Defining sine, cosine and tangent, then their three reciprocals

0:00A right triangle has three sides, and once you pick an acute angle they have names: the opposite leg, the adjacent leg, and the hypotenuse. Now a counting question. How many ratios can you build by putting one of those over another? Three choices on top, two underneath each, since a side over itself is always one and tells you nothing. Six quotients, and there is no seventh. I built that table on three hundred and thirty six right triangles, at both acute corners of each, and it had six entries every time.

0:40So the six named ratios ahead are not a selection from a longer list. They are the list. Take them one at a time. First: the opposite leg over the hypotenuse. That ratio is called the sine of the angle. The opposite leg is one of the two short sides; the hypotenuse is the longest of the three. So the top is always smaller than the bottom, and the sine of an acute angle is always less than one.

1:13That held at every one of the six hundred and seventy two acute corners I checked. These are quotients of lengths, so they are pure numbers with no unit, and this one is held below one by the shape of the triangle. Second. Keep the hypotenuse underneath and slide the top from one leg to the other: the adjacent leg over the hypotenuse. That is the cosine. Same denominator, different numerator, and nothing else changed.

1:46The same bound applies for the same reason: the adjacent leg is also shorter than the hypotenuse, so the cosine is under one too, on all six hundred and seventy two corners. Third, and this one is different. Measure the two legs against each other: the opposite leg over the adjacent leg. That is the tangent, and notice what is missing. The hypotenuse is not in it at all, which makes it the simplest of the three to build.

2:21It is also the one that is not bounded: nothing stops the opposite leg from being longer than the adjacent one. Over my three hundred and thirty six triangles the tangent came out bigger than one at two hundred and sixty four corners, and smaller than one at another two hundred and sixty four. At a hundred and forty four corners it was exactly one. Those are the triangles whose two legs are equal, and they cause trouble later.

2:54Now the other three, and there is no new geometry at all. You take each of the three fractions and turn it upside down. The sine was opposite over hypotenuse. Turn it over: hypotenuse over opposite. That is the cosecant. The cosine was adjacent over hypotenuse. Turned over, hypotenuse over adjacent: the secant. And the tangent was opposite over adjacent. Turned over: the cotangent. Three plus three is six, and the counting is closed: every quotient has exactly one name, and no quotient has two.

3:36Since sine and cosine are under one, their reciprocals must be over one, at every one of the six hundred and seventy two corners. Here is the part that trips everybody. Three of the six names carry a prefix — cosine, cosecant, cotangent — and three do not: sine, secant, tangent. It is tempting to read that as a grouping. It is not one. Sine pairs with cosecant. Cosine pairs with secant. Tangent pairs with cotangent. I did not assume that: I multiplied every one of the six by every other and kept the products that came to one, and that is what the search reported.

4:23Now count the prefixes in each pair. Sine and cosecant: one. Cosine and secant: one. Tangent and cotangent: one. Exactly one member of every pair carries it. Over five hundred and twenty eight corners that is one thousand five hundred and eighty four pairs, and the count came out one every single time. So the prefix marks out a set of three that cuts straight across the pairing, never landing on both members and never on neither.

4:57Guess the partner by adding or removing the prefix and you get two of the six right, and those two by accident. There is no rule to recover. Learn the three pairs and stop reading the spelling. Now a relation worth deriving rather than memorising. Divide the sine by the cosine: on top, opposite over hypotenuse; underneath, adjacent over hypotenuse. Dividing by a fraction is multiplying by its reciprocal, so this is opposite over hypotenuse, times hypotenuse over adjacent.

5:39And there is the hypotenuse, once on top and once underneath. It cancels, and what is left is opposite over adjacent, which is the tangent. The reason the hypotenuse is absent from the tangent is that it was in both places at once and cancelled itself out. The division gives the tangent at all six hundred and seventy two corners; multiplying instead gives it at none, and subtracting at none. Run it the other way up and you get the cotangent.

6:14But be careful. The tangent is not built from the sine and the cosine; it was built from two legs. The relation is a consequence, not a definition. A word about the writing, because two things about it are examined. When you write s-i-n and then the angle, those three letters are one symbol. They are not a quantity called sin multiplied by an angle. Here is why. If it were a product, dividing by the angle would leave the same number every time.

