Squeezing 30°, 45° and 60° out of two special triangles

0:00Somewhere in your work you will need the sine of thirty degrees, and the answer will be exactly one half. Not nearly one half. Exactly. Ask for the sine of thirty one degrees and there is no such answer. There is a decimal that goes on forever. So what makes thirty different? Nothing about the number itself. What is different is that there is a triangle you can BUILD with a thirty degree angle in it.

0:30Build a figure and you know its proportions exactly, because you put them there. Measure a figure and you know its proportions to whatever your ruler is worth. That is the whole reason this short list of angles exists. There are two constructions in this video. Between them they hand over every exact value you are expected to know. Here is the first one, and it starts with almost nothing.

0:59Draw a right angle. Now set one of the other two corners to forty five degrees. That is the only choice I get to make. Watch what happens to the third corner. The three angles of any triangle add to a straight angle. Ninety is spoken for. Forty five is spoken for. There are forty five degrees left, and nowhere else for them to go. So the third corner is forty five as well. I did not choose it. It was forced.

1:31That is the first thing worth noticing about this triangle. It has a kind of symmetry I did not ask for. Equal angles face equal sides. So the two legs of this triangle are the same length. I will call that length a, and I will not say what a is, because it is about to leave. Now the third side. Pythagoras: a squared plus a squared. That is two a squared, so the longest side is a times the square root of two.

2:03There is the first root of the video, and notice where it came from. Not from an angle. From adding a square to itself. I built a hundred and forty four right triangles with whole number legs, and asked in each one whether the two acute angles were equal. They were equal in exactly twelve of them, and those twelve were exactly the ones with equal legs. Not one triangle had equal legs and unequal angles. Not one had equal angles and unequal legs.

2:38So the forty five triangle is not an assumption here. It is something the family picks out on its own. Every side is labelled now, so reading the six ratios off is just looking. Sine is the side facing the angle over the longest side. That is a over a root two. The a cancels, which is the point of never saying what a was. One over root two, which I will write as root two over two. Same number, tidier bottom.

3:10Cosine is the side touching the angle over the longest side. But the side touching and the side facing are the same length. So cosine of forty five is also root two over two. The two are equal here, and only here. Tangent is one leg over the other leg. Equal over equal is one. And the other three are these three turned upside down: root two, root two, and one.

3:40Six exact values, and every one of them fell out of a picture with two equal legs in it. The second construction gets you thirty and sixty, and it does not start from a right angle at all. It starts from the most symmetric triangle there is. All three sides equal. If all three sides are equal then all three angles are equal, and three equal angles adding to a straight angle are sixty degrees each.

4:11That is sixty, already, for free. But there is no right angle in this picture, and the six ratios only mean anything in a right triangle. So I am going to make one. Drop a perpendicular from the top corner to the side underneath. That single line does three things, and only one is obvious. The obvious thing is that I now have two right triangles. The less obvious thing is where the foot of that perpendicular landed. It landed exactly halfway along.

4:46That is worth proving rather than believing. The two halves share the perpendicular. Their long sides are equal, because all three sides of the original are equal. And each has a right angle at the foot. That makes the two halves identical. Not similar. Identical. And identical figures have identical parts, so the two pieces of the bottom side are equal, and the angle at the top has been cut into two equal pieces.

5:19I checked the landing point on sixty six triangles, computing where the perpendicular actually meets the line rather than assuming it. It landed halfway in fifty four of them, and those fifty four were exactly the ones with two equal sides coming down from the top corner. In the other twelve it landed somewhere else. So the halving is not a property of perpendiculars. It is a property of this kind of triangle, and that is why the construction had to start where it did.

5:55Now take one of the halves on its own. The top angle was sixty and got cut in two, so it is thirty. The bottom left angle is still sixty. The foot is ninety. Thirty, sixty, ninety, and I did not measure any of them. Now the sides. Call the original side two a, so the halving gives whole numbers to talk about. The long side of the half is a full side of the original, so it is two a.

6:26The short side is half of a full side, so it is a. There is the factor of two that everybody remembers and almost nobody can explain. It is there because this is half of something. The third side by Pythagoras: four a squared minus a squared is three a squared, so that side is a root three. The second root of the video, and again it came from a subtraction, not from an angle.

