Picking the one ratio that links what you know to what you want

0:00Two figures, the same angle in both. On the left, a tower. You stand fifteen metres from its foot and look up at the top, at sixty degrees. You want the height. On the right, a ladder against a pole, also at sixty degrees. This time you want the length of the ladder. Same angle, same picture, the same three sides. But the two problems do not use the same ratio, and the difference is not a matter of taste. It is forced.

0:32By the end you will read which ratio a figure wants without thinking about it. Start with the triangle and forget the numbers. A right triangle has three sides, and once you pick one of its two acute corners, each side gets a name. The side facing the right angle is the hypotenuse. The side facing the corner you picked is the opposite side. And the side running from that corner to the right angle is the adjacent side.

1:05Now, how many ways can you pick two of those three? Three ways, and each pairing carries two names, because you can put either side on top. Opposite over adjacent is the tangent, and the other way up is the cotangent. Opposite over hypotenuse is the sine, and upside down it is the cosecant. Adjacent over hypotenuse is the cosine, and upside down, the secant. Six names. Three pairs of sides. That is the whole vocabulary.

1:41Opposite and adjacent are not properties of a side. They are properties of a side relative to the corner you stand at. Here is the same triangle, unchanged, read from its other acute corner. The hypotenuse is still the hypotenuse. It faces the right angle, and the right angle has not moved. But the two legs have swapped names. I checked that on seventy two right triangles, turned, mirrored and shifted. Every time: the legs trade, the hypotenuse keeps its name.

2:18And at the right angle itself the naming refuses to answer, because there is nothing there for those two words to mean. So the first thing you write on a figure is not a number. It is which corner you work from. Now the rule, and it is one sentence. Say what you have and say what you want. That pair of sides names the ratio. Is that a rule, or just a habit? To find out I wrote a routine that cannot be told what was wanted.

2:51You hand it one equation - a named quotient and its value - and one measured side. It either solves for something, or it cannot. Then I ran it five thousand one hundred and eighty four times: every triangle, every corner, every pair of sides, every one of the six names. The verdict comes out two, two and two. For any pair of sides, exactly two of the six solve for the side you wanted: a ratio and its reciprocal.

3:24Exactly two more solve for the other side instead - a correct answer to a question nobody asked. And exactly two cannot be used at all, because the side you measured does not appear in them. One equation, two unknowns. Every length it handed back matched the distance actually measured on the triangle - all three thousand four hundred and fifty six. Feed the same routine the ratio from the triangle's other acute corner, and every one of them comes back wrong.

3:59So the pair of sides really does settle it. The only freedom left is which one goes on top. Your tower is a particular tower. Why is a number written down for it in advance? Because the ratio does not depend on the triangle. It depends only on the angle. Here is that, drawn rather than asserted. One ray out of a corner, at some fixed slope. Drop a perpendicular to the ground at six different distances along it.

4:32Six right triangles, sharing the angle by construction. I never measured it. I used the same ray. Their hypotenuses come out at six different lengths, so the triangles really are different sizes. And all six give the same six quotients. Not close. Equal. Then six different rays, and no two ever agreed on a single one of the six. So a quotient is a function of the angle and nothing else. That is what makes a table possible.

5:06Be precise about what that covers, because it is not everything. Take a triangle and turn it, mirror it, slide it, scale it up or down. A hundred and forty four times over, and not one of the six quotients moved. Mirroring even reverses the order the corners run in, and the ratios do not care. Now stretch it along one axis only. Twice as wide, the same height. The right angle survives that, so it is still a right triangle. And every one of the six quotients changes, in all thirty six readings.

5:45So the ratio is invariant under similarity, and only under similarity. Same shape, any size, anywhere, either way round. Back to the tower. Letter it: the top, the foot, and the point where you stand. The right angle is at the foot, because the tower is vertical and the ground is level. You are at the corner with the sixty degrees, so work from there. The fifteen metres joins your corner to the right angle, so it is the adjacent side. The height faces you across the triangle, so it is the opposite side.

6:27Adjacent and opposite. That pair names the tangent. So the height is fifteen times the tangent of sixty, and the tangent of sixty is root three, exactly. The tower is fifteen root three metres. Exact, and finished. Notice what did not happen. Nobody chose the tangent. The two sides chose it. Now do that again with the cotangent. The cotangent is the adjacent over the opposite, so cotangent of sixty equals fifteen over the height.

