Reading the graph of an inverse off the original by swapping the axes

0:00Here is a picture you already know, and a second picture you can get from it without drawing anything new. Start with one point. The sine of a twelfth of a turn is one half. So the point whose horizontal position is pi by six and whose height is one half sits on the sine curve. Now exchange those two numbers. Take the point whose horizontal position is one half and whose height is pi by six.

0:31That point is on the graph of the inverse sine. Not approximately — exactly, and you can check it: the inverse sine of one half is pi by six, which is what the height says. That is the whole idea, and it is worth stating as a rule about points rather than about pictures. Every point on a function's graph has a partner on the inverse's graph, with its two numbers the other way round.

1:00The picture follows from the points, never the other way about. Get that order wrong and you end up believing the drawing instead of checking it. In this video the drawing is checked. Every panel you will see is the original curve's plotted points with their two coordinates exchanged, and every claim about what the panels show was measured on those exchanged points. Before any of that is allowed, one condition has to hold. The function has to be invertible on the piece you are drawing.

1:38That is not a formality. If two different inputs shared an output, then exchanging the coordinates would give two points at the same horizontal position with different heights, and no function has two heights at one place. So the restriction work comes first and the picture comes second. On each of the six ratios, exactly one of two intervals survives — from minus a quarter turn to a quarter turn, or from zero to a half turn.

2:12Each of the six is one-one on exactly one of those two. Not both, not neither. One. Everything drawn from here on is drawn on those promoted intervals and on the copies of them a half turn apart, and the pictures are honest only because that check was done first. Which is also the answer to a question students ask about these pictures: no, the graph does not prove the inverse exists. The restriction did that, by an argument about values.

2:46The picture is how you remember the result. It is also where a wrong restriction becomes visible, which is a real use — but it is not the proof. Now do the exchange to a whole curve rather than one point. Take the sine over its promoted interval. Angle runs along the bottom from minus a quarter turn to a quarter turn; height runs from minus one to one. Exchange every point's two numbers. What was along the bottom is now up the side, and what was up the side is now along the bottom.

3:26The picture stands up. The widest the original panel gets is a quarter turn across and one unit tall; after the exchange it is one unit across and a quarter turn tall. Exactly swapped. That single sentence is the one thing you must hold on to when reading any of these panels. On an original panel the angle is along the bottom. On an inverse panel the angle is up the side.

3:57It is the commonest way to misread one of these drawings, and it is not subtle once you have been told. Look at which axis carries the angles, every single time, before you read anything else off the picture. Nothing else changed. Not one point was added, not one removed, not one moved to a new place. The same points are on the paper; the paper has been read differently.

4:27There is a second way to describe the very same operation, and it is worth having both. Exchanging a point's two numbers is a reflection — a mirroring in the line through the origin where the height always equals the horizontal position. Call it the diagonal. That is a geometric claim, so here it is measured rather than asserted, and measured two ways at once for every point. First: the midpoint between a point and its exchanged partner lies on the diagonal. Second: the segment joining them meets the diagonal square on.

5:08Over four thousand and one plotted points of the promoted sine, the number of midpoints off the line is zero, and the number of segments not square to it is zero. Two conditions, not three. A third was written first — that the segment's two halves are the same length — and then removed, because a midpoint on the line with a segment square to the line already forces equal halves. It could never have failed on its own, so it was measuring nothing.

5:42And these checks can fail, which is what makes passing them mean anything. Put a mirror that simply turns the height upside down through the same two checks and it fails both, four thousand times each. So the swap and the mirror are one construction described twice. Use whichever you find easier to see. The clearest single picture in this whole topic puts three things on one pair of axes.

6:12The sine over its promoted interval. The inverse sine. And the diagonal running through the origin between them. Now the pairing is visible rather than described. Pick any point on the sine, drop the perpendicular onto the diagonal, carry on the same distance again, and you land on the inverse. Do it with pi by six and one half and you land on one half and pi by six, which is the point we started this video with.

6:44There is one more thing this picture makes obvious, and it is easy to miss. The two curves cross the diagonal, they do not cross each other anywhere else, and the diagonal is not a third curve in the family. It is the mirror, and mirrors are not part of what they reflect. From here the method never changes. Restrict, exchange, read. So the rest of this video is about reading — because the pictures use a convention, and if you cannot decode it, twelve panels teach twelve unrelated things instead of one.

7:24Here is the convention, and it is carried entirely by how thickly a line is drawn. The heavy stroke is the promoted branch — the piece that answers when you write the inverse symbol with no branch named. The thin curve running away above and below is the other branches. And here is the part that gets misread: those are not scribble, and they are not rejected working. Each thin piece is a genuine inverse function on its own interval. Take five branches of each of the six families — thirty branches in all — and put every one through the same test the promoted one passed.

8:09The largest number of angles sharing a single value, across all thirty, is one. Every branch is one-one. Not one of them fails the test that the heavy one passes. They also reach exactly as far across as the heavy one does: for each family, the five branches all cover the same horizontal span. So the heavy stroke does not mark the only correct piece. It marks the agreed one. That is a completely different claim, and drawing only the heavy part would have been dishonest.

8:46Two more marks to know. Short dashed ticks are leaders running from an axis label to the curve — they are labelling, not mathematics. Long dashed horizontal lines are something else entirely, and they get their own section. Take the first pair: the inverse sine and the inverse cosine. Both are single unbroken strokes. One arc each, no gaps. Both are squeezed into a narrow vertical strip. The furthest either of them reaches from the middle is one unit, and neither of them draws a single point beyond that.

