Which side is opposite and which is adjacent depends on the angle you pick

0:00Here is a tall tower. You are standing on the ground some way from its foot, and you would like to know how high it is. You are not going to climb it with a tape measure, so the height is a length you cannot reach. Second picture. You are on a balcony above a river, looking across at a building on the far bank. How wide is the river? You are not swimming across with a tape measure either.

0:27Third picture. A balloon is up in the air and you are on the ground watching it. How far up is it? Three questions, and in every one the length is somewhere you are not. But look at what you can do in all three. You can stand still, and you can look up. How far you tilt your eye to see the top of the tower is something you can measure from where you stand.

0:52So in each picture there is a length out of reach and an angle that is not. Draw the sight line in, and the same shape appears every time. The ground from you to the foot of the tower, the tower going up, and the sight line closing the figure. And the corner where the tower meets the ground is a right angle. The tower is interchangeable. The balloon is interchangeable. The triangle is not.

1:23So the shape worth studying is the right triangle, because one of its three angles is known before you measure anything. The corner at the foot is ninety degrees, given by the picture and not by an instrument. The three angles of any triangle add to a hundred and eighty, and ninety are already spent. So the other two share the ninety that is left, which means each of them is smaller than a right angle. Both are acute.

1:54And it means something stronger: knowing one of them tells you the other. They are tied together. One angle known for free, and two more that determine each other. That is a shape you can get numbers out of. Now the sides. There are three of them, and exactly one has a name that belongs to it. The side facing the right angle is called the hypotenuse. It is the one side that does not touch the square corner.

2:25The right angle pins it. You do not have to choose anything to know which side it is. It is also the longest of the three, and that is not a coincidence. I checked the two readings against each other on three hundred and thirty six right triangles. The side facing the right angle and the longest side were the same side in every one. Strictly longest, too, not merely tied. To be sure that word was doing work, I ran the same test on sixteen triangles whose two longest sides are equal, and there it said no every time.

3:06So the hypotenuse is settled. The other two sides are the interesting ones. Here is the triangle drawn cleanly. Right angle at B, and the other two corners are A and C. Stand at A, and mark the angle there with an arc. The two remaining sides run from A to B, and from B to C. The side from B to C does not touch A at all. It sits across the triangle facing the angle, and it is called the side opposite to A.

3:39The side from A to B runs out of the corner you are standing in, forming one arm of the angle. That one is called the side adjacent to A. Now, two sides touch the corner at A: the leg AB, and the hypotenuse. That is true at every acute corner of every one of the three hundred and thirty six triangles, six hundred and seventy two corners in all.

4:07But the hypotenuse already has a name, so it is never in the running. Strike it out and one side is left, and it is exactly the leg the word adjacent picks. Six hundred and seventy two times out of six hundred and seventy two. Now do not redraw anything. Same triangle, same ink, same three lengths on the page. Just move the arc from A to C and stand there instead.

4:37The side from A to B no longer touches you. It is now across the triangle, facing C. It has become the opposite side. And the side from B to C, which was the opposite side a moment ago, now runs out of the corner you are standing in. It has become the adjacent side. The two legs traded names, and not one line of the drawing moved. What survived? The right angle at B is still at B. And the hypotenuse is still the hypotenuse.

5:11It is pinned to the right angle, and the right angle did not care which acute corner you walked to. On all three hundred and thirty six triangles it stayed put. So here is the thing this topic is really about. Opposite is not a property of a side. The segment from B to C is opposite to A and adjacent to C, at the same time, in the same picture, with no contradiction.

5:41The name does not belong to the side. It belongs to the pairing of a side with an angle. Which is why you cannot say the opposite side of a right triangle. That phrase names nothing until you say opposite to what. A right triangle with no angle chosen has a hypotenuse and two legs that are not yet called anything. And that is why every ratio in trigonometry is written with an angle attached to it. The angle is not decoration. Without it the words on top and underneath do not refer.

