One distance calculation on the unit circle yields the cosine of a difference

0:00You know the cosine of sixty degrees, and you know the cosine of thirty. What is the cosine of the two of them put together? There is a long list of identities for angles built out of other angles, and it looks like a list to be memorised. It is not. Exactly one thing on it has to be earned from geometry. Two more are carried in from earlier, ready made.

0:26Every other line is that one earned result, rearranged. So this is a video about a single measurement. Here is the entire toolkit, and it is small. A point on a circle of radius one, whose two coordinates are the cosine and the sine of its arc. The fact that those two coordinates, squared, add to one. The distance formula between two points. And two facts about what a minus sign on the input does.

0:57Nothing else gets in until the one thing is proved. The obvious guess is that the cosine of a sum is the sum of the cosines. Kill it now, in one line. Take both angles to be a quarter turn. The cosine of a quarter turn is nothing at all, so the guess offers nothing plus nothing. Zero. But a quarter turn and a quarter turn together make a half turn, and the cosine of a half turn is minus one.

1:30Zero is not minus one. The guess is dead. Now, how dead? Put twenty four points evenly round the rim, one every fifteen degrees, and try the guess on every pair of them. That is five hundred and seventy six pairs. It fails on five hundred and fifty eight of them. Which means it is accidentally right on eighteen. Sixty degrees and three hundred is one of the eighteen; both sides come out at one.

2:02So a worked example that agrees is not a proof of anything at all. Draw the circle of radius one, and mark where it crosses to the right. That mark is where every arc starts. Call the first angle A and the second B. Now four points, and each one is fixed by nothing but its arc from that mark. Walk round by A: first point. Walk round by A and B together: second point.

2:32Walk round by B the other way, which is minus B: third point. And the starting mark itself is the fourth. That is the whole figure. Four points, and the arcs that put them there. Notice what has not happened. Nothing has been dropped inside the circle, no right angle has been used, and no lengths have been compared yet. Every one of the four was placed by walking round the rim a stated distance, and by nothing else.

3:04Now write down the coordinates of all four, which costs nothing. The definition says the point at an arc of A sits at cosine A across, sine A up. So that is the first point. The second is the cosine and the sine of A plus B. The third is the cosine and the sine of minus B. And here the two carried in facts do their first job. The cosine does not notice the minus sign, so across it is just cosine B.

3:36The sine does notice, and turns over, so up it is minus sine B. The fourth point, the starting mark, is one across and nothing up. This is the step where the definition earns its keep. If cosine and sine were still ratios of sides in a right angled triangle, this could not be done. There is no right angled triangle here carrying an angle of A plus B, and there is certainly none carrying an angle of minus B.

4:07The coordinate definition does not mind; it will name any point on the rim. Here is the move the whole proof rests on. Join the first point to the third, and the second point to the fourth. Two chords. The claim is that those two chords are exactly the same length. Not because the picture looks symmetric. Because of a congruence, and here is the reason in full. Take the triangle made by the centre, the first point and the third.

4:40And the triangle made by the centre, the second point and the fourth. Two triangles, sharing exactly one corner, which is the centre. Two sides of each are radii, so all four of those sides are one. That is two sides matching. Now the angle at the centre. In the first triangle it runs from the first point back to the mark and on to the third, which is A and then B.

5:09In the second it runs from the second point to the mark, which is A plus B as well. Two sides and the included angle: the triangles are congruent, and the two chords must match. Checked at all five hundred and seventy six pairs, the two arcs at the centre agree every single time. Now measure that first chord, with nothing but the distance formula. Difference across, squared, plus difference up, squared.

5:40Across it is cosine A minus cosine B. Up it is sine A plus sine B, because the third point sat at minus sine B. Square both brackets and expand. Out come four squares, and some cross terms. Look at the four squares. The first two are the squared coordinates of one point on the circle, and they add to one. The other two are the squared coordinates of another point on the circle, and they add to one as well.

6:11So the four of them add to two. Here is the trap worth avoiding. That two is not there because there are two points. It is there because one identity was used twice, once on A and once on B. Every two in this derivation can be traced to something, and that is what makes the proof repeatable. What is left is two, minus twice the quantity cosine A cosine B minus sine A sine B.

6:42Now the second chord, the same way, and faster. It runs from the point at A plus B to the starting mark, which is one across and nothing up. Across, cosine of A plus B, minus one. Up, sine of A plus B, minus nothing. Square and expand. The squared cosine and the squared sine of A plus B belong to one point on the circle, so those two add to one.

