Deeper understand of $\cos$ and $\sin$ I was hoping to understand $\sin$ and $\cos$ a bit more in detail regarding how they came about, not so much about the history but rather and the relationships of these functions with nature and other math topics. In school and college (depending on what your major is), we focus just on the mechanical aspects of $\sin$ and $\cos$. But I'm still having a hard time wrapping my head around how these functions came about.
On the unit circle, why does $\sin$ correspond to the $y$ coordinate on a point on the circle and $\cos$ correspond to the $x$ coordinate(was this just chosen by someone a long time ago)? Or why the vertical distance from the center of the circle to the tip of the line gives us the amplitude of the sine wave as shown on this site? Or why is there a relationship between these two trig functions and a circle? Lastly, are $\sin$ and $\cos$ found in cyclic parts of nature, or are these functions used as good approximations regarding cyclic behaviors in nature?
Sorry if the questions are open-ended, it's just that I've read some wiki articles and visited some other random websites, but I still wasn't able to find answers to these questions. Thanks for your time and help!
 A: Let's look at the most basic definition of $\sin x$ and $\cos x$, using a right-angled triangle.

$\sin x=\frac{\text{Opposite}}{\text{Hypotenuse}}$ and $\cos x=\frac{\text{Adjacent}}{\text{Hypotenuse}}$
This gives us a clear geometric interpretation of what we mean by $\sin x$ and $\cos x$. If we want to compute the sides of a triangle (or the angles of a triangle), then these definitions of $\sin x$ and $\cos x$ will suffice. $\sin x$ and $\cos x$ are useful because if we know what the value of the angle $x$, then the ratios stays the same regardless of how big or small the right-angled triangle is.
At this point, it is unclear why we would need to know what $\sin(100°)$ is. Indeed, using the above definition, $\sin(100°)$ doesn't even have a defined value. It is only when we start using a unit circle that $\sin(100°)$ is meaningful. Why would even try to define $\sin(100°)$? Well, it's not immediately clear. At some point in history, mathematicians were scratching their heads and wondering what the use of defining the $\sqrt{-1} \text{ as } i$ was (if this makes your head spin, don't worry!). Again, it's not immediately clear. But often, when we extend the definition of something, its uses becomes apparent later on. (Incidentally, $i$ is now used routinely in physics and engineering.)
$\sin(100°)$ is similar to $i$ in that it now has many real-world applications. According to Wikipedia, 'the sine function can be used to model phenomena such as sound and light waves'. It is much easier to extend the arguments of $\sin x$ and $\cos x$ to all real numbers, than to stick rigidly to the definition provided by the right-angled triangle. Therefore, while $\sin(100°)$ is certainly an abstract concept at first, ultimately it is very useful.
Now that we have established the motivation for extending the arguments of $\sin x$ and $\cos x$, all that is left to answer is exactly how. Fortunately, there are many sources that explain this. One of your questions was about why the unit circle definition is used, and why $\sin x$ is simply the $y-$coordinate. In this Wikipedia article, all is explained. Let me know if you have any questions.
And, we can keep on going. Remember when I mentioned $i$, the $\sqrt{-1}$ earlier? That's exactly the kind of argument that modern defintions of $\sin x$ and $\cos x$ can accommodate. Again, more abstract, but more useful. I'll spare you the details, but just remember, that there is always a rationale behind what we do in mathematics. No one ever just decided that $\sin x$ was something without good reason for it.
