**An Open Letter to Students Wondering, “Why Do We Use Radians Instead of Degrees?”**

Let me start with a question of my own. Did you know that you speak Babylonian?

You may not think that you do – you can’t ask for directions to a Babylonian bathroom, or order lunch off of a Babylonian menu, or ask a good-looking Babylonian out for coffee. But you’ve inherited Babylon’s legacy nevertheless.

You’ve probably noticed that our systems of counting and measuring are largely base 10. There are 10 years in a decade, 10 decades in a century, and 10 centuries in a millennium. But take a look at shorter time lengths: We don’t carve up the day into 10 hours, each 100 minutes long, even though this would make perfect sense. Instead, we divide our time up by multiples of 60 – 60 seconds in a minute, 60 minutes in an hour.

Why do we do this? Because our Babylonian uncles used a base 60 number system, and we’ve been following their example ever since.

Babylon left another relic in our system of measurements, one that requires a little deeper explanation, and which lies at the heart of your question (yes, I’m getting there): the degree.

***

Angles are everywhere. They form when light bounces off of a mirror, when a bird dives towards the water, when a lamp casts a shadow on a distant wall. And mathematicians, being the go-getters that they are, need a way to measure angles.

It’s clear from looking at two angles which one is bigger:

But without any numbers, it’s hard to say how much bigger. That’s where angle measurements come in.

We can think of any angle as a piece of a circle. For example, a right angle (like the corner of a rectangle) is one quarter of a circle, and the angle in an equilateral triangle takes up one sixth of a circle:

So, how do we measure angles? Well, the Babylonians had an idea. They decided to cut a circle into 360 pieces, and call one of those pieces a degree.

This makes it easy to talk about the size of an angle. Since there are 360 degrees in a whole circle, a right angle will be one-quarter as many degrees – i.e., 90^{o}.

Most people learn this system so well, and at such a young age, that it becomes second nature. Skateboarders pull 540^{o}’s, a businessman looking to change plans will call for a 180^{o}, and at one point in our political past, Americans chanted “Fifty-four forty or fight!” to demand that our border with Canada be set at 54^{o}40’ of latitude (where 1’ = 1/60^{th} of a degree).

What this familiarity disguises is that the number 360 is totally arbitrary, chosen simply because the Babylonians preferred multiples of 60. Why not divide the circle into 100 pieces, or 5 pieces, or 400 pieces? (In fact, the 400-piece system does exist – 1/400^{th} of a circle is called a gradian, and it’s that weird mode on your calculator that you never use.) There’s no mathematical reason to pick 360, or any specific number, for that matter. It’s fundamentally just a matter of taste.

***

As it turns out, there’s a better way. But to explore it, we’ve got to lift ourselves out of this rut of thinking. No more dividing the circle into some arbitrary number of units.

Instead, let’s draw a whole circle, and throw in a radius for good measure.

Now, let’s take that radius and wrap it around the outside of the circle. See how it forms an angle? We’ll call that angle 1 radian.

Here’s an angle that’s 2 radians, for comparison.

And here’s an angle that’s 3 radians. Notice that it’s almost half a circle (what we used to call 180^{o}), but not quite.

This raises a question: how many radians are there in a circle? Draw it out, and you’ll find that it’s not a nice, round integer:

Looks like a little more than 6. But how much more?

For that, we’ll need to dredge up a geometric formula relating the circumference and the radius:

C = 2πr

In other words, the circumference is equal to the length of the radius times 2π (where π is roughly 3.14. For a better – but still imperfect – approximation, try this).

This tells us exactly how many radians there are in a circle: 2π!

Knowing this, we can now convert between radians and degrees – just as we can convert between miles and kilometers, or Fahrenheit and Celsius. Radians become a perfectly valid, usable measure of angles.

***

But I know you’re not satisfied with that. You’re sharp-witted and wary of being made to learn new things. You want to know: What was wrong with degrees? Arbitrary or not, they let us work with integers ( like 30^{o}) instead of nasty ratios involving irrational numbers (like π/6). Degrees are warm, friendly, familiar. Why ditch them in favor of this bizarre radian?

