I've looked over the answers so far, and I it very likely with all that 'accurate' information, you may well have the answer you need. If not, I can sympathise with your dilemma so if I may, I can make a few basic suggestions from my initial "experience" with Fourier Calculus. It can be confusing and may not address 'fundamental question':
Unfortunately, I just cannot see why.
As I get this, the emphasis is on "why
"? In that spirit, I suggest that the issue is firstly that you may be a visually oriented person, I take my lead on that because you 'get' the diagram above. So you understand that the COSINE() function of an angle, is a ratio of two sides of any right-angle triangle: Adjacent and Hypotenuse.
Related, geometrically, to the angle with the adjacent adjacent to the sloping side (hypotenuse).
If I can suggest the "lazy way" first; the fraction created from the ratio between these sides as the corner-angle, alpha, changes is-a *curve*in Euclidean space. So if you take a grid and plot sides: (x, y) at the centre for angle alpha ...
- x = (Length of the Adjacent side)
- y - (Length of the Hypotenuse side)
On an axis ... as you change the angle you will see a Cosine curve traced at the opposite point. Common tools and web graphics use this to make 'magical' sprials and stuff. As a boy I had a Spirograph ... it creates patters. ALL of these are based on the same rations from a circle/triangle. It is a kind of magic, just like the Greeks thought. Beauty may be truth.
Why? If you don't have a Spirograph - You can makes some cool substitutes.
- Create a triangle shape with three flat lengths of wood.
- You can ensure the triangle retains rectangle with a nail or tight screw.
- Leave the other two sides - FREE to move.
When you attach a pen to the arms of this right-angle, and move the sides(arms); you will trace curves. Congratulations, they are cos or sin curves!
Once upon a time 'science' like Geometry included philosophy and 'manners' for conduct. We don't live in that world any longer, do we? The aim of Geometry is to consider every thing (material item) objectively in 3-Dimensional space. At that time, 3,00 years ago Science was a kind of magic.
The "Smooth CURVES" you asked about, were considered part of the bounty or reward to those of enquiring minds. Recall that is is like, 2,000 years before Picasso -- Euclid had abstract art. And pretty cool stuff too!