I have been thinking about all of the different ways that I have encountered sine and cosine in my studies. There are no courses on trigonometry at my school, so perhaps that's why I feel like something is missing, something that ties all of these ideas together.
- The Unit Circle. The unit circle is a way to organize all possible right triangles up to similarity. The sine and cosine can be defined as ratios of sides of these right triangles, though in practice the sine becomes the vertical component and the cosine the horizontal. What I want to know is how do I relate this beautiful diagram to the other constructions of sine and cosine.
The orthonormal basis of the solution space to the differential equation: $y'' = -y$. In some sense this is as good of a definition as the unit circle.
Taylor Series. These grow clearly out of the differential equation above. How could I connect Taylor series to the concept of the unit circle? I think of partial Taylor series as "better and better" approximations of these function centered at zero (or maybe some place else, I guess it would not matter) -- so how is that concept linked to what $y'' = -y$ says about these equations?
Lastly, are there any other major representations I should also consider?