Where each function is defined, what values it reaches, and how it repeats

0:00Six functions. One point, walking round a circle of radius one. Two of them are just the coordinates of that point: the reach across, and the height up. The other four are quotients built out of those two, and nothing else. Now four questions get asked about each of the six. Which inputs does it accept? Which values does it reach? Which way does it go as the point moves?

0:29And how far along the input line before it starts over? That is twenty-four answers, and it looks like a page to memorise. It is not. Every one of those answers was already decided the moment the definition was written down. Nothing here is new information. It is the definition, read back to you. Start with which inputs are allowed. The input is how far round you have walked. Pick any distance at all, forwards or backwards, and you land somewhere.

1:05There is no arc length that fails to put you on the circle. And wherever you land, the point has a reach and it has a height. So the height, which is the sine, has a value at every input there is. And so does the reach, which is the cosine. Not because anyone decided to be generous. Because reading a coordinate off a point that exists cannot fail. Two of the six are finished already, and we have not done any work.

1:36The other four are quotients, and a quotient has one way to fail. Its bottom can be zero. The tangent is the height divided by the reach. So the tangent has no value exactly where the reach is zero, and nowhere else. That is the top of the circle and the bottom of it. The cotangent is the other way up: reach divided by height. So it goes missing where the height is zero, which is the two sides.

2:07The secant inverts the reach, so it fails where the tangent fails. The cosecant inverts the height, so it fails where the cotangent fails. Four functions, four sets of missing inputs, and only two different sets between them. None of that is a new fact. It is the zero set of a denominator, which the definition handed over for free. Here is the one that catches people. The tangent is not missing at the multiples of a half turn.

2:40That is the cotangent's list. Swap them and you will be answering the wrong question all afternoon. Now look at what that exclusion actually is. On the circle, only four points are ever refused by anything. The top, the bottom, and the two sides. Four points, out of a whole circle. But on the input line the same four points come back every single turn. So the tangent's missing inputs go on forever in both directions.

3:13And they are evenly spaced. The gap from one to the next is always the same: half a turn. The same is true of the cotangent's, and the secant's, and the cosecant's. One tiny fact on the circle, stretched out into an infinite list on the line. That is the whole difference between the two pictures. Next question: which values does each one reach? For the height and the reach, draw the square that just contains the circle.

3:46Its sides sit at one and at minus one, both ways. The circle touches that square and never crosses it. So no point has a height above one, or below minus one. That is not an observation about a picture. The reach squared plus the height squared is one, and a square is never negative. So the height squared is at most one. That is the bound, and it comes straight out of the circle's own equation.

4:15Now the other direction. Name any height you like between minus one and one. There is a point on the circle at exactly that height, and you can build it. Ask for a height of one and a millionth, though, and the construction refuses. Not approximately refuses. There is nothing to build. The cosecant is one divided by the height. The height is never more than one away from zero.

4:45So one divided by it is never less than one away from zero. A small number inverts to a big one, and there is no way round it. Which means the cosecant misses the entire middle. Everything strictly between minus one and one is unreachable. Ask for a cosecant of one half and the construction refuses. And here is the part worth noticing. It refuses for exactly the reason a height of two refuses.

5:17A cosecant of one half would need a height of two, and there is no such point. It is the same refusal, arriving from the other side. The secant does the same to the reach. And the other end has no ceiling. The height gets as close to zero as you like without ever being zero. So its reciprocal runs past any number you care to name. That leaves the tangent and the cotangent.

5:47These two reach everything. Name a value - zero, or a million, or minus a millionth - and a point exists whose tangent is exactly that. The construction never refuses, for any value at all. You can see why on the circle. In the first region the height climbs from nothing up to one while the reach falls from one down to nothing. So their quotient starts at zero and grows without any ceiling.

6:17And the signs in the next region make it run over the negatives. So six functions, and only three different sets of reachable values between them. Two that stay inside the bound. Two that abandon the middle. Two that take everything. Third question: which way does each one go? Take the first region, and walk the point from the right-hand side up to the top. The height starts at zero and climbs to one.