6:49The sine of thirty degrees is a half; divide by thirty and you get one sixtieth. The sine of ninety degrees is one; divide by ninety and you get one ninetieth. Different number. Over every angle in the family where the sine is exact I got seven different values. A genuine product put through the same test gives one value, always the same. So the three letters do not come apart. On their own they name an instruction, not a number, and it means nothing until an angle is supplied.

7:28The second piece of notation. You will often see the square tucked in front of the angle rather than around the whole thing. That is a space-saving convention meaning the square of the ratio: work the ratio out first, then square the answer. It does not mean square the angle. Those are different operations and they give different answers. Take thirty degrees. The sine is a half, so the square of the sine is a quarter. Now square the angle instead: thirty squared is nine hundred degrees, which lands where a hundred and eighty does, and the sine there is zero.

8:10A quarter and zero. Not the same number. I tested that at twelve angles where both answers come out exactly. They disagreed at ten of the twelve, and the two that agreed are angles where the ratio itself is zero, so there was nothing for a difference to show up in. One more warning. A superscript minus one does not mean the reciprocal. The reciprocal already has a name: cosecant, secant or cotangent. That superscript names something else entirely, which comes later.

8:48Let us make all six concrete. Right angle at B, the side from A to B is twenty four, and the side from B to C is seven. Twenty four squared is five hundred and seventy six, seven squared is forty nine, and those add to six hundred and twenty five. The hypotenuse is twenty five. Stand at A. Opposite is seven, adjacent is twenty four, hypotenuse twenty five. Sine of A is seven over twenty five. Cosine of A is twenty four over twenty five. Tangent of A is seven over twenty four.

9:28Now turn each one over. Cosecant is twenty five over seven, secant is twenty five over twenty four, cotangent is twenty four over seven. You worked none of the last three out. You wrote the first three down and inverted them, and each of the last three times the one it inverts gives exactly one. Now do the whole thing again from the other corner. Nothing on the page moves, but the two legs trade names, so all six answers move with them.

10:03Sine of C is twenty four over twenty five. Cosine of C is seven over twenty five. Tangent of C is twenty four over seven. Compare that with the list at A. Sine and cosine have swapped, tangent and cotangent have swapped, and so have secant and cosecant. I compared each ratio at one acute angle against all six at the other and let the search report the matching. It reported that same permutation on all two hundred and sixty four triangles with unequal legs.

10:41On the seventy two with equal legs it could report nothing definite, and that is honest rather than a failure: four of the six ratios collapse into two values there, and no comparison of numbers can tell you which name is which. Which is why an equal-legged triangle is the wrong place to learn any of this. Here is that swap earning its keep. A right triangle, right angle at Q. The side from P to Q is twelve, and the hypotenuse from P to R is thirteen.

11:19Thirteen squared is a hundred and sixty nine, twelve squared is a hundred and forty four, and the difference is twenty five. So the side from Q to R is five. The question: the tangent at P, minus the cotangent at R. From P, the opposite leg is Q to R, which is five, and the adjacent leg is P to Q, which is twelve. Tangent at P is five over twelve.

11:49From R the roles are exchanged: P to Q is the opposite leg now, Q to R the adjacent one. Cotangent is adjacent over opposite, so that is five over twelve as well. Same number, so the difference is zero and no arithmetic was needed at the end. The sine at P and the cosine at R do the same thing: both are five over thirteen. But the same name at the two angles does not cancel. Tangent at P minus tangent at R is minus one hundred and nineteen over sixty.

12:30One property all six share: they carry no unit, and they do not depend on how big the triangle is. I scaled all three hundred and thirty six triangles by five different factors. One thousand six hundred and eighty rescalings, and not one of the six ratios moved on any of them. Meanwhile a plain length moved on all one thousand six hundred and eighty, which is what makes the first fact worth stating.

13:00So these ratios read the shape of the triangle, not its size. Finally, where the word came from. Aryabhata, writing in Sanskrit around the year five hundred, was working with half of a chord of a circle. He called it ardha-jya: half-chord. In use it shortened to jya, then jiva. Arabic writers carried it across as a sound rather than a translation. Then it was rendered into Latin as sinus, a word meaning a curve or a fold, and from sinus we get the English sine.

13:39The three-letter abbreviation came much later, from Edmund Gunter, an English professor of astronomy. So the symbol you write has a Sanskrit half-chord, an Arabic transliteration and a Latin mistranslation folded into three letters. And what it names is three ratios, plus a rule for turning them over.