6:21Now here is the trap, and it is the reason this looks harder than it is. Take a right triangle whose two legs are the same length, and move from one acute corner to the other. The legs still trade. But they are the same length, so nothing you can measure changes. The swap happens and leaves no trace. So suppose a student reads the labels off one drawing and remembers them as belonging to the sides. On the equal-legged triangle, that gives the right answer at both corners.

6:57I tested exactly that. Of the three hundred and thirty six triangles, seventy two have equal legs, and on all seventy two the memorised labels agree at the other corner. On the other two hundred and sixty four, they disagree at the other corner. Every single one. The equal-legged triangle is the one case where the mistake is invisible. So it is the worst possible triangle to learn the naming from.

7:30One more idea to get rid of. Opposite does not mean long, and adjacent does not mean short. Which leg is longer is a real question with a clean answer, but it is not about the words. The leg opposite an angle is the longer of the two exactly when that angle is the larger of the two acute angles. Both mixed cases came up empty: no triangle where the bigger angle faces the shorter leg, and none where the smaller angle faces the longer one. Two hundred and sixty four each way, and nothing in between.

8:10On the equal-legged ones neither angle is larger and neither leg is longer, so the question simply does not arise. So if you stand at the smaller acute angle, the side opposite you is the short one. The name did not change. Your viewpoint did. Let us put numbers on it. Right angle at B, the side from A to B is twenty four, and the side from B to C is seven.

8:40The hypotenuse comes from the two legs: twenty four squared is five hundred and seventy six, seven squared is forty nine, and those add to six hundred and twenty five, which is twenty five squared. So the hypotenuse is twenty five. Stand at A. Opposite is seven, adjacent is twenty four, hypotenuse twenty five. Opposite over hypotenuse is seven over twenty five. That ratio is called the sine of A. Adjacent over hypotenuse is twenty four over twenty five, the cosine of A.

9:20Now walk to C. Opposite is twenty four, adjacent is seven, and the hypotenuse is still twenty five. So the sine of C is twenty four over twenty five, and the cosine of C is seven over twenty five. Look at what happened. The sine at A and the cosine at C are the same number, and so are the cosine at A and the sine at C. The two legs traded, so the two answers traded. That is the swap, written in numbers.

9:54That pairing is not an accident of these two numbers. It is forced by the ninety degrees. The two acute angles add to a right angle, so each one is what is left of ninety when you take the other away. They are complements. And the opposite leg of one of them is the adjacent leg of the other, by the naming we just did. So one angle's sine has to be the other angle's cosine.

10:23That held on all three hundred and thirty six right triangles. And to be sure the right angle was carrying the argument, I ran the same comparison on twenty four triangles with no right angle in them at all. It held on none of them. Zero out of twenty four. So this is a fact about right triangles, not about arithmetic. One more thing before the name. These are ratios of two lengths, so they carry no unit, and something follows from that.

10:59Triple the triangle. Seven becomes twenty one, twenty four becomes seventy two, and twenty five becomes seventy five. The hypotenuse squared goes from six hundred and twenty five to five thousand six hundred and twenty five, nine times as big. But seven over twenty five and twenty one over seventy five are the same number. The sine did not move, and nor did the cosine. Over eight hundred and forty rescalings, the ratios stayed exactly where they were, while a plain length moved every single time.

11:38And to check that was really about shape and not just about arithmetic being kind, I stretched one leg without the other, four hundred and twenty times. That keeps the right angle but changes the shape, and the ratios moved on all four hundred and twenty. So the ratios are reading the shape of the triangle, not its size. Which brings us to the name of the subject. It is built from three pieces of Greek.

12:09Tri, meaning three. Gon, meaning sides. Metron, meaning measure. Three sides, measured. The oldest surviving work on it was set down in Egypt and in Babylon, and early astronomers reached for it to estimate how far away the stars and planets are, which is a length nobody was ever going to reach. And what the subject relates is exactly what we set up at the start: the angles of a triangle on one side, and the lengths of its sides on the other.

12:45You measure the angle, because you can. It hands you the length, because you cannot. And the bridge between them is a ratio, written with an angle attached to it, because opposite and adjacent do not mean anything until you say where you are standing.