Which of the six stays positive where, and what happens at the quarter turns

0:00Here is a table you are going to be told to memorise. Six functions down the side, four regions across the top, twenty-four cells of pluses and minuses between them. It is not twenty-four things. Eight of those cells carry all the information and the other sixteen are forced by them, and the eight come from two. By the end of this you should be able to rebuild any cell of it from a picture, in a couple of seconds, without having seen the table since.

0:34Start where all six of them came from. One point on the circle of radius one. It has two coordinates. The first one is the cosine. The second one is the sine. The tangent is the second divided by the first. The cotangent is the first divided by the second. The secant is one over the first, and the cosecant is one over the second. So every one of the six is built out of the same two numbers, using nothing but division, and the sign of any of them is decided by the signs of those two numbers. Two.

1:14There is no third piece of information in the table, which is why it cannot hold twenty-four independent facts. Before the table, one small picture that pays for itself many times over. Take the point for some arc, up in the first region. Now take the point for the same arc measured the other way. Same distance, opposite direction. It sits directly below the first one. The two are mirror images in the horizontal axis: turning through an arc and turning back through it land you at the same sideways reach and the opposite height.

1:56So write down the two coordinates of each. The first point is a, b. The second point is a, minus b. The first coordinate did not move at all. The second one turned over. Now read that off as a statement about the functions. The first coordinate is the cosine. It did not change. So the cosine of minus the arc is the cosine of the arc. The second coordinate is the sine. It reversed.

2:28So the sine of minus the arc is minus the sine of the arc. Two results, from one reflection. Now the trap, which is to say that negating the input negates both. It does not. Only the height turned over, because the mirror was the horizontal axis and mirrors do not move things sideways. I checked it by walking the circle in twenty-four steps. At all twenty-four the reach was unchanged and the height reversed.

3:00And the control: how many of those twenty-four had an unchanged height? Two. The two where the height was already nothing, so reversing it changed nothing. One more free result before the table. Draw the square that just contains the circle. It runs from minus one to one, both ways. Every point of the circle is inside that square. It has to be, because the circle has radius one. So both coordinates of every point are between minus one and one.

3:37Which is to say, the sine of any real number lies between minus one and one, and so does the cosine. No exceptions. I checked both coordinates at all twenty-four positions of a turn and all twenty-four sat inside. The ends are reached, though. Two of the twenty-four have a reach of exactly one or minus one, and two have a height of exactly one or minus one. So if a line of working produces a sine of one point two, it is wrong, and you need not find the mistake to know it is there. No point on that circle has a height of one point two.

4:20Now the four regions. The two axes cut the plane into four, and each of the four is described by the signs of the two coordinates in it. Up and to the right, both positive. Up and to the left, the reach negative and the height positive. Down and to the left, both negative. Down and to the right, the reach positive again and the height not. Four regions, four different pairs of signs.

4:49But notice how they are described. Strictly between one quarter turn and the next. Not up to and including, so the axes are in none of them. Draw them with gaps where the axes are, so the exclusion is something you can see rather than something you have to remember. Of the twenty-four positions in a turn, twenty land inside a region, five in each. The other four land on an axis, and no region will hold them.

5:21Now fill in the table, and start with the two rows that carry everything. The sine is the second coordinate, so the sine's row is just the sign of the height, region by region. Up and to the right, the height is positive. Up and to the left, still positive. Down and to the left, negative. Down and to the right, negative. Plus, plus, minus, minus. Read straight off the picture.

5:50The cosine is the first coordinate, so its row is the sign of the reach. Positive, negative, negative, positive. That is the whole of the first two rows, and neither of them was remembered. You can rebuild them by pointing at where the point is. Eight cells done, every one a statement about a coordinate rather than about a function. The tangent is the height divided by the reach. So its sign is the sign of the height times the sign of the reach.

6:25Not divided. Times. For signs the two do the same thing, because a positive over a negative is negative and so is a positive times a negative. So take the two rows you have and multiply them, cell by cell. Plus times plus is plus. Plus times minus is minus. Minus times minus is plus. Minus times plus is minus. Plus, minus, plus, minus. That third region is the one people get wrong. Both coordinates are negative there, so the tangent is positive. Two negatives.

