Shifts by a quarter, a half and a whole turn all fall out of the same two results

0:00Here is a block of eight formulas that most people meet as a table to be memorised. A quarter turn on. Half a turn back. Half a turn on. A whole turn back. Each of the four, twice: once for the first coordinate, once for the second. Eight lines, each carrying a function name and a sign, and no visible reason for either. So people learn them, and then get them wrong under pressure, because two things are changing at once and the table does not say which.

0:34Here is the question nobody asks about it. Why does a quarter turn change which function you end up with, while a half turn never does? That is not a quirk of the table. That is the entire content of the table. And once you can answer it there is nothing left to memorise, because there is one expansion, four pairs of numbers, and a rule for which of two products dies.

1:01Let me show you the machine. Start with the only numbers this whole topic ever substitutes. Go round the circle and stop at the places where one of the two coordinates is nothing at all. At the start: one across, nothing up. A quarter turn on: nothing across, one up. Half a turn: minus one across, nothing up. Three quarters: nothing across, minus one up. Exactly four of the twenty four points have a coordinate equal to nothing.

1:36And none of the four has both equal to nothing, which would be a point that is not on the circle at all. Now notice which one dies where, because this is the answer to the question. At the start, and at half a turn, it is the second coordinate that is nothing. At a quarter turn, and at three quarters, it is the first. Two of the four each way.

2:01Hold on to that split. Now take the difference form, which you already have. The first coordinate of a difference is the product of the two first coordinates, plus the product of the two seconds. Put a quarter turn in for the first angle, and any angle you like for the second. Watch what happens. The first coordinate of a quarter turn is nothing, so the first product is nothing times something.

2:32The whole term dies. The second coordinate of a quarter turn is one, so the second product is one times the second coordinate of your angle, untouched. And that is the answer: the first coordinate of a quarter turn less an angle is the second coordinate of that angle. One substitution killed one term and left the other one alone. That single move is the one everything else in this topic reuses.

3:02Checked at every one of the twenty four inputs, it holds at all of them. Now the companion, and it needs no new work whatsoever. Take the line you just proved, and feed it a quarter turn less the angle, instead of the angle. A quarter turn, less a quarter turn less the angle, is the angle back again. So the same line now reads the other way about: the second coordinate of a quarter turn less an angle is the first coordinate of that angle.

3:36The result was applied to itself, and no new geometry appeared anywhere. There is a reason that is legitimate rather than lucky. The substitution that does it sends the twenty four inputs onto the twenty four inputs, one for one. It reaches all of them. So proving the first line at every input has proved the second at every input too. And do it twice and you are back where you started, at all twenty four.

4:07That is what makes the companion free. Those two lines together are saying something that a picture says faster. Two angles that add up to a quarter turn have their two coordinates exchanged. Draw the circle, and draw the diagonal through the centre at half a right angle. Take any point on the rim and reflect it in that diagonal. Reflecting in that particular line swaps the two coordinates: across becomes up, up becomes across.

4:41And the point it lands on is exactly the point at a quarter turn less the arc. All twenty four of them do this. Not most of them. All of them. So the two results are not two facts. They are one reflection, read twice. And the word people use for a pair of angles that add to a quarter turn is suddenly doing real work, because it is naming that reflection.

5:09Now the second coordinate of a sum, which people expect to need new geometry. It does not. Watch the assembly. Step one: the second coordinate of anything is the first coordinate a quarter turn back from it. That is the result we just built, read from right to left. Step two: regroup that input as a quarter turn less the first angle, less the second angle. Nothing was rewritten there; it is the same point named a different way.

5:43Step three: that is now a first coordinate of a difference, so expand it. Step four: convert the two quarter turn pieces back, using the same result a second time. And out falls the first sine times the second cosine, plus the first cosine times the second sine. Four steps, three earlier results, no picture. The checker keeps a log of what each step leaned on: four entries, three different results, the difference form used once and the exchange result used twice.

6:19Nowhere in that log does the answer appear. The derivation never used itself, and that is checked rather than promised. All four steps agree at every one of the five hundred and seventy six pairs. The difference version now costs one line. Put the second angle in turned round. The first coordinate does not notice a minus sign on the input, and the second coordinate turns over. So the plus in the middle becomes a minus, and nothing else moves at all.

