Cutting the domain down until the function is one-one, and what a branch is

0:00None of the six trigonometric ratios has an inverse on the set of inputs it arrives with, and the reason is always the same: too many inputs share an output. So here is the repair, and it is one sentence. Keep the rule. Shrink the set of inputs. Nothing about the function's values changes. The sine of pi by six is still one half, and every identity you already know still holds at every input that survives.

0:32What changes is only which inputs you are allowed to hand it. Do that well and the shrunk function has an inverse. Do it carelessly and it does not, and this topic is almost entirely about the difference. Because the six lists of intervals that come out of this look like six arbitrary conventions, and they are not. Where the cuts fall is forced. What genuinely is a convention is which of the surviving pieces gets promoted to be the standard one, and that distinction is worth keeping straight.

1:11A surviving piece has to meet two demands, and they are separate demands. First, the ratio has to be one-one on it. No two inputs in the piece may share an output, because the undoing map would then have to choose. Second, the piece has to keep reaching everything. The ratio's whole range has to still be covered, because the undoing map is handed values from that range and must have an answer for each.

1:41Drop the first demand and the inverse cannot be a function. Drop the second and the inverse cannot be defined on the range you claimed for it. So every candidate piece in this video is put to both tests, separately, and the tests share no working. The first walks the piece and buckets its points by output: how many points, how many distinct values, and the largest number of points landing on any one value. One-one means that last number is one.

2:16The second walks the declared range instead. For each value in it, produce an angle that should land there, then check two things: that the angle is actually in the piece you kept, and that the ratio really sends it to that value. That routine is capable of failing. Handed a producer that returns the target itself as though a ratio were an angle, it reports 41 tried and 40 landing astray. Handed a piece only half as wide as it needs, it reports 20 of 41 witnesses landing outside the piece.

2:56So when it comes back with nothing wrong, that is a reading. Start with the sine, and take the closed interval from minus half pi to half pi. Across it the sine climbs steadily from minus one to one and never turns around. Walk 121 points of it. There are 121 distinct values and the biggest bucket holds one point. So it is one-one. Now the second demand, over 41 values spread across minus one to one. All 41 tried, nothing that no angle could be named for, nothing landing outside the piece, nothing landing astray.

3:42Both demands hold, so this piece works. Now look at the two ends, because they are doing real work. At minus half pi the sine is minus one. At half pi it is one. Those are the smallest and largest values the sine ever takes. So if you dropped the two ends and kept only the open interval, the piece would no longer reach two of the 41 values. That is why the brackets are closed here. It is not tidiness. Dropping either end loses a value the inverse would then have no answer for.

4:21Now the thing students usually miss. That piece is not the only one that works. Slide it a half turn to the left and it works. Slide it a half turn to the right and it works. Take five consecutive members of that family, each a half turn wide, running end to end. All five are one-one, and none of them leaves any of the 41 values outside its span.

4:48And there is nothing special about five. The family runs on for ever in both directions. The reason is easy to see on the wave: between one turning point and the next, the sine goes up once, or down once, and never repeats. Consecutive odd multiples of half pi bracket exactly one rise or one fall. So the piece we kept was not forced. Every member of that family is a legitimate answer.

5:20But slide by a quarter turn instead of a half, and it breaks: not one of the six ratios stays one-one when its piece is shifted that way. So the family is a family of specific pieces, not of any piece of the right width. Two words now, because they are what all of this has been building to. Each admissible piece gives you one inverse function, and each of those is called a branch.

5:50There are infinitely many branches, one for each member of the family, and they are all perfectly good functions. They just give different answers. Ask for the angle whose sine is one half and one branch says pi by six, the next says five pi by six, the next says pi by six plus a full turn. All correct. All different. Which is exactly why a convention is needed. So one branch is promoted, and it is called the principal value branch. For the sine it is the one whose outputs run from minus half pi to half pi.

6:30That word principal is doing all the work: it means chosen, not forced. And notice which way round the sets now sit. The inverse sine takes a number from minus one to one and returns an angle in that interval. The interval of angles has become the output side. The cosine gets exactly the same treatment, anchored a quarter turn away. The piece kept runs from nought to pi, closed at both ends.

7:03On it the cosine falls steadily from one down to minus one, never repeating: 121 points, 121 distinct values, biggest bucket one. And the same 41 values are all covered, with nothing astray and nothing outside. The ends are closed for the same reason as the sine's. At nought the cosine is one and at pi it is minus one, and those are its extreme values. Open the interval and you lose two of the 41.

7:39Its family behaves the same way too: five consecutive members, all five one-one, none of them leaving a value uncovered. So it is one move made twice. The only difference is where the piece is anchored, and that difference is a quarter turn, which is exactly the offset between the two waves. If you can see why the sine's piece is centred on nought and the cosine's starts there, you have understood the pattern that runs through the rest of this.

8:14Now the cosecant and the secant, which do something that looks strange until you see the reason. The cosecant's piece is the closed interval from minus half pi to half pi with the point nought taken out of the middle. That hole is not there to make it one-one. It is there because the cosecant has no value at nought at all: the sine is nought there, and the cosecant is one over the sine.

