Sine and tangent go back to the definition, because no rule so far reaches them

0:00By now you have a small machine that differentiates almost anything. A constant. The variable itself. Any whole-number power. And then sums, differences, products and quotients of whatever you already have. Feed it a polynomial and it answers. Feed it one polynomial over another and it answers. That is a closed room. Everything the machine builds is a ratio of polynomials, because that is all those four operations can make out of powers of x.

0:35And the sine is not in the room. So this is the moment the method has to change - not because a harder trick is needed, but because there is no trick. You go back to the definition. Before we do that, it is worth seeing that the sine really is outside, rather than just hard to get at. Here is the cheapest possible reason. Count the places a function is zero.

1:03A ratio of polynomials is zero only where its top is zero, and a polynomial of degree n has at most n roots. Finitely many. Always. The sine is zero over and over, forever. Sweep one stretch of the line and count the sign changes - each one forces a crossing between the two points either side of it. Over that stretch the sine changes sign fifteen times. The cosine, sixteen. And x squared plus one, which is a perfectly good ratio of polynomials, changes sign zero times.

1:43Fifteen crossings means no ratio of polynomials with degree under fifteen can be the sine - and lengthening the sweep pushes that number up without limit. There is no finite combination. The room has no door. So: the definition, applied to the sine. The quotient is sine of x plus h, minus sine of x, all over h. And now look at what you are actually holding. Two sines, subtracted.

2:16You cannot expand them - a sine of a sum is not a sum of sines. You cannot cancel anything. Left as it is, sending h to nothing gives you nothing over nothing, which is not an answer. Something has to reshape that difference into a product, because a product is the only thing you can take apart. The identity that does it turns a difference of two sines into a product.

2:45Applied here, it gives: two, times a cosine of x plus half of h, times a sine of half of h. Two things happened. A difference became a product. And a half appeared, in both places. That half is the whole reason this works, and it is worth being precise that it was not chosen. Here is that precision, measured. Take every place and every step size in a grid - fifty-six pairs - and ask whether the reshaped product really does bracket the same number as the difference it replaced.

3:24The reshaping with the halved argument: agrees fifty-six times out of fifty-six. The same reshaping with its sine and its cosine exchanged: agrees zero times out of fifty-six. The same reshaping with the halving simply left out: also zero out of fifty-six. So the half is not bookkeeping and it is not a trick. It is the only thing that is true. Now divide that product by h and see what you have.

3:57A cosine of x plus half of h - which is going to settle on the cosine of x as h shrinks, and needs nothing special. And a sine of half of h, divided by h. Rewrite the bottom as half of h too, and put the leftover half out in front, and the second factor becomes a sine of something, divided by that same something. That is not a new problem. That is the limit that was proved earlier by squeezing a sine between two things either side of it - the one that comes from the geometry of a circle.

4:36Watch it close. At a step of a half, the sine over its own argument reads zero point nine five eight. At a sixteenth, zero point nine nine nine three. At one part in two hundred and fifty-six, zero point nine nine nine nine nine seven. At one part in four thousand and ninety-six, zero point nine nine nine nine nine nine nine. It is going to one. Put the two together. A cosine of x, times one.

5:08The derivative of the sine is the cosine. But that is a claim, so let us not take it. Let us measure it. At each of seven places, build the difference quotient with no rule of any kind involved, walk the step in by halving, and ask when the quotient goes inside a narrow band around the cosine at that place - and stays inside for every smaller step after it.

5:35Seven places. Settles at all seven. Never settles at none of them. And it takes between seven and nine halvings, depending on the place. Nothing here was told the answer; it was walked in on. Seven out of seven only means something if failing was available. So here are three near misses, scored on the identical line. The cosine with a minus sign in front: settles at zero of the seven.

6:07The sine itself, on the theory that a function might be its own rate of change: zero of seven. The sine negated: zero of seven. Three wrong answers, none of which ever settles anywhere. Which is what makes the seven a measurement. There is one thing hiding inside that limit, and it is worth dragging out. The sine over its own argument goes to one only if the argument is in radians.

6:39In degrees the very same computation runs, and the very same limit exists - but it is not one. It is zero point zero one seven four five three, which is what one degree is when you measure it the natural way. So in degrees, the derivative of the sine is not the cosine. It is the cosine multiplied by that number. Every formula in this topic quietly assumes radians. That is not a convention chosen for tidiness. It is the only measure in which the limit is one, and the reason the formulas are clean.

7:20The cosine goes the same way, and it is worth doing because exactly one thing changes. The corresponding identity turns a difference of two cosines into a product as well - but this time it produces a sine of the shifted point, and it produces it with a minus sign in front. The half-argument factor is identical. The same imported limit closes it the same way. So the derivative of the cosine is minus the sine.

7:52Measured the same way: seven places, settles at all seven. The sine without the minus: zero of seven. The cosine itself: zero of seven. And the walk is a different walk - six to nine halvings, not seven to nine. The two computations are alike. They are not the same computation. Now the tangent - and here something worth noticing happens: the route goes back to the definition a second time, rather than reaching for a rule.