Here’s the best answer I can give you: Degrees are fine for everyday measurements. But Trigonometry marks a turning point in math, when the student lifts his gaze from the everyday towards larger, more distant ideas. You begin exploring basic relationships, deep symmetries, the kinds of patterns that make the universe tick. And to navigate that terrain, you need a notion of angles that’s more natural, more fundamental, than slicing up the circle into an arbitrary number of pieces. The number π, strange though it may seem, lies at the heart of mathematics. The number 360 doesn’t. Clinging to that Babylonian artifact will only distract you and obscure the elegant truths you’re searching for.

***

**Just like you, I learned to speak Babylonian long before I encountered radians. And for years, Babylonian remained my native tongue – to give an angle in radians first required an act of mental translation. So if you’re resistant towards radians, nostalgic for Babylon, I can sympathize.**

Still, when push comes to shove, radians can take you places that degrees simply can’t. That’s why, when my Trig students give an angle in degrees instead of radians, I tell them: “I’m sorry, I don’t speak Babylonian.”

I hope this helps. When in doubt, remember: 180^{o }= π radians.

Best,

Ben

Nice explanation, though as a physicist I must point out your missing unit at the end. 180 degrees = pi *radians*

Thanks! My hazy memory is that radians can be treated as unit-less in a lot of contexts… although I can’t remember why at the moment.

EDIT: You’re right that the “rad” is necessary. Radians are unitless because they’re arc length divided by radius length (so the units of length cancel). But the same is true of degrees (which are arc length divided by 1/360th of circumference). So including the units is the right move. Will edit accordingly.

Angle is the ratio of length of an arc in a circle and of rsdiud its radius., Both quantities have the dimension of Length. 1 Radian subtends arc of length of the radius.

It’s probably worth noting that our year has 365 days, but the Babylonian year had 365 days (split into 12 months of 30 days each plus 5 extra days for holidays) which may have influenced why they chose 360 as the number of degrees in a circle. 360 is also convenient because it divides so many ways into nice whole unit portions.

Yeah, the many factors of 360 do make it nice to use. I wonder if that’s part of the reason gradians never caught on – 400 has only two prime factors, whereas 360 has three.

The Babylonian version of the year sounds kinda nice – I’d love to have equal-length months, with five bonus days.

I want to say that a former Roman calendar had something like that: 12 months of 30 days, then a week-long festival to get back on track (on that note why ISN’T Leap Day a Federal holiday?). Or maybe that was Egyptian. The question then becomes…why did anyone ever pick SEVEN as the divisor for weeks?

With no evidence whatsoever, I’d guess the thought behind the 1/400th has to do with the historical definition of a meter. Originally defined by a study from the French Academy of Science in the 18th century as a universally human unit to be embraced by all countries of the world:

–1 meter = 1 ten millionth of a quarter meridian

–1 meter = 1/40,000,000 of the (meridianal) circumference of the earth

–100 km = 1/400th of the circumference of the earth.

So now you can sail your ships (or fight the Canadians) over lovely gradian arc lengths of 100 km, and each lon-lat grid also has an area of 1 hectare (ignoring curvature).

Of course, although a solid effort for the 1790’s, the measurements weren’t quite perfect (the meter was never redefined, so modern measurement places the Earth’s meridianal circumference at 40,008 km). And, more to the point, the Earth isn’t perfectly spherical (equatorial circumference is 40,075 km).

Ah, cool. I’d never thought about the gradian as corresponding to nice arc lengths going around the world.

There’s also a quicker (though maybe no more satisfying) explanation for the 400: it makes a right angle 100 gradians, which feels suitably metric.

As for the seven-day week, Wikipedia is as stumped as the rest of us. It’s a quarter of a lunar cycle, which I guess is nice if you’re on a lunar calendar, but Wikipedia pokes some holes in that theory, too.

http://en.wikipedia.org/wiki/Seven-day_week

I believe the meter is defined as being the length of a pendulum with period of one second (two seconds?).

One effective interactive way to introduce radian measure is to use your body to measure the angle subtended by a distant object. To make the measurement, extend your arm and sight along your thumb. Of course, our arms and thumbs are different sizes, so to standardize the measurement, students can be prodded to realize that they need to measure the ratio of the height of the object (along the thumb) to the distance from the eye to the thumb. This ratio gives the angle in radians. For instance, if my thumb is 60 cm from my eye and my thumbnail is 2 cm long, an object that appears the same size as my thumbnail subtends an angle of 2 cm / 60 cm = 1/30 radian. Though I could convert that result to degrees, it’s more useful in radians. If the object I’m measuring is about a football field’s length away, I can find its height directly from my angle measurement: 1/30 x 100m, or about 3.3 m high. Presented and used this way, radians are natural, useful, and memorable, making it interesting for students to solve the challenging question of how these natural angle units relate to angle measurements in degrees.

