Step right up, Ladies and Gentlemen! Get your ticket to see the True Oddities of the Natural World! Do not be taken in by the Shameful Forgeries at Inferior Circuses, here you will see Genuine Curiosities from the Mists of Time! Beside these Archaic Functions, the Two-Headed Horse or Whatever is a Mere Bagatelle!
So, why do we bother having a sine and a cosine? They’re the same thing, one just happens to be shifted. Whatever angle you plug into cosine, you’ll get the same result if you plug that angle plus 90 degrees into the sine. Why not just scrap one of them entirely and just teach the idea of plugging in the equivalent angle?
And certainly you could. They are the same function, up to that shift. But in many other senses, they’re different functions entirely. They have opposite parity properties, they satisfy differential equations with entirely different initial conditions, their Taylor series expansions don’t share a single term, they’re entangled with the theory of the complex numbers in different ways, and so on. In that sense the fact that these two very separate functions are really the same is a deep property of mathematics.
As such, we have the sine and cosine and mathematicians would generally be horrified at the suggestion that we scrap our notation that treats them independently. But this hasn’t been true for every trig function in the past. Some permutations of sine and cosine were originally useful for some obscure (usually navigational) purpose, but as their practical utility waned they were forgotten. Since they didn’t really have independently important mathematical properties of their own, they’re now written only in terms of the “standard” sine and cosine functions. Just for today, we’ll reanimate a few of them Frankenstein-style and see what they look like:
Back before electronic calculators, it was pretty irritating to have to square small numbers. Since the square of the sine pops up in trig very frequently, it was useful to have tables of the versine pre-computed in a book. For the same reason there was also the vercosine, which is the same function with the sine squared switched out for cosine squared.
Just like sine and cosine are shifted from each other, the coversine is just the versine shifted. There’s also the covercosine, which is shifted from the vercosine in the same way.
The haversine is just the versine divided by 2. It found its use mostly in navigation, where it can be used to find the distances between two points on a sphere – such as the earth. The haversine has its own twin, the havercosine, which is the vercosine divided by 2. I’m getting tongue-tied just typing this.
I promise I’m not making these up. Like its cousin the haversine, the hacoversine is mainly useful for doing trig on a sphere when you don’t have a calculator handy to square the familiar trig functions. Does it have a twin like the other functions above? Does a bear, uh, eat berries in the woods? Of course it does. The hacoversine has a twin which is just the covercosine divided by 2. It’s called, I am sorry to say, the hacovercosine.
Let’s just say I’m happy to stick with the good old sine and cosine.