8:28Form the quotient. Two tangents subtracted, over h. Put them over a common denominator, which is the product of the two cosines. The numerator becomes a sine times a cosine, minus a cosine times a sine. And that is a sine of a difference - of the difference between the two angles, which is just h. So the whole thing is a sine of h, over h, times one over the product of the two cosines.

8:58The first factor is the same imported limit again, going to one. The second settles on one over the cosine squared. The derivative of the tangent is the secant squared. There is a shorter route, and it is worth being clear about why it is not the one just taken. The tangent is a sine over a cosine, and you have a rule for a quotient. Apply it and the numerator comes out as a cosine squared plus a sine squared, over a cosine squared.

9:34The top is one. So the answer is one over the cosine squared - the secant squared, again. Shorter. Correct. And unavailable if the question says from first principle, because the quotient rule needs the derivatives of the sine and the cosine, which the definition had to supply first. Both routes were scored here against the raw difference quotient, separately, at all seven places. Both settle at seven out of seven. The secant unsquared, on the same line: zero of seven.

10:12The two routes meet because of one identity, and it is the one worth keeping in view. A sine squared plus a cosine squared is one. At every place tested, the number of times that failed was zero. That is the step that turns the quotient rule's numerator into a one, and it is the same fact that lets the first-principle route recognise its numerator as a sine of a difference.

10:41Two arguments, one identity doing the work in both. One more thing the measurement shows. The tangent takes longer to settle than either the sine or the cosine - nine to fourteen halvings against seven to nine. And it takes longest at the place closest to where its own cosine vanishes. The nearer the trouble, the slower the approach. The rest of the family needs no new ideas at all.

11:11The secant is one over the cosine. The cosecant is one over the sine. The cotangent is a cosine over a sine. Each is a quotient of things you now have, so each goes through the quotient rule. The secant gives secant times tangent. The cosecant gives minus cosecant times cotangent. The cotangent gives minus cosecant squared. All three were walked in on and all three settle at seven of seven.

11:44And the cotangent with the minus left off - the single most common slip in the family - settles at zero of seven. Worth saying plainly: of this whole list, only the sine and the cosine are results handed over. The rest are things you are expected to be able to produce. Now a confusion that is built into the material, because two worked examples sit near each other and use one identity in opposite directions.

12:16The first differentiates a sine squared. By the product rule that is twice sine times cosine - which can be rewritten as the sine of twice the angle. The second differentiates the sine of twice the angle. Rewritten as twice sine times cosine first, the product rule gives twice a cosine squared minus a sine squared. So one problem's answer looks like the other problem's question. That is where the confusion comes from, and it is worth killing with a measurement.

12:53The squared sine, walked in against twice sine cosine: settles at seven of seven. Against the other answer: zero of seven. The doubled angle, walked in against twice cosine squared minus sine squared: seven of seven. Against the other answer: zero of seven. They are two different functions with two different rates of change. The identity relates the functions. It does not relate their derivatives. Last: where these things stop existing, and how that is visible rather than asserted.

13:32The tangent and the secant are built by dividing by a cosine, so they fail wherever the cosine is zero. The cotangent and the cosecant divide by a sine, so they fail where the sine is zero. The sine and the cosine themselves never fail, and neither do their derivatives. Walk in toward one of those places and watch. A tenth of the way out, the tangent reads about ten and the secant squared reads about a hundred. A ten-thousandth of the way out, the tangent reads about ten thousand and the secant squared reads about a hundred million.

14:13The function grows like one over the distance. The derivative grows like one over the square of it. The rate of change runs away faster than the thing changing. Put six bounds to each, increasing by a factor of ten, the smallest of them small enough that even a cosine clears it. The number the tangent never gets past is zero. The number the secant squared never gets past is zero. And a cosine, walking in to the very same place, fails to get past five of the six.

14:52Step back and look at what actually did the work. One identity to turn a difference into a product. One limit, imported from a squeeze on the geometry of a circle, to finish it. That is the entire entry into the family. Everything after it - the cosine, the tangent twice over, the secant, the cosecant, the cotangent - is assembly out of rules you already had. And the entry had to be made through the definition, because no combination of the four rules ever reaches a function with infinitely many zeros.

15:32The definition is not the slow way you use before you learn the fast way. It is the way in. So the derivative of the sine is the cosine, and the derivative of the cosine is minus the sine, and you have watched both of them arrive rather than been handed them. Both were scored against the raw definition at seven places with no rule involved, and the near misses were scored on the identical line and never settled anywhere.

16:05The half-argument was shown to be forced, fifty-six times out of fifty-six, with both of the ways of getting it wrong failing every time. The radians were shown to matter, because in degrees the same limit comes out at zero point zero one seven four five three instead of one. And the number the whole thing turns on was never written down here. It was found, by asking where a cosine crosses nothing: one point five seven zero seven nine six, and twice that, three point one four one five nine two.

16:44What is worth keeping is not the list of derivatives. It is that the list has one door, and the door is the definition.