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I think this is just something I've grown used to but can't remember any proof.

When differentiating and integrating with trigonometric functions, we require angles to be taken in radians. Why does it work then and only then?

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Degrees aren't really numbers that you can do calculations with. Radians, on the other hand, are –  JChau Mar 21 at 8:40
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A nice discussion: teachingcalculus.wordpress.com/2012/10/12/951 –  Siminore Mar 21 at 8:48
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I read somewhere on MO, "Degrees are from the watcher's perspective, radians from the walker's". –  geodude Mar 21 at 8:49
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$\sin x \approx x$ for small $x$ is only true in radians. –  Hans Lundmark Mar 21 at 9:15
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To define degrees you have to pull the number $360$ as an artificial unit out of a hat, whereas $2\pi$ is already there. –  Christian Blatter Mar 21 at 9:18

8 Answers 8

up vote 21 down vote accepted

Radians make it possible to relate a linear measure and an angle measure. A unit circle is a circle whose radius is one unit. The one unit radius is the same as one unit along the circumference. Wrap a number line counter-clockwise around a unit circle starting with zero at (1, 0). The length of the arc subtended by the central angle becomes the radian measure of the angle.

From Why Radians? | Teaching Calculus

We are therefore comparing like with like the length of a radius and and the length of an arc subtended by an angle $L = R \cdot \theta$ where $L$ is the arc length, $R$ is the radius and $\theta$ is the angle measured in radians.

We could of course do calculus in degrees but we would have to introduce awkward scaling factors.

The degree has no direct link to a circle but was chosen arbitrarily as a unit to measure angles: Presumably its $360^o$ because 360 divides nicely by a lot of numbers.

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I prefer this answer because it gets at the heart of why radians are unitless, not just that they're good because they are. –  Almo Mar 21 at 15:00
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I read that "360" comes from how a base 60 numbering system was used in babylon –  Joe the Person Mar 21 at 22:17
    
Excellent answer, this is exactly what I was looking for. –  naslundx Mar 21 at 23:55
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360 was also used in China or India (or both?) in part because there are roughly 360 days in a year. –  R R Mar 22 at 0:11
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@Alom: But being unitless is not the point. Degrees are also unitless since they are a scalar multiple of a radian. The point is that one wants to avoid a multiplicative constant in derivatives, just like the exponential function $e^x$ is selected for this property. –  Marc van Leeuwen Mar 22 at 10:31

To make commenters' points explicit, the "degrees-mode trig functions" functions $\cos^\circ$ and $\sin^\circ$ satisfy the awkward identities $$ (\cos^\circ)' = -\frac{\pi}{180} \sin^\circ,\qquad (\sin^\circ)' = \frac{\pi}{180} \cos^\circ, $$ with all that implies about every formula involving the derivative or antiderivative of a trig function (reduction formulas for the integral of a power of a trig function, power series representations, etc., etc.).


Added: Regarding Yves Daoust's comment, I read the question, "Why does it work [if angles are taken in radians] and only then?", as asking, "Why do the derivative formulas for $\sin$ and $\cos$ take their familiar form when (and only when) $\sin$ and $\cos$ are $2\pi$-periodic (rather than $360$-periodic)?" If this interpretation is correct, and if one accepts that one full turn of a circle is both $360$ units of one type (degrees) and $2\pi$ of another (radians), then the above formulas are equivalent to $\sin' = \cos$ and $\cos' = -\sin$, and (I believe) do justify "why" we use the $2\pi$-periodic functions $\cos$ and $\sin$ in calculus rather than $\cos^\circ$ and $\sin^\circ$.

Of course, it's possible naslundx was asking "why" in a deeper sense, i.e., for precise definitions of "cosine and sine in radians mode" and a proof that $\cos' = -\sin$ and $\sin' = \cos$ for these functions.

To address this possibility: In my view, it's most convenient to define cosine and sine analytically (i.e., not to define them geometrically), as solutions of the second-order initial-value problems \begin{align*} \cos'' + \cos &= 0 & \cos 0 &= 1 & \cos' 0 = 0, \\ \sin'' + \sin &= 0 & \sin 0 &= 0 & \sin' 0 = 1. \end{align*} (To say the least, not everyone shares this view!) From these ODEs, it's easy to establish the characterization: $$ y'' + y = 0,\quad y(0) = a,\ y'(0) = b\quad\text{iff}\quad y = a\cos + b\sin. $$ One quickly gets $\cos' = -\sin$ and $\sin' = \cos$, the angle-sum formulas, power series representations, and periodicity (obtaining an analytic definition of $\pi$). After this, it's trivial to see $\mathbf{x}(\theta) = (\cos \theta, \sin \theta)$ is a unit-speed parametrization of the unit circle (its velocity $\mathbf{x}'(\theta) = (\sin\theta, -\cos\theta)$ is obviously a unit vector). Consequently, $\theta$ may be viewed as defining a numerical measurement of "angle" coinciding with "arc length along the unit circle", and $2\pi$ units of this measure equals one full turn.