6:54Counting the sum form's own four steps, that is six, and two of the six are the turning round. Both of the new steps agree at all five hundred and seventy six pairs. That is now the third time in this video that one substitution has done the whole job. Now the table itself, one entry at a time, out of the same machine. A quarter turn forward. Substitute a quarter turn for the first angle in the sum form.

7:27The first coordinate: nothing times the first coordinate here, minus one times the second. The product carrying the shift's first coordinate dies. What survives is the one carrying the other coordinate of your input, and that is precisely why the name changes. So the first coordinate a quarter turn on is minus the second coordinate here. Not just a sign. A different function. Now the second coordinate: one times the first, plus nothing times the second.

8:03Again the product carrying the shift's first coordinate dies, and what survives is the first coordinate of your input, untouched. Two entries, both derived, neither one remembered, and each holds at all twenty four inputs. Now half a turn, and watch what is different. The first coordinate of half a turn is minus one, and the second is nothing. So this time it is the other product that dies: the one carrying the shift's second coordinate.

8:37And that changes everything, because the product left standing is now the one carrying the same coordinate of your input. Half a turn back, first coordinate: minus one times the first coordinate. Reversed, and still a first coordinate. Half a turn back, second coordinate: the second coordinate, untouched. Half a turn forward, the same substitution with a plus, and both of them reverse. Four more entries. And in not one of them did a name change.

9:12The signs moved. The functions stayed exactly where they were. And a whole turn back, which is the same thing as turning the input round. The first coordinate does not notice; the second turns over. Two more entries, and they are not new information. You already knew the coordinates come back to themselves after a whole turn, because that is what going round a circle means. Deriving it again through the expansion is a consistency check and nothing more.

9:47That is eight entries. Every one produced. Not one recalled. Now the answer to the question from the beginning, and it is worth being exact about. Take every shift there is on these twenty four points: go any number of steps round, having first turned the input round or not. That is forty eight shifts, and they are forty eight genuinely different maps. The one expansion reproduces every single one of them.

10:18For the first coordinate that is one thousand one hundred and fifty two shifts and inputs, with no exception, and the same again for the second. But only some of them are tidy. A shift is tidy when one of the two products is nothing at every input, so the answer comes down to a single term. How many of the forty eight are tidy? Eight. None of the forty eight kills both products, which is just as well, because that would be a shift standing somewhere off the circle.

10:53And of those eight, how many change which function you end up with? Four. The other four keep it. Here is the rule those eight obey: the names change places exactly when the shift's own first coordinate is the one that is nothing. How many of the eight disagree with that rule? Zero. That is the whole table, in one sentence. The other four functions come from these two by dividing or by inverting, and they cost no new work either.

11:29Take the quotient of the second coordinate by the first. Half a turn on, both coordinates reverse, so the quotient is unchanged: the two minus signs cancel. That has a consequence worth saying out loud. The two coordinates take a whole turn to come back to themselves. Twenty four steps, and not one step fewer. The quotient comes back after twelve. It repeats twice as often, and this is the reason for it, not a separate fact about it.

12:02The quotient also refuses to exist at two of the twenty four inputs, where the first coordinate is nothing and there is nothing to divide by. Invert instead of dividing and the same machine answers again: the reciprocal of the first coordinate turns over after half a turn, at all twenty two inputs where it exists. Now the argument that settles it. The usual list names five different maps: the four of the block, and the exchange pair's one.

12:37There are forty eight. So the list leaves forty three of them unnamed. That would not matter if the unnamed ones never came up. They come up immediately. An eighth of a turn, forwards and backwards. Three eighths of a turn. Three quarters of a turn, forwards and backwards. Five maps, and every single one of them turns up in the questions you get set. How many of the five does that list name?

13:06None of them. Zero out of five. And how many does the one expansion produce anyway, at every input, for both coordinates? All ten. A student holding the table is stuck at the first question that moves by an eighth of a turn. A student holding the machine is not. So here is what those eight formulas actually are. One expansion, which you already had before this video started. Four pairs of numbers, which are only where the circle crosses the two axes.

13:43And one observation. At each of those four places one of the two coordinates is nothing, so one product dies, and the product left standing decides both the name and the sign. When the shift's first coordinate is the one that dies, the survivor carries the other coordinate of your input, and the names swap. When it is the second, each term keeps its own function, and the only thing left to change is a sign.

14:14That is it. That is the table. There are not eight facts here, and there are not sixteen once the other four functions arrive. There is one machine, and you can run it as often as you like. Including on the shifts the list leaves out.