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:
The Versine:
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.
The Coversine:
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:
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.
The Hacoversine:
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.
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Sunday function is great. Could you do some fucntions which are everywhere continuous but nowhere differentiable? Or other examples of interesting pathological cases which are mathematically possible, but that don't make any physical sense?
Thanks! I've done some pathological cases in the past - in fact both the Weierstrass and Blancmange functions. Both their entries are pretty short though, so they may be worth revisiting. Another of my favorites is a function that's continuous at one and only one point. I'll try to do some more pathological anomalies in future posts!
On this topic, it isn't completely clear to me why we make kids memorize what the secant, cosecant, and cotangent are. It seems that which functions we require kids to memorize and which we don't are accidents of history rather than products of good pedagogy.
Incidentally, I was aware previously of the versin, coversin, haversin but only vaguely aware of the others. This leads to some horrible ideas for trivia questions at some point...
Let +v and +u be two 3-d unit vectors with angle theta between them.
Suppose that a spin-1/2 particle has spin in the +u direction. You measure its spin in the +v direction so that the result is either +v or -v. The haversine gives the probability that the result of the measurement is -v.
The way I learned about sine and cosine (sinus and cosinus we call them) they were defined from a right-angled triangle. The sine of one of the angles was the length of the opposite side divided by the length of the hypothenuse, and the cosine was the length of the adjacent side divided by the hypothenuse. As long as trigonometry was about triangles it made lots of sense to have both of them :-)
Also, thinking like this helped me many times to remember the relationship between sin, cos and tan.
I'm one of those really old farts who used haversines in navigation during World War II. I still have in a basement file cabinet my copy of Bowditch with its haversine tables.
As the 15th century waned the One True Church expressed strong opinions on the use of Roman numerals vs. devilish cyphers for calculation. Protestants not yet having a franchise, captains sailing out of sight of land were predominantly Muslims, Jews, and pagans. One suspects Columbus had square and compass friends whose maps and navigation toolboxes qualified him for being with his crew in a Spanish gaol rather than sailing westward.
@7,
That's not really true, AFAIK. In fact, Pope Sylvester II was one of the early advocates of Arabic numerals in the West (in the 980s!).
If there was official opposition to Arabic numerals from the Church/Government (which were hard to tell apart at the time), I'd wager it had to do with contracts and business transparency.
On a more mathy note, many of the "exotic" trig functions have nice geometric interpretations, just like sine and cosine. Wikipedia has a picture with some of them here: http://en.wikipedia.org/wiki/Trigonometry
Ah, @8 saved me some typing. Some years ago I ran across the obscure fact that the "main" functions are sine, tangent, and secant. If you isolate those 3 in the diagram in that wiki article, you can see why, and how the co functions complement them. A clean (and more traditional) version with just the 3 main functions is on the wiki "Trigonometric functions" page. (It also shows how the co functions are reciprocals of them.) That circular geometrical picture gets lost when the focus shifts to right triangles and secant takes on a different meaning in calculus.
And thanks for that info about the versine.
"That's a hacoversine, buddy." ha ha ha ha ha ha :) (Yes, this horrendous pun is my sole contribution here ;)
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