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Does this really explain why it "works" with radians ? –  Yves Daoust Mar 21 at 11:07
    
If radians aren't really units, why do we have things like "rad/s" in physics? –  Lazar Ljubenović Mar 21 at 11:51
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@LazarLjubenović for convenience and transparency. Just like we use $N\cdot m$ for torque instead of $J$, which is equivalent. –  Ruslan Mar 21 at 11:57
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Ruslan, that's begging the question. If you treat angles as a dimension, then torque is N m / rad while energy is still N m. There are valid reasons but it isn't as clear-cut as it's sometimes made out to be. –  Doug McClean Mar 21 at 12:13
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@Ruslan: No, how can you say that on a Mathematics site! There is a world of difference between N⋅m and J; namely that one measures the magnitude of a pseudo-vector, and the other the magnitude of a vector. –  Pieter Geerkens Mar 22 at 2:42

It really comes down to the following limit: $$ \lim_{x\to 0} \frac{\sin(x)}{x} = 1 $$ Or in other words, "$\sin x \approx x$ for small $x$". As a consequence, we have $$ \frac{d}{dx}\sin x = \cos x, \qquad \frac{d}{dx}\cos x = -\sin x $$ For any other choice of angular unit, these derivatives require some sort of coefficient (such as $\pi/180$). In this sense, radians are the "natural" unit for an angle, as far as calculus is concerned.

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I am afraid that this is a circular argument, as the given limit is computed by the l'Hospital rule or Taylor expansion. –  Yves Daoust Mar 21 at 11:02
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@YvesDaoust that is very much not the case. This limit is computed using the squeeze theorem, and used to find the derivative. In fact, notice that $$ \lim_{x \to 0} \frac{\sin x}{x} = \lim_{h \to 0} \frac{\sin(0 + h) - \sin(0)}{h} $$ –  Omnomnomnom Mar 21 at 11:10
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@YvesDaoust see this answer –  Omnomnomnom Mar 21 at 11:28
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@YvesDaoust nothing about this is a circular argument. I am claiming that if we define sin and cos to be the geometric functions on angles measured in radians, then all the nice analytic properties of sin and cos can be derived. If we chose a different unit for angles, we would get different analytic properties. Each of the properties I've given can be derived from the geometric definitions of sin and cos. –  Omnomnomnom Mar 21 at 16:09
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The function $(\cos, \sin): \mathbb R \to S^1$ can be defined as a parametrization of the circle by arclength, starting at $(1, 0)$ for $t=0$, and moving counterclockwise. Of course you need a little calculus to make sense of this definition, but nothing circular (other than, you know, the circle itself). –  Dustan Levenstein Mar 21 at 16:46

Assume that the formula $\sin'_r(x)=\cos_r(x)$ is true for some angular unit, let "$r$". For another angular unit, let "$d$", there is a conversion factor, let $\lambda_{d\rightarrow r}$, and we can write:

$$\sin_d'(x)=\sin_r'(\lambda_{d\rightarrow r} x)=\lambda_{d\rightarrow r}\cos_r(\lambda_{d\rightarrow r} x)=\lambda_{d\rightarrow r}\cos_d(x).$$

So the derivation formula can only be simple ($\lambda=1$) for the specific angular unit $r$, which we use to call radians.

But how do we know how much is a radian ?

Using $\sin_r'(x)=\cos_r(x)$ (and in turn $\cos_r'(x)=-\sin_r(x)$) allows to derive various Taylor-McLaurin series expansions, among which that of the arc tangent, and eventually leads to the Gregory-Leibnitz formula. This defines the constant $\pi$ and shows that an eighth of a turn (angle of the isosceles right triangle) is $\frac{\pi}{4}$ radians, equivalent to 45 degrees (by definition of the degrees).

In the end, $\lambda_{d\rightarrow r}=\frac{\pi}{180}$ and $\sin_d'(x)=\frac{\pi}{180}\cos_d(x)$.

