Coordinates on the unit circle extend the ratios to every real number

0:00Here is a question that sounds perfectly reasonable and, as things stand, cannot even be asked. What is the sine of a hundred and twenty degrees? Try to answer it with the definition you already have. Sine is the side opposite, divided by the hypotenuse. So draw the right triangle. You cannot. The three angles of a triangle add to a straight angle, and one of them is already a right angle.

0:31That leaves ninety degrees to share between the other two. So both of them are less than ninety. Every one of them. Always. The old definition does not give a wrong answer for a hundred and twenty. It has nothing to say at all. And the same is true of a right angle itself, of zero, and of any negative number. Of the eight angles in the standard table, the old definition can be asked about exactly three.

1:04So we are not going to extend that definition. We are going to replace it. Here is the replacement, and it is quick. Draw the axes. Draw the circle of radius one, centred at the origin. Put the angle at the origin, with one arm along the positive horizontal axis. The other arm cuts the circle at exactly one point. Call it P. That is the whole construction. And notice what it does not need.

1:36It does not need the angle to be small. Turn as far as you like, the arm still cuts the circle somewhere. It does not need the angle to be positive. Turn the other way and you land somewhere too. Every real number puts P somewhere. There is no angle this construction refuses. Now the definition itself, and it is shorter than you are expecting. P is a point in the plane, so it has two coordinates.

2:07Call the first one a and the second one b. The cosine of the angle is a. The sine of the angle is b. That is it. That is the definition. Not a ratio of two sides. The first coordinate is given the name cosine, and the second coordinate is given the name sine. They were already there. All we did was name them. And because a point exists for every real number, so does a value for each function.

2:38The moment you write that sentence down, both functions are defined on the entire number line. Now, that should worry you slightly. You already know the sine of thirty degrees. If this new definition gives a different answer, we have a problem. So check it, in the one place where both definitions apply. Take the angle small enough to be inside a triangle. Drop a perpendicular from P down to the horizontal axis, and call the foot M.

3:12Now look at the triangle you have just made. O, M and P. It has a right angle at M. Its hypotenuse is the radius, which is one. The side adjacent to the angle has length a. The side opposite has length b. So the old definition says the cosine is a over one, and the sine is b over one. Which is a, and b. Exactly what the new definition just said.

3:41I checked the three angles where both definitions can speak, and computed the old side ratios from the triangles themselves. Half an equilateral triangle for thirty and sixty. An isosceles right triangle for forty-five. Every one of them agrees with the coordinate, exactly, with no rounding anywhere. Nothing you already knew has changed. The new definition simply keeps going after the old one stops. Now watch the first thing you get for free.

4:18Go back to that little triangle. Right angle at M, hypotenuse one, legs a and b. Pythagoras says a squared plus b squared is one. But a is the cosine and b is the sine. So cosine squared plus sine squared is one. There it is. The most famous identity in the subject, and it took one line. But look again at what that line actually says. A squared plus b squared equals one is the equation of the circle of radius one.

4:51It is not a fact about triangles that happens to be true. It is the circle, written down. Which is why it holds everywhere, with no exceptions and no small print. I checked it at every one of the twenty-four positions in a turn, and at seventy-two more across the three turns after that. Not one failure, anywhere. Every point on the circle satisfies it, and every point here is on the circle.

5:23Now let us collect. Walk P round to each axis in turn and just read the coordinates. Start at zero. P sits at one, nothing. So the cosine of zero is one and the sine of zero is nothing. A quarter turn. P is at the top. Nothing, one. Cosine nothing, sine one. A half turn. P is at the left. Minus one, nothing. Three quarters. P is at the bottom. Nothing, minus one.

5:54A whole turn, and P is back where it started. One, nothing. Five positions, ten values, and no work at all beyond reading a picture. These are called the quadrantal angles, and the name is a trap. Quadrantal does not mean in a quadrant. It means on the boundary between two of them. The arm lies flat along an axis, which is exactly why one of the two coordinates is always nothing there.

6:25And one more thing worth noticing. Five positions, but only four distinct points. The whole turn brought P back to where it started. That last observation answers a question you will be asked a great deal. Where is the sine equal to nothing? The sine is the second coordinate. So the question is: where does P have height nothing? On the horizontal axis. Both ends of it. That is zero, and a half turn, and a whole turn, and a turn and a half, and so on forever in both directions.

7:03Every whole number of half turns. That is the complete answer, and it took one look at the picture. Now the cosine. The cosine is the first coordinate. So where is P directly above or below the origin? On the vertical axis. A quarter turn, three quarters, and every odd quarter turn after that. Two families, and they never overlap, because P cannot be on both axes at once unless it is at the origin, and it never is.

