equal to one, why? they are going to ask and the teacher isn't going to answer because the way to justify this is with the idea of the trigonometric functions being the coordinates of a point in a circle, but the teacher didn't teach that, he teached this arbitrary relationships on a triangle.

The child gets to uni, where the professor tells the class about the idea of trigonometric functions being the x and y coordinates of a point in a circle and he finally understands the "fundamental identity of trigonometry" is just saying that a point on a circle satisfies the circle equation, something that is obvious, the class continues and the professor do some proofs of trigonometric identities and then just hands some tricks to solve problems involving trignometry, this way of presenting trigonometry is better than the one presented in school but it has it's shortcomings. The principal shortcoming is that it does not present the motivation behind those identities and the tricks for solving the problems, and that the notation is the same as the one in school.

As you may guess, the story of that child is my story and I want to also point how this can be solved as well as point to the way I think is better to think about trigonometry. The sine function is just a relationship between an angle and the y coordinate of point in a circle of radius 1, and the cosine function is just a relationship between the x coordinate of a point in a circle of radius 1. Every other trigonometric function can be defined as some operation combining this 2 operations.

First of all I will tackle the problem with the multiplicative inverses trigonometric functions (\(\csc x, \sec x, \cot x\)), why did we put names to this functions? I really don't know but probably in ancient times it had it's uses, but now in this current day and age it just makes more obtuse the notation, if we just write it as \(\frac{1}{\sin x}\) for the cosecant or just (\(\sin x)^{-1}\) it would be a lot cleaner and consistent with other notations.

Another big problem I have with trigonometry notation is the square of the functions, why do we write \(\sin^{2} x\) for the square of \(\sin x\), that notation should be reserved to a double application of the sine function, \(\sin(\sin(x))\), and this would make a lot of sense because we also have that the inverse function for the trigonometric functions is \(\sin^{-1} x\), in this case we are saying we want the inverse of sin, not the multiplicative inverse, that's the problem with this inconsistent notation. If you have \(\sin^{2} (\sin^{-1}(x))\) it isn't equal to \(\sin x\), in this case it is equal to \(x^{2}\), but we are putting exponents to the functions, not to the values.

I guess we can say that the problem with trigonometry is that is badly taught and that notation is incosistent with what we have learned before, making it a pain in the ass to re learn

Hope you enjoyed.