9:28That is not a shortage of ink. Those two functions accept only the numbers from minus one to one, so there is nothing to draw outside the strip. The picture stops because the mathematics stops. Inside the strip both curves are climbing steadily. The inverse sine runs from a height of minus a quarter turn at the far left up to a quarter turn at the far right. The inverse cosine runs the other way, from a height of a half turn at the left down to zero at the right — because its promoted interval is the one anchored at zero rather than the one anchored below it.

10:11Which is why one of them answers a negative number with a negative angle and the other answers it with an obtuse one, and now you can see that fact rather than remember it. Two panels, one shape between them, and the entire difference is which interval was promoted. Now the pair that looks strangest on the page: the inverse cosecant and the inverse secant. Each is drawn as two separate arcs. The count is not a matter of opinion — walk along the plotted points and count how many times the pen has to leave the paper, and the answer is two for these and one for the other four.

10:56And between the two arcs there is an empty vertical strip. Neither of these panels puts a single point anywhere strictly between minus one and one, on any of their branches. Those are two different facts with two different causes, and merging them is the commonest error on these two pictures. The gap in the middle of the curve, the reason the pen lifts, is a height taken out of the promoted interval. For the inverse cosecant that missing height is zero; for the inverse secant it is a quarter turn.

11:34The empty strip is about width, not height. As the height creeps towards that missing value, the curve runs away sideways without limit — it never comes back to the middle, it leaves. The edges of the strip come from somewhere else again: from the two ends of the promoted interval, which are kept. The inner tip of each arc sits at height minus a quarter turn and a quarter turn for the inverse cosecant, and at zero and a half turn for the inverse secant.

12:08One hole in the heights, one hole in the widths, and each with its own reason. Say both out loud when you draw these. The last pair is the calmest to look at and the easiest to misread. The inverse tangent is a single heavy S through the origin, and the inverse cotangent is a single heavy S too. One stroke each. Neither is confined to a strip. Both of them run out as far as the panel goes and would keep going — every value there is gets an answer.

12:45Above and below the heavy S sit thin copies of it, one for each of the other branches, and they are stacked a half turn apart all the way up and down. But the heavy S does something the earlier panels never did. It flattens. Follow it out to the right and it keeps climbing, but by less and less, as though something were holding it down at a particular height.

13:11Something is. There is a level it approaches and never reaches, and on these panels that level is drawn as a long dashed horizontal line. Those dashed lines are the only marks on any of these pictures that are neither the curve nor a label, and almost nobody is told what they are. So that is the next thing to settle. A dashed level is a height at which the ratio underneath has no value at all.

13:44That single sentence covers all four panels that have them, and it explains why the levels sit where they do. For the inverse tangent and the inverse secant they fall at odd multiples of a quarter turn. For the inverse cotangent and the inverse cosecant they fall at whole multiples of a half turn. Both patterns hold across every branch, not just the promoted one. Two of the six families have no such height anywhere in their promoted interval, and their panels have no dashed lines at all — which is a check on the rule rather than an exception to it.

14:26Now, approaching a level and reaching it are two different statements, so they are measured separately. Widen the panel to a horizontal position of ten, then a hundred, then a thousand, then ten thousand, and watch the gap between the curve and the level for the inverse tangent. The gaps are about a tenth, then a hundredth, then a thousandth, then a ten-thousandth. Shrinking every time. And never once zero.

14:56Now the two things a dashed level is not. It is not a gridline. And it is not where the branch ends — which matters, because for two of the four families the dashed level sits strictly inside the promoted interval rather than at its edge. On the inverse secant the level runs right through the middle of the promoted branch. Read it as a boundary and you will put the branch in the wrong place, which is exactly the mistake these lines cause.

15:30Now use one of these panels for what it is for. Stand at the horizontal position one half on the inverse sine panel and go straight up. The first thing you meet is the heavy stroke, at a height of pi by six. That is the principal value, read off a picture. Keep going up and you meet the curve again, this time a thin part, at a height of five pi by six.

15:57That second height is not a mistake and it is not an approximation. Its sine really is one half — the same one half you started from. Over three turns either side of zero there are six such heights, and the promoted branch keeps exactly one of them. Two crossings on one vertical line is the clearest picture in this whole topic of what the promotion is actually for. Without it the question has six answers; with it, one.

16:30The same reading works anywhere. Stand at minus one half and the promoted crossing is at minus pi by six — below zero, as that row always answers a negative number. And on the inverse tangent panel, standing at one gives a promoted height of pi by four, with six crossings in the same window and the same single survivor. So what have twelve panels actually settled? They have settled nothing, and that is not a criticism.

17:05Every fact these pictures show was established before any of them was drawn — by restricting each ratio until it stopped repeating, and by checking that the restricted piece still reached everything. What the pictures do is different and still worth having. They make the results memorable, they make the six promoted intervals visible in one glance, and they make a wrong branch obvious the moment you sketch it. That last one is a genuine use. If you place the inverse tangent's promoted branch on the wrong side of a dashed level, the picture looks wrong immediately, long before the arithmetic catches you out.

17:50But a drawing is only ever as honest as the decisions behind it, so hold the three that these drawings encode. The heavy stroke is the promoted branch. The thin curve is the other branches, every one of them a genuine inverse on its own interval. And the long dashed lines are heights the curve approaches and never reaches. With those three, every one of these panels reads off as a consequence of a decision already taken. Without them, they are twelve unrelated pictures — and that is the difference between a page you can rebuild and a page you have to memorise.