I like that. Radians are built around arc length – a quantity we rarely use in trig classes after week 1. But this is a nice way to make finding arc length, as you say, “natural, useful, and memorable.”

Reblogged this on Rajnie's Blog and commented:

Had my Mathematics teacher taught me like this, I would have developed affinity towards the subject. 🙂

It’s so hard to think in terms of degrees now that I’ve been in higher-level math courses for the last few years! All those physicists out there need to standardize to radians. (OK, I’m just kidding.)

Another good grump is the whole pi vs. tau issue: Other than the formula for area of a circle, why not use tau instead of pi? It would be a lot simpler… 🙂

Yeah, I like tau. It would make the area formula less elegant, but it’d make the formula for circumference more elegant (tau * r), among the other improvements.

I think it makes the formula for the area of a circle more natural … once you are willing to consider the area an infinite number of triangles summed around a point (1/2 * base * height).

Agreed it is a little clunkier; in the same way that the formula for a triangle has an extra term compared to a rectangle. Although in my mind, the formula should really be 1/2 * (2 * pi * r) * r ; so 1/2 * (tau * r) * r is still the more elegant.

I’m actually in Trig right now, and Radians are what we started learning.

While I do see pie a lot simpler (mainly in places where Tau/2 are pretty ugly), Tau would be very great in, say *RADIANS!* 1/4 Tau radians are one fourth of a whole circle.

I also see Tau being great in finding the area of a circle. 1/2 Tau for half of a circle, 3/2 tau for three halves, and so on.

When I was a trigonometry student, I was convinced that we should all ditch radians AND degrees, and just use the “turn” as our unit of angle. A quarter of a turn, a half of a turn, 3/7 of a turn – what could be easier?

Only recently did I realize what you pointed out in one of the drawings here: if you were using any units other than radians, cos(x) would not be the derivative of sin(x). If x were in “turns,” the derivative of sin(x) would be 2*pi*cos(x). The solutions to f” = -f, the functions that you and I know simply as sin(x) and cos(x), would be called cos(x/(2pi)) and sin(x/(2pi)). And Euler’s Formula would be e^(i*2*pi*x) = cos(x) + i*sin(x).

…Actually, Euler’s Formula doesn’t look too bad in its new form. Especially if you’re using tau: e^(i*tau*x) = cos(x) + i*sin(x). The polar form of a complex number would be re^(i*tau*theta). It seems to me like all the math we do now would still be doable with “turns” instead of radians. Just a little bit uglier.

I wouldn’t be surprised if there was another planet in our galaxy full of sentient beings who adopted turns instead of radians as the conventional angle unit of higher math, because it seemed like the obvious thing to do at the time. Now, centuries later, one of their mathematicians has written up a Radian Manifesto explaining why everything would be so much prettier if they would just start using this weird alternate unit of angle instead of the turn. But of course it’s far too late for them to change: everyone uses turns as the assumed unit for trig functions in all mathematical publications, and every calculus student has memorized that the derivative of sin(x) is tau*cos(x), and how on earth could they make the transition now? We humans should pat ourselves on the back for not making that mistake.

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ily man haha

I guess we could celebrate tau day at a time closer to a solstice. But pi day is close to an equinox….

yeah i pretty much just read this because of my math theacher #boredoutofmymind

There should be a reason for 360 degrees . The Babylonians could have a assumed that the earth rotated 360 times (made 360 days) for 1 revolution around the sun.(Or we might have slowed down by 5.25 days) Hence applying the concept of earth circling around the sun to circles.

I taking Math 4 (swedish system) this semester and it’s all about trigonometry. We just started with radians and my question is the opposite; why don’t we abandon the 360 degree system altogether? As a former skate- and snowboarder I think it would be even cooler to say “I just did a 2π indie” or “π frontside ollie”. Regarding extreme sports you never do any other rotation than n*π so it wouldn’t be confusing at all but rather more streamlined phrases with less syllables since you’d only have to say π, 2π, 3π etc.

You should make it happen! If it catches on among Swedish snowboarders – presumably the coolest people on Earth – I don’t see why it wouldn’t spread everywhere.

