I expected to always find cos and sin functions to be identical to each other with the only exception being that their phases will differ by π/2
So what I am trying to say here is that we should always expect to find:
f(sin(θ)) = f(cos(θ - π/2))
However, discounting the phase shift, you should always expect the shapes of the curves to be identical. Surely any cos function should be found to be identical in form (shape) to its sibling sinusoidal equivalent, only shifted along by π/2 radians, right?
However, to my astonishment, I discovered a scenario when this was not the case! What I expect to be a universal law of logic, was breached.
Consider the following curves:
Graphing these functions (for example, on https://www.desmos.com/calculator), you will find the two curves differ in appearance!! They are not a phase shifted mirror of one another!!
I find this quite unexpected and surprising and am quite amazed by this discovery!
Dissecting the cos equation, I can see where it diverges, but it is unclear to me how I was wrong to assume sine and cosines operations would always yield the same shape only phase shifted? It seems counterintuitive that this assumption is not universally always true. Your thoughts please.