My understanding is that $\cos$ is defined by the value of $x$ as you trace the graph of a circle counterclockwise, starting at the point $(1, 0)$. Similarly, $\sin$ traces the $y$ value. I understand HOW the trigonometric functions work. The issue that has been gnawing at me for years is WHY they are defined that way. There's probably a totally reasonable explanation, I know the history of Trig goes back thousands of years, but I don't understand the reasoning behind defining the Trig functions with the most arbitrary, least intuitive possible rules.
If I had been the person to invent $\cos$ and $\sin$, I would have defined them by starting at the topmost point of a circle, and trace it clockwise. Is that not the most intuitive method? Maybe it's just a modern preference, but it seems to me that we humans like to read things left-to-right, and yet the trig functions are defined starting from a circle's right-most point. Furthermore, $\cos$ and $\sin$ start at $x = 1$ on a circle's graph. Why not start at $x = 0$?
I think this is why so many people have no intuitive understanding of $\sin$ and $\cos$, and why many students get through high school and college by simple rote memorization of what a handful of $x$ values evaluate to in $\sin$ or $\cos$.
Best Answer
It's because clockwise angles are taken to be negative while counterclockwise angles are positive. Thus to define the functions, we work on positive angles, hence counterclockwise.
You may ask, why are counterclockwise angles positive and not the other way round?
The answer lies in our choice of complex plane.
Our choice while defining complex plane was-
As a result, multiplication by $e^{i\theta}$ rotates the complex plane by the angle $\theta$ counterclockwise.
So, to avoid negative signs in our problems, we say that multiplication by $e^{i\theta}$ rotates the plane by angle $\theta$, not $-\theta$, in other words, we adopt the counterclockwise direction as positive.