The exchange rate between the two units, and the arc-length rule it buys

0:00We now have two ways of reporting the size of an angle, and they disagree about every single angle. So there has to be an exchange rate between them. Here is the whole of it, and it is one sentence long. Take a half turn. In the old unit, a half turn is one hundred and eighty. In the new one, a half turn is pi. That is the entire relationship.

0:26One angle, read twice, giving two numbers. And an equation between two numbers can be divided either way you like. Everything anybody ever does with these two units is that one line, rearranged. Not two rules to keep apart. One equation, turned whichever way the question needs. So divide it, and watch both answers fall out of the same line. Divide both sides by one hundred and eighty. One degree is pi over one hundred and eighty radians.

1:02Now go back to the same line and divide both sides by pi instead. One radian is one hundred and eighty over pi degrees. Two multipliers, and neither one was remembered. Both were produced, on the spot, from the sentence about the half turn. And here is the check that settles which way up you want it. Multiply the two together. Pi over one hundred and eighty, times one hundred and eighty over pi.

1:33Everything cancels and you get one. They are reciprocals, which is exactly what an exchange rate and its inverse should be. If you are ever unsure, do not try to recall it. Write the half turn down and divide. Now the obvious question. How big is a radian, in the unit you grew up with? One hundred and eighty, divided by pi. And here is where something important happens. To turn that into a number you have to choose a value for pi.

2:08Take pi as twenty-two sevenths, the usual working value. One hundred and eighty divided by twenty-two sevenths is twelve hundred and sixty over twenty-two. Which is fifty-seven point two seven two seven, going on. Now take the whole part off. Fifty-seven degrees. The leftover, times sixty, is sixteen point three six. So one radian is about fifty-seven degrees and sixteen minutes. Hold on to that number, because we are about to break it.

2:44Do the same division again, but this time do not round pi at all. Keep it exact, and squeeze the answer between two fractions as tightly as you like. One radian comes out as fifty-seven degrees, seventeen minutes, forty-five seconds. Look at the minutes. Sixteen became seventeen. The two answers disagree in the second figure, not in some digit far down the line. That is not a rounding wobble. That is the working value showing up in the answer.

3:19Twenty-two sevenths is not pi. It sits above pi by about one and a quarter thousandths. Three point one four is not pi either. It sits below, by about one and a half thousandths. Of the two, the fraction is the closer. One degree, the other way, is about nought point nought one seven four six radians. With exact pi it is nought point nought one seven four five. So when a question tells you which value to use, that is not politeness. The value you use is part of the answer.

3:58Some angles come up so often that it is worth having both readings side by side. Seven of them. Thirty degrees is pi over six. Forty-five is pi over four. Sixty is pi over three. Ninety degrees is pi over two. One hundred and eighty is pi. Two hundred and seventy is three pi over two. Three hundred and sixty is two pi. There is a trick people teach for rebuilding that list, and it is worth being careful with.

4:32The trick says: take one hundred and eighty and divide it by the number sitting under the pi. For the first five, it works. For the last two, it does not, because three pi over two and two pi do not have a bare one on top. Five out of seven is a bad rule. Here is the one that never fails. Multiply the coefficient of pi by one hundred and eighty.

4:59Three halves times one hundred and eighty is two hundred and seventy. Two times one hundred and eighty is three hundred and sixty. Same equation as before, and it covers all seven. Before we spend any of this, one piece of housekeeping about how angles get written down. The convention is short and it catches people out. A little circle after the number means the number is a degree measure.

5:29No circle at all means the number is a radian measure. The word radian is usually just left off. So a bare pi is not three point one four of anything. It is a half turn. A bare pi over four is forty-five degrees. And a bare four is four radians. Four radians is about two hundred and twenty-nine degrees. Four degrees is four degrees. The two differ by a factor of more than fifty-seven.

6:00One missing circle, and the answer is out by more than fifty-seven times. First worked example, and it turns on one thing only. Convert forty degrees and twenty minutes into radians. The dangerous move is to read that as forty point two. It is not. A minute is a sixtieth of a degree, not a tenth. Twenty minutes is twenty sixtieths, which is a third of a degree. So the angle is forty and a third degrees. As a single fraction, one hundred and twenty-one over three.

6:38Now apply the multiplier we derived. Multiply by pi over one hundred and eighty. One hundred and twenty-one over three, times pi over one hundred and eighty. The three and the one hundred and eighty give five hundred and forty. One hundred and twenty-one pi over five hundred and forty radians. Notice what we did not need. No value for pi at all. Going this direction, pi stays a symbol and the answer stays exact.

7:13Now run it the other way, which is where the work is. Convert six radians into degrees, with pi taken as twenty-two sevenths. Six times one hundred and eighty, divided by twenty-two sevenths. That is seven thousand five hundred and sixty over twenty-two. Which is three hundred and forty-three degrees, and seven elevenths left over. Now the same step twice. Take the whole part. Three hundred and forty-three degrees. Multiply the leftover by sixty. Seven elevenths of sixty is thirty-eight and two elevenths.

7:53Take the whole part again. Thirty-eight minutes. Multiply that leftover by sixty. Two elevenths of sixty is about eleven. Three hundred and forty-three degrees, thirty-eight minutes, eleven seconds. Take the whole part, keep sixtieths of the rest, do it again. That is the entire procedure. And now do the same conversion without rounding pi. You get three hundred and forty-three degrees, forty-six minutes, twenty-nine seconds. The degrees agree. The minutes are eight apart.