7:03Rearrange, and the height is fifteen divided by the cotangent of sixty, which is one over root three. That gives fifteen root three. The same number, of course - it is the same equation written upside down. That held in every one of the eleven steps in this video. Two ratios answer, and the two never disagreed about the length. What they disagree about is the arithmetic. One puts the unknown on top, so you multiply. The other puts it underneath, so you divide.

7:40That is the only reason to prefer one. Not correctness. Convenience. A ratio and its reciprocal are not two methods. They are one method with the unknown parked in two different places. The ladder next, and this is where reaching for the tangent out of habit will cost you. The pole is five metres tall, the climber must reach a point one point three metres below the top, and the ladder stands at sixty degrees.

8:12First a step that is not trigonometry at all: five minus one point three is three point seven. That is the height to reach. Now name the sides. Three point seven runs up the pole, facing the sixty degree corner, so it is the opposite side. The ladder faces the right angle at the foot of the pole, so the ladder is the hypotenuse. Opposite and hypotenuse. That pair names the sine.

8:43So the ladder is three point seven divided by the sine of sixty, which is seven point four over root three. Reach for the tangent here and you get a real length. It is just not the ladder. It is the distance from the ladder's foot to the pole's foot - which happens to be the second question about this figure. The figure has not changed. Only the wanted side has, and the ratio changed with it.

9:13Opposite and adjacent now, so tangent or cotangent: three point seven over root three, about two point one four metres. One figure, two questions, two ratios, and no redrawing. Two more, because sometimes the ratio does not finish the job. An observer one and a half metres tall stands twenty eight and a half metres from a chimney and reads forty five degrees to the top. Adjacent known, opposite wanted, so the tangent again, and the tangent of forty five is one. The triangle gives twenty eight and a half metres.

9:55But that is the height above her eye, not above the ground. The chimney is thirty metres, and the last metre and a half came off the figure, not off any ratio. The broken tree is the same trap, doubled. A storm snaps a tree so the top bends over and touches the ground eight metres from the trunk, making thirty degrees there. The piece that fell is the hypotenuse, the stump still standing is the opposite side: two different pairs, so two ratios, on one figure.

10:31The fallen piece is sixteen over root three and the stump is eight over root three, so the piece that fell is exactly twice the piece standing. The tree was the sum of the two: eight root three metres, about thirteen point eight six. Of the ten problems here, eight ask for a side of the triangle and two ask for a sum. Check which before you stop. One last thing, and it surprised me.

11:01You will often be told to take root three as one point seven three. Try that on the tower. Fifteen times one point seven three is twenty five point nine five. But fifteen root three can also be written as forty five divided by root three. Exactly the same number. Forty five divided by one point seven three is twenty six point zero one. The true value is twenty five point nine eight. Neither substitution reached it, and the two land on opposite sides of it.

11:38That is not an accident. One point seven three squared is two point nine nine two nine, which is under three. The decimal is a shade too small, so multiplying by it undershoots and dividing by it overshoots. I read all seven root three answers in this video to two decimals, both ways. In four of the seven, neither route gives the correctly rounded value. With one point seven three two instead, both routes are right on every one of the ten answers.

12:13So keep the surd while you work, substitute once at the end, and use the extra digit. Here is the whole thing as a checklist you can run in ten seconds on a figure you have never seen. One. Find the right angle and mark it, then pick the corner whose angle you know. Two. Name the sides from that corner: hypotenuse facing the right angle, opposite facing you, adjacent joining you to the right angle.

12:45Three. Say what you have and say what you want. Two names. Four. That pair is the ratio. Take whichever way up puts the unknown on top. Five. Solve, keep the surd, then ask whether the side you found is really the thing that was wanted. Try it. A kite is sixty metres up on a taut string at sixty degrees to the ground, and you want the length of the string.

13:12Sixty is opposite the angle, the string is the hypotenuse, and that pair names the sine. Sixty over the sine of sixty is forty root three metres, about sixty nine point two eight. You never had to remember which ratio goes with kites. You read it off the two sides.