The “Here’s the best answer I can give you:” didn’t really seam like an answer, therefore I still feel like radians aren’t a good measurement, they use to many fractions like the imperial system, degrees are much easier to use precisely. Why make a revolution 6 radian plus another other small? Seems harder and more arbitrary than degrees.

Why do newer posts appear at the bottom? Just as arbitrary as radians I suppose.

Now I get it, it isn’t arbitrary, “the arc length of 1 radian = the radius”, but still, comments should go on top.

hi

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david is awesome

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ᕦ( ͡° ͜ʖ ͡°)ᕤ

So basically, they are useless.

good work.

And here I was thinking it was because 365 days in a year. You know, that huge circle around the sun that the earth makes? Sure, those 5 extra days are a pain to deal with, but as I recall, weren’t there 5 holy days to compensate? There is some varied evidence to suggest that they knew the earth orbited the sun. So why not 1 day equals 1 degree?

Reblogged this on Holliday's Inner Workings has relocated to seasonalchats.com! and commented:

While re-exploring and relearning much of Geometry this year to teach it to my high school students, I myself wondered why we learn and use radians. This post captured my attention and is really interesting and insightful. Thanks Ben for the awesome work!

How do I get to know Ben?

Thank’s for the explanation. I couldn’t relate to the figure 360 for any logical reason,I wondered why we didn’t use 10 as a base unit.

Why not turns, though? 1 is definitely not an arbitrary number.

That being said, I can’t wait to see someone make an extremely inefficient angle measurement system using the golden ratio just for the heck of it.

Ben,

I will be teaching trig next year and this was a great introduction! I might have to use this with my students!

Thank you for the creative, simple explanation!

Ms. Hoegh

The reason for having 360 degrees in a circle is not because it is related to 60, but because of the fact that Babylonian thought the year is 360 days. A circle was imagined as a year and each day one degree on the circle.

Excellent note, its worth to be an essay or seminar topic for school maths class.

but this whole thing is true only bcoz they divided into 360 parts ..if i divide in some other no. of parts.. results would be different? why so?

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Hello Sir

Great Article![Actually ‘convincing article’ is more appropriate.]

I always searched to find something of this sort.

I have a question,though not literally a ‘question’………

You mentioned that radian measure is more ‘natural’,which helps in working with things where degrees create chaos.I think you also meant that radians help in more mathematically abstract works.Will you please give me some examples to justify your ‘naturalness’ point?

Thank you

Keep doing the great work!!

Hi Anandmay,

I tried to respond to the “natural” part in my comment above, about using your body (or any makeshift measuring device) to measure small angles (for instance, the height of a distant object). To use your body, extend your arm, stick your thumb up so it appears alongside the distant object, and measure both the apparent height of the object at your thumb (e.g., 2 cm) and the distance from your eye to your thumb (e.g., 60 cm). The ratio of these two measurements is the angle measure in radians, and can be used directly to find the unknown object’s height if you know its distance.

This “natural” use of your own body is related to another aspect of the “natural” character of radian measure: it works because for small angles, sin(x) is approximately x (technically, as x approaches 0, sin(x) approaches x). This fact is only true if x is measured in radians, and is intimately tied to the realization that radians are much easier to work with in trigonometry, calcullus, etc.

Hello Sir

As a student, I see quite an interesting way of introducing radian measure.

But I regret I am not able to understand this:

…….. if my thumb is 60 cm from my eye and my thumbnail is 2 cm long, an object that appears the same size as my thumbnail subtends an angle of 2 cm / 60 cm = 1/30 radian……..

Here,i don’t understand: The object subtends 1/30 radian angle WITH??[i.e.,with what??]

Here i am facing a little bit confusion.

Will you please help me clear the concept?

Thank you

However if we use Tau…

I completely agree with using the logical radian units instead of the arbitrary (and antiquated) degree unit. However, I take issue with your Fact-Radian-Degree picture.

Isn’t the derivative of sinx with respect to x simply cosx, regardless of units? For instance, let x=(pi/180)y, where y is the same angle in degrees. Let’s just call the pi/180 conversion factor C for convenience. Now sin(x) = sin(Cy). To take the derivative w.r.t. x, we must adjust the differential: dx = d(Cy) = Cdy. Thus, d/dx = d/Cdy = 1/C*d/dy. So d/dx[sin(x)] = 1/C*d/dy[sin(Cy)] = 1/C*Ccos(Cy) = cos(Cy). (Which of course equals cos(x)).