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Can we be sure that $sin'(r) = cos(r)$ (where $\lambda = 1$) actually has a solution, or do we just try and see that it actually does? –  naslundx Mar 22 at 22:40
    
Actually, this is the way to define the $sin$ and $cos$ functions analytically. Starting from $dy/dx=y$ in complex numbers, we integrate $dy/y=dx$ and define the logarithm $ln(y)=x$ and its inverse, the exponential $y=e^x$ (such that $e^x:=e^{Re(x)}(\cos Im(x)+j.\sin Im(x)$). This defines both $e$, the base of natural logarithms, and $2\pi$, the "base" of "natural" trigonometric functions. –  Yves Daoust Mar 23 at 9:41

Simple answer: radian is not really a unit, it's an absence of one. Degree, on the other hand, is. Working with dimensionful quantities in calculus is the last thing you would want to do (unless you're into kinky things) :)

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As someone pointed out, degrees are also unitless since they are just a scalar multiple of radians. –  naslundx Mar 22 at 23:15

In Calculus, sine and cosine are defined via the exponential function, meaning that $ \cos x = \mathrm{Re}\{e^{ix}\}$ and $ \sin x = \mathrm{Im}\{e^{ix}\}$ and as you know, $e^0 = e^{i2\pi} = 1$ which means that $360^{\circ}$ which is the full circle corresponds to $2\pi$. For further reference see here.

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That's not the original definition though. –  naslundx Mar 23 at 8:54
    
I know of two possible definitions: using a right-angled triangle or, what I thought of, using the unit circle. If you think of the unit circle as $e^{iz}, z=0..2\pi$ in the complex plane, sine and cosine can exactly be defined like I did. Then, connection between degree and radian become clear. –  namsap Mar 23 at 21:16

It is similar to why SI units are used for scientific calculations.

Many people will notice some quantity which they want to measure, for example length or angle. They will come up with repeatable ways of measuring these, designed to be easy to perform. For example the cubit and the foot are based on body parts. Degrees were chosen because they make division easy for many numbers (this is closely related to our measurement of time). The Babylonians used such number systems extensively.

After a long time, some pattern may be spotted. For example, we can calculate the energy used to move something by multiplying the force applied over the distance travelled. This lets us say things like "Energy in calories is proportional to a force in pounds multiplied by a distance in feet", but it doesn't give us an equation. To turn this into an equation requires a constant of proportionality, which will usually be difficult to remember because the units were chosen arbitrarily.

Later on, some bright spark will create a new unit, defined by these patterns. For example, in SI units the energy in Joules is equal to the force in Newtons multiplied by the distance in metres. The constant is 1 by design.

The same is true of radians. Degrees get horrible numbers everywhere because 360 divisions is arbitrary. We can make the equations nicer by changing the number of divisions to a pattern, like Tau (the circumference of a circle / its radius). A system with Tau divisions, instead of 360, makes all of the equations nicer, just like SI units do. That system is Radians.

Note that Tau = 2 * Pi, since Pi is circumference / diameter and diameter = 2 * radius.

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SI units are up to humans to choose and define as we will. Pi defines itself outside of a human's choice. –  R R Mar 22 at 0:17
    
The vast majority of SI units are not up to humans, since there are only a few degrees of freedom in the system. If humans define the metre, second, Coulomb and kilogram, then we 'automatically' get definitions for Newtons, litres, Joules, Watts, Hertz, Bequerels, Amperes, Volts, etc. without any choice. The same happens with radians, except there are no degrees of freedom; as you say, "Pi defines itself". –  Warbo Mar 24 at 9:35

Another perspective, in my opinion the most "proper" one:

degree is just a named numerical constant that equals ${}^\circ=\frac{\pi}{180}$. So when you read $180^\circ$ you are actually multiplying by that constant! In a similar fashion, conversion into degrees just means that you divide & multiply by a degree and carry out the division, leaving the multiplication unevaluated. In that sense, the degree is just like SI prefixes (kilo, mega, etc.). It just makes the notation more convenient, but it ultimately has no deeper meaning.

The trig functions themselves are, as all other functions, defined for pure number arguments. For instance, through power series $\sin x=\sum\frac{(-1)^n x^{2n+1}}{(2n+1)!}$, and there are no degrees implied in these formulas, it doesn't even imply an angle. It is just a function, it takes any numerical argument.

"Angles" may also appear in formulas outside trig functions (also common in physics). In that case, you cannot survive simply by saying that you used special degree-versions of the trigonometric functions. For instance, the function $${\rm sinc}(x)=\frac{\sin x}{x} $$ take both $x$ as pure numbers.

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