8:42The two ends stay in, because the cosecant is minus one at the left end and one at the right, and those are values the inverse has to be able to return. Walk that punctured piece: 120 points, all of them values the cosecant actually has, and a biggest bucket of one. Then coverage, over 40 values from outside the interval minus one to one: all 40 tried, none refused, none astray.

9:15The secant is the mirror of this, and here is the trap. The secant loses half pi, not nought. Its piece runs from nought to pi with half pi taken out, and half pi is where the cosine vanishes, so it is where the secant breaks. Each interval loses the point where its own function breaks, and the two functions break in different places. That is the whole difference between them.

9:43Same readings for the secant: 120 points, biggest bucket one, 40 targets covered with nothing astray. The tangent and the cotangent throw both ends away instead. The tangent's piece is from minus half pi to half pi with neither end included. The ends cannot be kept, because the tangent has no value at either of them — the cosine is nought there. And nothing is lost by dropping them. Ask the piece to hold minus half pi or half pi and it says no; ask the 41 targets along the line whether they are all reached, and every one of them is, with nothing outside and nothing astray.

10:32That is the difference from the sine in one line. The sine's ends carry values that would be lost. The tangent's ends carry no values at all. 119 points on the tangent's piece, 119 distinct values, biggest bucket one. The cotangent is the same, shifted a quarter turn: from nought to pi, neither end included, because the sine vanishes at both. 119 points, biggest bucket one, all 41 targets covered.

11:08So two ratios keep both ends, two throw both away, and two keep both ends but lose a point in the middle. Which gives three bracket shapes, and each has its own cause. Read them and you can reconstruct all six restrictions from the brackets alone. A closed end means the ratio has a value there and that value is needed. Sine and cosine, both ends. An open end means the ratio has no value there at all. Tangent and cotangent, both ends.

11:45A hole in the middle means the same thing as an open end — no value — it just happens to fall inside rather than at the edge. Cosecant and secant. Three shapes, and the six promoted pieces use exactly three between them. So the bracket is never a stylistic choice. It is a report on what the function is doing at that input. And that is worth saying plainly, because a student who thinks brackets are house style will copy the wrong one and then wonder why their inverse is undefined somewhere.

12:24Now the sentence that organises all six lists at once. Every cut falls either at a turning point or at a break. That is not a coincidence: those are the only two places a function can stop being one-one. Either it turns around and starts retracing values it has already taken, or it breaks and starts again somewhere else. So test it. Take a window of three turns and find every cut point of every one of the six families.

12:57The sine, the tangent and the cosecant cut at the odd multiples of half pi — 6 of them in that window. The cosine, the cotangent and the secant cut at the whole multiples of pi — 7 of them. No angle is in both lists. The two patterns interleave. Now classify each cut by looking at the ratio just either side of it. Does it break, does it turn around, or does it do neither?

13:27The sine turns at all of its cuts. So does the cosine. So do the secant and the cosecant at the ends of their pieces. The tangent breaks at all of its cuts, and so does the cotangent, and so do the secant and the cosecant at their holes. Across every cut and every hole of all six families, the number that are neither a turn nor a break is nought.

13:53And this file can return neither: a twelfth of a turn away from a cut, the sine is neither turning nor breaking. So nought is a reading, not a definition. Two pieces that do not work, because the rule of thumb students reach for is width, and width is not the test. First: the sine on the closed interval from nought to pi. That is exactly as wide as the piece we kept — half a turn.

14:25Walk 121 points of it and you get 61 distinct values, with a biggest bucket of two. Here is the collision, in the open: the sine of pi by six is one half, and so is the sine of five pi by six. Both of those angles are inside this interval. So the first demand fails. And the second fails too: the sine never goes below nought here, so 20 of the 41 values are outside the range this piece can reach.

14:59Right width, wrong anchor, both demands broken at once. Second: a piece that is honestly too small. Take the sine on nought to half pi. It is perfectly one-one there — no complaints on the first demand at all. But run the coverage walk. Of the 41 targets, 20 produce an angle that is not in the piece. It cannot reach the negative values, because it has none of the inputs that would produce them.

15:29So one demand can hold while the other fails, in both directions. That is why they are checked separately, and why a piece is never judged by its width. So what has the shrinking bought, and what has quietly been chosen for us? Bought: six functions that are now one-one and still cover everything they claimed, so each has a genuine inverse. That was forced — every cut sits where the graph turns or breaks, and it could not sit anywhere else.

16:03Chosen: which member of each infinite family gets promoted. Nothing in the mathematics prefers the piece from minus half pi to half pi over the one a half turn along. That choice is a convention, made so that everyone writing down an inverse sine writes down the same number. And both halves have to be said. A student who thinks the intervals were forced will never check a proposed one. A student who thinks any interval of the right width will do has not checked either.

16:39Check both demands, separately, every time. Thirty pieces were checked that way here, five from each of the six families, and all thirty passed — and not one of them contains a point where its own ratio has no value. The next question is what these inverse functions actually look like, now that each of them has been given a domain and a branch to live in.