6:55I ran the same halving on forty two triangles that have two equal sides but are not the fully symmetric one. The factor of two never once appeared. One picture, three labelled sides, and the thirty degree corner at the top. The side facing thirty is the short one, a. The longest side is two a. So sine of thirty is a over two a, which is one half. The a leaves again.

7:25Cosine of thirty uses the side touching it, a root three, over two a. That is root three over two. Tangent is a over a root three, which tidies to root three over three. Turn all three over and you get two, two root three over three, and root three. Six more exact values, off the same figure, and one half is now explained. It is one half because a is half of two a.

7:56Here is the part people learn twice and only need to learn once. I have not finished with that picture. There is another acute corner in it, and it is sixty degrees. Same triangle, same three sides. All that changes is which side is facing me and which side I am sitting on. From down here the short side a is the one I am touching, and a root three is the one facing me.

8:25So sine of sixty is root three over two, and cosine of sixty is one half. Those are the thirty answers with sine and cosine trading places. Tangent and cotangent trade too. Secant and cosecant trade. That is not a coincidence and it is not special to thirty and sixty. It happens whenever two angles add to ninety. I took two hundred and eighty eight corners from that family of triangles and compared every ordered pair of them, more than eighty thousand comparisons.

9:02The three trades held for a pair exactly when the two angles added to a right angle. One thousand seven hundred and twenty eight pairs did. Every other pair failed all three. So learning the thirty column and the trade gives you the sixty column for nothing. Step back and look at what the two constructions actually produced. One triangle with two equal legs, which gave a root two. One triangle that is half of a symmetric one, which gave a root three.

9:38Every one of the eighteen values carries a root two, a root three, or no root at all. Nothing else turns up. That is not a rule about angles. It is a consequence of those two figures being the only ones we built. Build a different right triangle and you get different roots. Legs of one and two, and the answers arrive carrying a root five. Nothing is wrong with root five. It is just that no simple construction hands you the angle that goes with it.

10:12And this is the reason the list of exact angles is short. Not because the other angles are badly behaved, but because we have run out of figures we can draw. Now the values do some work. A right triangle, one side known, one angle known, and a side wanted. The vertical side is five centimetres. The angle at the far corner is thirty degrees. Find the other two sides.

10:43The skill here is not the table. The skill is choosing which of the six connects what I have to what I want. I have the side facing thirty. I want the side touching it. Facing over touching is the tangent, so touching over facing is the cotangent. Cotangent of thirty is root three, so the bottom side is five root three centimetres. For the longest side I want longest over facing, which is the cosecant. Cosecant of thirty is two, so that side is ten.

11:19And here is a check that costs nothing. Twenty five plus seventy five is one hundred, which is ten squared. Pythagoras agrees with the ratios, by a route that used none of them. Notice also what happened to the size. The moment a real length went in, the answers came out in centimetres. A ratio fixes the shape; one length fixes everything else. Now run it the other way. Two sides known, and the angles wanted.

11:53The short side is three, the longest is six. Facing over longest is the sine, and three over six is one half. So I go looking for an angle whose sine is one half, and I know one. Thirty. The remaining angle is sixty. One number, and the whole triangle is pinned down. Harder version. Two angles are hiding. The sine of their difference is one half, the cosine of their sum is one half, and the sum is no more than a right angle.

12:27The difference must be thirty. The sum must be sixty. Two unknowns, two equations, and the angles are forty five and fifteen. Do not skip that last condition about the sum. It is doing real work. I searched every whole number of degrees all the way round the circle. A cosine of one half happens at sixty degrees and it happens again at three hundred. Without the condition, forty five and fifteen is not the only answer. A hundred and sixty five and a hundred and thirty five also fit.

13:05The condition is what makes the reading unique, and a question that leaves it out has more than one right answer. So here is the finished table, and I want to be careful about what it is. It is not the thing to learn. It is what falls out of two drawings. One right angle and one forty five degree corner gives you the middle column. One symmetric triangle cut in half gives you the outer two, and each of those is the other one traded.

13:38If you can draw those two figures you can put this table back on a blank page in under a minute. The trade is worth saying once more, because it is half the table. Sine and cosine swap, tangent and cotangent swap, secant and cosecant swap, whenever the two angles add to ninety. Two constructions. Eighteen exact numbers. Nothing measured, and nothing remembered that could not be rebuilt.