7:37Across four turns in each direction I found the sine vanishing nine times and the cosine eight. Hold on to those two families, because in about two minutes they come back wearing different hats. Next, something the picture makes almost too obvious. Take P wherever it is. Send it round one complete extra turn. Where does it end up? Exactly where it started. That is what a complete turn means.

8:09And if the point has not moved, its coordinates have not moved either. So the cosine and the sine are unchanged by adding a whole turn. Or by adding two. Or by subtracting one. I checked this by actually walking it, at all twenty-four positions in a turn, forwards and backwards. Twenty-four out of twenty-four, unchanged, every time. And the control matters here. Add anything short of a whole turn and the point does move.

8:42One step of the walk moves it at all twenty-four positions. Half a turn moves it at all twenty-four. So it is genuinely the whole turn doing the work, and not the adding. This is why you can be handed an angle far outside one turn and still answer. Here is the point where the old picture quietly stops being helpful. In a triangle, a and b were the lengths of two sides.

9:11Lengths are not negative. There is no such thing as a side of length minus a half. But a and b are not lengths any more. They are coordinates. Send P round into the second quadrant, up and to the left. Its height is still positive. But it is now to the left of the origin, so its first coordinate is negative. The cosine has just gone negative, and no negative side was ever drawn.

9:41That is not a patch. It is the definition working exactly as written. Of the twenty-four positions in a turn, eleven have a negative first coordinate and eleven have a negative second. A definition built on lengths could never have produced a single one of those values. So far we have two functions. There are six. The other four are not new ideas. Every one of them is built out of the two we already have.

10:14The tangent is the sine divided by the cosine. The cotangent is the other way up. Cosine divided by sine. The secant is one over the cosine. And the cosecant is one over the sine. Notice how they pair off. Two of them are built on the cosine, sitting underneath. Tangent and secant. Two of them are built on the sine, sitting underneath. Cotangent and cosecant. That pairing is the only thing you need to remember about them.

10:48Because it tells you, without any further thought, exactly where each one is going to break. A quotient breaks in exactly one way. When the thing underneath is nothing. So take the two functions built on the cosine, and ask where the cosine is nothing. We answered that four minutes ago. The odd quarter turns. So the tangent and the secant have no value at a quarter turn, or at three quarters, or at any odd quarter turn ever.

11:21Not a large value. Not infinity. No value. And the two built on the sine break where the sine is nothing, which is every whole number of half turns. Four functions, and only two lists of excluded inputs between them. And we did not have to find those lists. We already had them. They are precisely the two zero families from earlier, read a second time. In one turn, the tangent has a value at twenty-two of the twenty-four positions, and the two it refuses are exactly the two where the cosine vanishes.

11:59Same two for the secant. The other two for the cotangent and the cosecant. One picture, asked twice, answers a question about four different functions. Now the two identities that usually arrive as things to memorise. One plus tangent squared is secant squared. One plus cotangent squared is cosecant squared. They are not extra facts. They are the first identity, divided. Start with cosine squared plus sine squared equals one.

12:34Divide every term by cosine squared. The first term becomes one. The second becomes tangent squared. The right-hand side becomes secant squared. There is the first of them, in one step. Now go back and divide the same line by sine squared instead, and the second one falls out the same way. But look at what the division carried in with it. You cannot divide by nothing. So the first one is only available where the cosine is not nothing.

13:07Which is to say, not at an odd quarter turn. Where, as it happens, neither side of it exists anyway. I checked both across a full turn. Each holds at twenty-two positions and fails at none. At the remaining two, there is nothing to check, because neither side has a value. The first identity needed no condition because nothing was divided to get it. These two do, because something was.

13:39Which brings us to the table everybody is told to memorise. Eight columns. Sine along one row, cosine along the next, tangent along the third. You do not need to learn it as twenty-four separate facts, because it checks itself. Every entry in the tangent row is the entry above it divided by the entry above that. I checked all six of the tangent cells that have a value. Every one of them is its own sine over its own cosine.

14:13And the two cells that refuse to hold a value? They fall exactly where the cosine row reads nothing. Both of them. Nowhere else. Those cells are not blank and they do not say infinity. They say there is no such number. One more thing hiding in that table. It has eight columns, but it only names seven different points. The first column and the last are the same place, because a whole turn brings you back.

14:43So the row you thought was the answer to eight questions is really the answer to seven. Step back and look at what the swap actually did. We gave up a definition that needed a triangle. In exchange, we got a definition that needs a point, and a point is always there. Everything after that was reading. The most famous identity in the subject turned out to be the equation of the circle.

15:14Ten values came from walking to four places and looking. The zeros were the two axes. The repetition was the fact that a circle closes. The four remaining functions were quotients, and their broken inputs were the zeros we already had. None of that was memorised. All of it was read off one picture. And the thing that made it possible was refusing to patch the old definition, and replacing it instead.