I was reading something on the Internet the other day, and I swear I came across a reference to an alternative sine function [which I now cannot find any mention of].
The usual sine function starts at a zero-crossing. The corresponding cosine function is a quarter-wave out of phase, starting at a peak.
The function I came across seems to start exactly half-way between these two points. This has the fascinating consequence that the corresponding "co"-function is just the additive inverse of the function itself. Similarly, the derivatives of this function and it's co-function are all just the function itself, or its negation.
Did I dream the entire thing, or has anybody else heard of this function before? Does it have a well-known name?
Update: It appears I have my facts slightly confused. It appears the function I'm thinking of is
$$f(x)=2^{-\frac12} (\sin x + \cos x) = \sin \left( x + \frac{\pi}4 \right)$$
I had in my head that $\langle f(t), -f(t) \rangle$ forms a circle --- but that clearly can't be right. However, $\langle f(t), f(-t) \rangle$ does appear to form a circle. So the "co"-function is't $-f(t)$, it's $f(-t)$. I believe $f'(t) = f(-t)$ and $f''(t)$ is therefore $-f(t)$. I'm not 100% sure.
Again, does this puppy have a name?