7:07Now the cotangent, which is the reach over the height. The same two signs, the other way up. And a quotient's sign does not care which way up it is written, so the cotangent row is identical to the tangent row. Not similar. The same four entries. That leaves the secant and the cosecant, and there is nothing to do. The secant is one over the cosine, and one over a positive number is positive while one over a negative number is negative.

7:43A reciprocal never changes a sign. So the secant's row is the cosine's row, copied straight down. And the cosecant's row is the sine's row, copied straight down. This is where the most common error in the table lives: people treat the cosecant as the opposite of the sine and flip it. It is the reciprocal, and one over minus a half is minus two, which is still negative. So there is the table. All twenty-four cells.

8:16Two rows read off the coordinates, two rows made by multiplying those, and two rows copied down unchanged. Eight cells of information, sixteen recovered by two rules, multiply or copy, and none of it stored. You will meet a mnemonic for this, and it is worth recognising when somebody says it. Going round the four regions in order, the functions that stay positive are: all of them, then the sine, then the tangent, then the cosine.

8:51All, sine, tangent, cosine. Now count what the table says. In the first region, how many of the six are positive? Six. In the second? Two. In the third? Two. In the fourth? Two. Six, two, two, two. And that adds to twelve, which is exactly half of twenty-four, so the table is evenly split. Which two survive the second region? The sine, and the cosecant that copies it. The third? The tangent and the cotangent, which are the same row.

9:26The fourth? The cosine and the secant that copies it. So the mnemonic is not naming one function in each region. It is naming a pair, shortened to the first name of the pair. It is a summary of the derivation, not a substitute for it, and it says nothing at all about what happens on an axis. Which brings us to the gaps. What is wrong with the quarter turns themselves?

9:55Stand on one. At the start of the turn the point is at one, nothing. So the height is nothing, which makes the sine nothing, and the tangent nothing as well. The cosecant and the cotangent both have the height underneath, so neither of them exists there at all. And the reach is one, so the cosine is one and the secant is one. Count that. Two of the six are nothing. Two of the six do not exist. Two of the six sit at one.

10:27I checked all four of the quarter turns and every one of them splits two, two and two the same way. Now look at what you would write in a column for such a point. Nothing is not a plus and it is not a minus, and a function that does not exist is neither either. A boundary point is not a mixture of signs. It is a different kind of answer, and that is why it has no column.

10:56Now run the whole thing backwards. You are handed one value and one region, and you have to produce the other five. First one. The cosine is minus three fifths, and the point is down and to the left. The secant is immediate: turn it upside down, minus five thirds. For the height, use the circle. Reach squared plus height squared is one, so the height squared is one minus nine twenty-fifths, which is sixteen twenty-fifths.

11:29So the height is plus or minus four fifths. The squaring lost the sign and the arithmetic cannot give it back. Down and to the left means the height is negative. Minus four fifths. Cosecant minus five fourths, tangent four thirds, cotangent three fourths. Second one, and it goes a different way. The cotangent is minus five twelfths, up and to the left. The tangent is minus twelve fifths. Now use the other identity: one plus the squared tangent is the squared secant.

12:07That gives a hundred and sixty-nine twenty-fifths, so the secant is plus or minus thirteen fifths, and up and to the left the reach is negative. Minus thirteen fifths. Cosine minus five thirteenths, sine twelve thirteenths, cosecant thirteen twelfths. Those two are not the same problem worked twice. One went through the circle and one through the squared tangent. Handed a cosine or a sine, go to the circle. Handed a tangent or a cotangent, go to the squared secant.

12:45I worked seven of these, thirty-five answers in all, and rebuilt the point every time to check the answers against the picture rather than against my own algebra. Three of the seven arrived as a tangent or a cotangent and had to take the second route. And in every single case, the value on its own was consistent with two different regions. All seven of them. Cover the region and each question has two answers, both correct.

13:18So the region is not decoration. It is half the question, and the sign table is what answers it. The sizes come from the identities. The signs come from knowing where the point is. Neither has to be memorised.