8:30Twenty-two sevenths cost you eight minutes of arc, and that is visible. So far the conversion has only cost us. Now it starts paying. An arc thirty-seven point four centimetres long sits under an angle of sixty degrees. What is the radius? Radius times angle gives the arc. That relation is the whole reason we changed units. But it works in radians and it works in nothing else. So convert first. Sixty degrees is pi over three.

9:06Now rearrange. The radius is the arc divided by the angle. Thirty-seven point four, divided by pi over three. Which is thirty-seven point four times three, over pi. With pi as twenty-two sevenths, that is one hundred and twelve point two, times seven, over twenty-two. Thirty-five point seven centimetres. Exactly, with no remainder. The exact answer sits between thirty-five point seven one and thirty-five point seven two. So the working value cost between a hundredth and a fiftieth of a centimetre, and it cost it downward.

9:48Convert, then use the relation. That order is the exercise. Next, a rotation turned into a distance. A watch has a minute hand one and a half centimetres long. How far does its tip travel in forty minutes? Start with the fraction of a turn. Forty minutes out of sixty is two thirds. Two thirds of a full turn is two hundred and forty degrees. Convert. Two hundred and forty degrees is four pi over three.

10:21Now the relation. Radius times angle. One and a half times four pi over three is two pi. With pi as three point one four, that is six point two eight centimetres. Two things about that answer matter. It is the length of the path, not the distance from start to finish. In a straight line the tip ends up about two point six centimetres from where it began. The path is nearly two and a half times the displacement.

10:52And one more. A real minute hand turns clockwise. Under our own sign rule, that angle is minus four pi over three. The question uses its size and quietly ignores its direction, which is fine as long as you know that. Now a question with no numbers in the answer at all. Two circles. One arc length, laid into both. In the first it opens an angle of sixty-five degrees. In the second, one hundred and ten.

11:27What is the ratio of the radii? Radius times angle is the arc, and the arcs are equal. So the first radius times sixty-five equals the second radius times one hundred and ten. Which means the radii are in the ratio one hundred and ten to sixty-five. In lowest terms, twenty-two to thirteen. Read that carefully. It is the wrong way round from most people's instinct. The bigger angle belongs to the smaller circle.

12:00Which makes sense. A fixed length of arc has to wrap further round a small circle to be used up. And notice what happened to the conversion. Convert both angles to radians first and you get exactly the same ratio, twenty-two to thirteen. The multiplier appears on both sides and cancels. That ratio could have been read straight off the degrees. A set of practice questions sorts itself into two piles.

12:33The first pile only pays the conversion. Twenty-five degrees into radians is five pi over thirty-six. Minus forty-seven degrees thirty minutes is the trap from earlier. It is not minus forty-seven point three. Thirty minutes is half a degree, so it is minus forty-seven and a half, and the answer is minus nineteen pi over seventy-two. Two hundred and forty degrees is four pi over three. Five hundred and twenty is twenty-six pi over nine.

13:08Going the other way, eleven sixteenths of a radian is thirty-nine degrees, twenty-two minutes, thirty seconds. Minus four radians is about minus two hundred and twenty-nine degrees, five minutes, twenty-seven seconds. But five pi over three and seven pi over six need no value for pi at all. They are three hundred degrees and two hundred and ten degrees, exactly. When the reading is a multiple of pi, the pi cancels and the answer is exact.

13:44There is also a wheel, turning three hundred and sixty times a minute. Three hundred and sixty turns is three hundred and sixty times two pi radians, spread over sixty seconds. Twelve pi radians every second. The second pile collects. A circle of radius one hundred, an arc of twenty-two. The angle is the arc over the radius. Eleven fiftieths of a radian. In degrees, that is twelve degrees and thirty-six minutes.

14:18A pendulum seventy-five long, tracing arcs of ten, fifteen and twenty-one. Same relation, three times. Two fifteenths, one fifth, and seven twenty-fifths of a radian. Two equal arcs opening sixty degrees and seventy-five degrees give radii in the ratio five to four. Bigger angle, smaller circle. Again. And then one question that is doing something else entirely. A circle of diameter forty, so radius twenty. A chord of length twenty. Find the shorter arc it cuts off.

14:58Look at the chord. It is twenty. The radius is twenty. So the chord and the two radii make a triangle with three equal sides. Every angle in it is sixty degrees. The angle at the centre is pi over three. Arc is radius times angle. Twenty times pi over three is twenty pi over three. No ratios, no tables, no trigonometry. Just noticing that two lengths were the same number.

15:31Step back and count what the exchange rate actually cost and bought. It cost us an ugly number. Fifty-seven degrees and change, for one radian. And it cost a decision, every time we convert a bare number, about which value of pi we pay with. What it bought is the relation we kept reaching for. Arc equals radius times angle. Two lengths and a bare number, with no multiplier dragged along behind them.

16:03Every one of those questions began by converting, and then the relation did the rest in one line. That is the trade. An awkward exchange rate, once, in return for a relation with nothing in it. And there is nothing to remember about the exchange rate itself. One angle. A half turn. One hundred and eighty in one unit, pi in the other. Divide it whichever way the question is facing.