So it’s not correct to have that factor in front of the derivative. There should indeed be a conversion factor on the inside with the argument/angle, but that is only if x is in radians to begin with. If you had defined x in terms of degrees, the radians expression would have been the unwieldy one.

In the next line, the picture of the graphs, you have also made a biased decision to show both graphs on the scale of radians. If your tic marks had represented 90 units or so instead of 1.2 units or so, the sine wave in degree units would look “nice” and the one in radians would be horribly scrunched.

My point is, I agree with you that radians are the preferred unit, but some of your arguments are fallacious and show a clear bias.

“So it’s not correct to have that factor in front of the derivative… I agree with you that radians are the preferred unit, but some of your arguments are fallacious and show a clear bias.”

Whoa there, cool your jets with that “your arguments are fallacious and show a clear bias” stuff. What is this, an internet comments section? That’s not how mathematicians should talk to each other. Plus, as you shall soon see… that factor really SHOULD be in front of the derivative! Here’s why:

In your proof, ultimately what you did was consider the usual sin(x) function (which interprets its argument as an angle in radians and returns the sine of that angle). You took its derivative with respect to x, incidentally making a substitution of cy for x along the way, where y is in degrees and c is the conversion factor, and you got cos(x). No one disputes that d/dx sin(x) is cos(x) when x is in radians, but that doesn’t answer the question of what the derivative of sin(x) would be if the argument were in degrees, not radians.

So let’s see what happens when we define sin(x) in terms of degrees rather than radians.

For clarity’s sake, we’ll rename the sin function as we know it radianSin(x) (since it’s the function which interprets its argument as an angle in radians and returns the sine of that angle). And let’s define a new function degreeSin(x) to be the function which interprets its argument as an angle in *degrees* and returns the sine of that angle. (Define radianCos(x) and degreeCos(x) similarly.)

So, for example, radianSin(pi/2) = 1, and degreeSin(90) = 1. Also, degreeSin(x) = radianSin(pi/180 x).

I claim that:

d/dx radianSin(x) = radianCos(x)

and

d/dx degreeSin(x) = pi/180 degreeCos(x).

The first claim just states that the derivative of the usual radian-based sin function is the usual radian-based cos function, which I’m sure you already believe.

Here’s the proof of the second claim:

d/dx degreeSin(x)

= d/dx radianSin(pi/180 x)

= pi/180 radianCos(pi/180 x)

= pi/180 degreeCos(x).

Behold! The factor stays!

Okay, so that proves it, but it doesn’t tell you intuitively why using radians eliminates the need for a scale factor. Here’s a sketch of the reason:

Imagine standing at the origin and swinging a unit-long rope with a tin can tied at the end counterclockwise around your head. The path of the can traces out the unit circle, and its position vector at a given angle theta is . (You can think of the string as its position vector.)

Here’s the question which this all revolves around (no pun intended): what’s the can’s velocity vector? In other words, what is ?

Well, the direction of the velocity vector is easy enough to figure out: the can is traveling tangent to the circle, so its velocity vector is rotated 90° counterclockwise from the string. So the can is moving in the direction of the unit the vector = .

But what’s the *magnitude* of the velocity vector? Well, it depends how fast you’re swinging the can (i.e. what’s dθ/dt?). It would be *nice* if the velocity vector was also a unit vector – that is, if you swung the can at just the right speed such that its velocity vector was the same length as the string: 1 unit. That way, = . Otherwise, if you spin too fast or too slow, the velocity vector will be multiplied by some scale factor greater than or less than 1 (like, for instance, pi/180).

But the velocity vector can only be a unit vector if you’re swinging the can at one unit of distance per unit of time: that is, if the can moves exactly one “string length” each unit of time.

The string length is the radius of the circle, so you must turn one radian per unit of time to make the can move one string length per unit time. In other words, when dθ/dt = 1 (or θ = t), = .

Tada!

Whoops, angle brackets are interpreted as html. Disregard this version and see below.

Here’s a more visual way to think about the derivative of sin x when x is in degrees. If you graph f(x)=sin x in radians and look at the slope of the graph at x=0, you see that the slope is 1 (corresponding to the fact that cos(0) = 1). When x is very close to 0, sin x is very close to x. For instance, sin(0.01)=0.0099998, only off by 2 parts in ten million.

