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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?

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Perhaps coversine? There are a lot of old trigonometric functions that fell out of popular use. For one, they can be expressed as linear combinations of the others. Also, they just didn't come up as much in investigations. Some others were versine, exsecant, and excosecant. – Bryan Jun 20 '14 at 20:14
$\sin(x + \frac{\pi}{4})$ meets the "starts between sine and cosine" property. – Zach Jun 20 '14 at 20:21
I hope it wasn't cosvnx:… – KCd Jun 20 '14 at 20:22
It can't satisfy the derivative properties unless it's just an exponential: $\frac{df}{dx} = \pm f \implies f ( x ) = C e^{\pm x}$ – Zach Jun 20 '14 at 20:38
@KCd: That's an absolutely hilarious article, thanks for the link! "Fotsin" and "fostin"? Gotta love it, +1 for you. – MPW Jun 20 '14 at 20:43

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