This could be a ******ed question, but I'll ask anyhow. So you graph sin(x) and cos(x) and get a couple nice little waves overlapping.
But there seems to be some room for 2 more overlapping waves in there to satisfy my concept of symmetry.
Sure you can just translate sine and cosine to fill in the gaps, but for that matter you could leave it as one wave and never "name" the other one. Say only do a sine and translate it over to get a cosine.
Is everything just a sine wave, and cosine is just a name we attached because it has some sort of geometric significance in relation to right triangles? Is there no significance of the missing 2 waves?
I'm just having a hard time visualizing the potential geometric significance of what these 2 other functions would be, so maybe someone could steer me in the right direction.
I reserve the right to say "Oh yeah, that's right" if I've just happened to forget this as some obvious result.
So, your question is, "Why do we go out of our way to label Sin and Cosine an not the other infinite number of periodic waves?"
I think there are a few reasons why Sin and Cos are nice. The most obvious answer is related to their trigonometric properties. Sin and Cosine have nice interpretations as components of a vector that is being spun around the unit circle which leads to all the triangle properties that we know and love. This is how most people come to understand Sin and Cos.
They are also nice because they have particularly simple definitions as power series.
The reason for having two of them could be related to the fact that we need two types of functions as basis for periodic functions in the fourier sense. If we pick Sin and Cos, one of which is odd and one of which is even, we can expand any periodic function as a linear combination of these functions with different frequencies. We don't need to add in another wave (like Sin(x + pi/4) or whatever) to do this.
I don't know. I think a lot of your question is related to semantics and definitions which won't necessarily be totally mathematically motivated. Hope that helped.
Your 4th paragraph, "everything is just a sine wave and cos is a special name", has some truth in it. In the geometric 'unit circle' interpretation, the sine, tangent, and secant are constructed on one angle - the (counterclockwise) angle between the x-axis and some ray of interest, and their co-functions on the (clockwise) angle between the y-axis and the same ray.
At the same time, it's true that there are a cycle of four functions whose fourth derivatives are equal to the original function, so it does make sense that you want to "see a family of four." The only nonconstant function whose first derivative equals itself is e^x; with second derivatives you have e^x and e^-x; you get additional solutions for third, fourth, etc derivatives from the complex third, fourth, etc roots of 1.