Now think about the slope of the sin graph when x is measured in degrees. If we use the same angle, .01 radian is about 0.572958 degrees. Sure enough, if you check sin(0.572958°) the result is still almost exactly 0.01. But this means the slope of this sine graph, near the origin, is 0.01/0.572958, which is very close to π/180. Thus the slope of sin x (in degrees) is π/180·cos x. That is, the derivative of sin x in degrees is not cos x, but rather π/180·cos x.

One last thought related to the graph: imagine using the same axes to graph sin x in radians and sin x in degrees. The period of the graph in radians is 2π, but the graph using degrees has the period stretched out to 360, resulting in much smaller values of the slope (derivative) due to the horizontal stretch.

“So it’s not correct to have that factor in front of the derivative… I agree with you that radians are the preferred unit, but some of your arguments are fallacious and show a clear bias.”

Whoa there, cool your jets with that “your arguments are fallacious and show a clear bias” stuff. What is this, an internet comments section? That’s not how mathematicians should talk to each other. Plus, as you shall soon see… that factor really SHOULD be in front of the derivative! Here’s why:

In your proof, ultimately what you did was consider the usual sin(x) function (which interprets its argument as an angle in radians and returns the sine of that angle). You took its derivative with respect to x, incidentally making a substitution of cy for x along the way, where y is in degrees and c is the conversion factor, and you got cos(x). No one disputes that d/dx sin(x) is cos(x) when x is in radians, but that doesn’t answer the question of what the derivative of sin(x) would be if the argument were in degrees, not radians.

So let’s see what happens when we define sin(x) in terms of degrees rather than radians.

For clarity’s sake, we’ll rename the sin function as we know it radianSin(x) (since it’s the function which interprets its argument as an angle in radians and returns the sine of that angle). And let’s define a new function degreeSin(x) to be the function which interprets its argument as an angle in *degrees* and returns the sine of that angle. (Define radianCos(x) and degreeCos(x) similarly.)

So, for example, radianSin(pi/2) = 1, and degreeSin(90) = 1. Also, degreeSin(x) = radianSin(pi/180 x).

I claim that:

d/dx radianSin(x) = radianCos(x)

and

d/dx degreeSin(x) = pi/180 degreeCos(x).

The first claim just states that the derivative of the usual radian-based sin function is the usual radian-based cos function, which I’m sure you already believe.

Here’s the proof of the second claim:

d/dx degreeSin(x)

= d/dx radianSin(pi/180 x)

= pi/180 radianCos(pi/180 x)

= pi/180 degreeCos(x).

Behold! The factor stays!

Okay, so that proves it, but it doesn’t tell you intuitively why using radians eliminates the need for a scale factor. Here’s a sketch of the reason:

Imagine standing at the origin and swinging a unit-long rope with a tin can tied at the end counterclockwise around your head. The path of the can traces out the unit circle, and its position vector at a given angle theta is (cos(θ), sin(θ)). (You can think of the string as its position vector.)

Here’s the question which this all revolves around (no pun intended): what’s the can’s velocity vector? In other words, what is (d/dt cos(θ), d/dt sin(θ))?

Well, the direction of the velocity vector is easy enough to figure out: the can is traveling tangent to the circle, so its velocity vector is rotated 90° counterclockwise from the string. So the can is moving in the direction of the unit the vector (cos(θ + 90°), d/dt sin(θ + 90°)) = (-sin(θ), cos(θ)).

But what’s the *magnitude* of the velocity vector? Well, it depends how fast you’re swinging the can (i.e. what’s dθ/dt?). It would be *nice* if the velocity vector was also a unit vector – that is, if you swung the can at just the right speed such that its velocity vector was the same length as the string: 1 unit. That way, (d/dt cos(θ), d/dt sin(θ)) = (-sin(θ), cos(θ)). Otherwise, if you spin too fast or too slow, the velocity vector will be (-sin(θ), cos(θ)) multiplied by some scale factor greater than or less than 1 (like, for instance, pi/180).

But the velocity vector can only be a unit vector if you’re swinging the can at one unit of distance per unit of time: that is, if the can moves exactly one “string length” each unit of time.

The string length is the radius of the circle, so you must turn one radian per unit of time to make the can move one string length per unit time. In other words, when dθ/dt = 1 (or θ = t), = (-sin(θ), cos(θ)).

So the components are equal: d/dt cos(θ) = -sin(θ) and d/dt sin(θ) = cos(θ).

Tada!