7:10Then there is a one from squaring the one. And from the cross term, minus twice the cosine of A plus B. Two, minus twice the cosine of A plus B. Same shape as before, and again every part of it came from somewhere nameable. Two expressions, measuring two chords that the congruence has already declared equal. So set them equal to each other. Two minus twice something, equals two minus twice something else.

7:43Subtract the two from each side; the twos go. Divide both sides by minus two; that goes as well. And what is left standing is the whole of it. The cosine of A plus B is cosine A cosine B, minus sine A sine B. That is the one result that needed geometry. And the geometry it needed was one distance, measured twice. Tried at all five hundred and seventy six pairs, it holds at every one of them.

8:16Now step back and ask what was actually measured. Those two chords came out equal, and yet the four points sat in four different places. The only thing the two pairs had in common was how far apart they were, going round the rim. That is the real content of the calculation. Take any two points on the circle at all. The square of the chord between them is two, minus twice the cosine of the difference of their arcs.

8:46Where they sit does not enter it anywhere. Only the gap. Slide them both round together, holding the gap fixed, and the chord does not change by a hair. Count it out on the twenty four points. Among all five hundred and seventy six pairs there are exactly thirteen different chord lengths. And there are exactly thirteen different gaps, once you accept that a gap of five one way is the same gap as five the other.

9:17Thirteen and thirteen is not a coincidence; it is the statement. The chord is a function of the separation, and of nothing else whatsoever. Which brings the second result, and it costs no geometry at all. In the line you just proved, replace B by minus B. The left side becomes the cosine of A minus B. On the right, the cosine of minus B is the cosine of B.

9:48That is the first fact carried in, and it is true at all twenty four points. The sine of minus B is minus the sine of B. That is the second, also true at all twenty four. So the minus sign in the middle meets a minus sign, and turns into a plus. The cosine of A minus B is cosine A cosine B, plus sine A sine B. Two facts carried in, one substitution, no picture.

10:18That is the pattern for the whole rest of the list. Run it on numbers. Sixty and thirty put together, which had better be ninety. Cosine sixty is a half and sine sixty is root three over two. Cosine thirty is root three over two and sine thirty is a half. Multiply the two cosines: root three over four. Multiply the two sines: root three over four as well. Subtract them, and there is nothing left, which is exactly the cosine of ninety degrees.

10:56Now the difference form, on the same two angles: add instead of subtracting. Root three over four plus root three over four is root three over two, and that is the cosine of thirty. Then one worth having in your pocket: fifteen degrees. Forty five minus thirty, straight into the difference form. And the number that comes out is exactly the first coordinate of the point that ruler and compass geometry actually constructs at fifteen degrees, by cutting the thirty degree arc in half.

11:33Two completely unrelated routes, landing on one number. One warning, because this one costs marks. Having seen the cosine of a sum, it is tempting to guess that the sine of a sum has the same shape with a plus in it. Sine A sine B plus cosine A cosine B. Try that at sixty and thirty. Root three over four plus root three over four, which is root three over two.

12:03But sixty and thirty together is ninety, and the sine of ninety degrees is one. Root three over two is not one. The guess has the right ingredients and the wrong arrangement. The true shape crosses them over instead of pairing them up: sine A cosine B plus cosine A sine B. That one does hold at all five hundred and seventy six pairs. But you do not get it by pattern matching from the one you already have.

12:34You get it by earning it, the same way. One honest limitation, said out loud rather than hidden. The congruence was argued on a drawing, and a drawing puts the four points in one particular arrangement. First point up in the upper left, second point down in the lower left, third point below the centre and just left of the vertical line, and the starting mark over on the right.

13:03So how many of the five hundred and seventy six pairs actually put the four points in that arrangement? Fifteen. Fifteen out of five hundred and seventy six. If the picture were the proof, the proof would cover fifteen cases and stop. But the algebra that came out of the picture never used the arrangement. It used coordinates, and a coordinate does not know where on the rim it is sitting.

13:32And the identity holds at every one of the five hundred and seventy six. The picture is a scaffold for the calculation. The calculation is the proof. So here is what happened. One distance was measured, in two different descriptions, and the two answers were set equal to each other. The twos cancelled, the factor cancelled, and an identity was left standing. Everything that follows on that long list comes out of it by substitution.

14:06Negate one input and the difference form falls out. Nothing new was measured for it, and nothing new was drawn. And the reason any of it works is the thing from the middle of this video. A chord on the unit circle knows only the separation of its two ends. Not where they are. How far apart they are. Which is why the cosine of a sum was ever going to be expressible in the first place.

14:35One measurement, done twice.