6:49So the sine rises across that region. The reach does the opposite - it starts at one and falls to zero. So the cosine falls. Keep walking into the second region. The height comes back down from one to zero. The reach keeps going down, from zero to minus one. Nothing here is being remembered. It is the point moving, and two coordinates being watched while it does. Do that for all six across all four regions and you get twenty-four entries.

7:24Twelve of them rise and twelve fall. But look at the first two rows side by side. The cosine's first region is the sine's second. The cosine's second is the sine's third. Every one of the cosine's four entries is the sine's entry for the next region along. The two rows are the same row, moved by one region. That is the quarter turn between them, showing up as a shift.

7:53And once those two rows exist, the other four are forced. A quotient's direction is decided by the two things it is a quotient of. You do not learn twenty-four entries. You learn one, and you watch a point move. Eight of those twenty-four entries have a plain number at each end. The other sixteen have an end where the values simply run away. That end gets written with an infinity symbol, and it is worth being precise about what it says.

8:26Ask the tangent for its value a quarter turn along and there is no value. The question is refused. The symbol is not the answer to it. What the symbol says is this: get close to that input and the values pass any number you name. Pass a million. Pass a million million. There is no ceiling, so there is no number sitting there waiting. And the two sides do not even agree.

8:55Come up to that input from one side and the tangent is enormous and positive. Come at it from the other and it is enormous and negative. A single value could not be both. That is why it is a hole with the curve running away on both edges, and not a missing dot. Now unroll all of it onto a flat picture. Put the input along the bottom and the value up the side.

9:24The sine leaves the origin, rises to one, comes back through zero, dips to minus one, and returns. The cosine is the same shape starting from one instead of zero. Two curves, one a quarter turn behind the other, which is the shift we just found. The tangent looks nothing like them. It climbs from below, races upward, and vanishes at a dashed line. Then it appears again at the bottom of the next stretch and does it all over.

9:56Those dashed lines sit exactly at the inputs where the reach is zero. The secant's dashed lines sit at the same places, and its curve never enters the strip between one and minus one. The cosecant's sit where the height is zero. Every feature of every curve was already in the table. The curve is not where the facts come from. It is where they become visible all at once.

10:25Last question: how far before it starts over? The obvious answer is a whole turn, and for four of the six that is right. Walk a full turn and the point is exactly where it was. Same reach, same height, so the sine, the cosine, the secant and the cosecant all repeat. But two of them come home sooner. The tangent starts over after only half a turn. And so does the cotangent.

10:55Half the distance, not a whole one. That is not an extra rule bolted on. Watch what half a turn does to the point. It sends it to the opposite point, straight through the centre. Both coordinates change sign. So after half a turn the height is the old height with a minus in front. And the reach is the old reach with a minus in front. For the sine, that is a change - it has turned over, and it has not repeated.

11:26But the tangent is one of those divided by the other. Minus something, divided by minus something else. The two minus signs cancel and the quotient is exactly what it was. That is the whole reason. One line, on the circle, no machinery. And the cotangent inherits it, because it is the tangent turned upside down. It also tells you why the sine cannot do the same trick. The sine is a coordinate, not a quotient.

11:59It has nothing to cancel against. Now the payoff, which is that awkward inputs stop being awkward. Take a sine at thirty-one thirds of pi. That is a long way out, and there is nothing to compute. Peel five whole turns off it and you land on pi over three. The answer is root three over two. Take a cosine at minus one thousand seven hundred and ten degrees. Add five turns and you are at ninety degrees, so the answer is zero.

12:34Notice what did not happen there. The minus sign on the input did not come out onto the answer. The cosine does not care which way you walked; only where you finished. The sine does care, and turns over. Five more of these and every one is the same single move. Three of the five have a negative input, and they are the second, the fourth and the fifth - not the last three, which is the trap.

13:02One is a tangent, and you might think its shorter repeat saves you a step. On that particular input it saves you nothing; both routes land on the same place. None of this was new. Where it lives, what it reaches, which way it leans, how far it runs - all four were settled the day the point was put on the circle.