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$$ \begin{array}{ccc} \sin{(\theta+180^{\circ})}=-\sin{\theta} & \cos{(\theta+180^{\circ})}=-\cos{\theta} & \tan{(\theta+180^{\circ})}=\tan{\theta} \\ \sin{(\theta+\pi)}=-\sin{\theta} & \cos{(\theta+\pi)}=-\cos{\theta} & \tan{(\theta+\pi)}=\tan{\theta} \end{array} $$

If I compare them, I will get $\pi=180^{\circ}$. Why? Isn't $\pi=3.142\ldots $? Can anyone prove this?

Rajdeep Sindhu
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Chin Huan
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    The degree and the radian are two different measurement units for angles. So $\pi$ is number and does not equal $180^o$. But $\pi$ radians does equal $180^o$ – Warren Hill Feb 16 '15 at 10:46
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    $1\text{ degree }=\frac{\pi}{180}\text{ radians}$. $1\text{ radian }=\frac{180}{\pi}\text{ degrees}$. – barak manos Feb 16 '15 at 10:55
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    Generic question about different units. Is 1 pound = 454 grams? – smci Feb 16 '15 at 12:17
  • Needs units. Degrees to radians conversion. There are 2π radians in a circle. There are 360° in a circle. Same sums apply, but for certain types of problem, working in radians 'works better'. – Sobrique Feb 16 '15 at 13:17
  • Sometimes we write $1.23^c $ to mean $1.23$ radians. This is usually omitted for radians that are multiples of $\pi $. – Karl Feb 16 '15 at 16:23
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    It might be helpful to you to understand more clearly what an angle really is. Imagine a slice of pie. The angle in radians is the length of the curved side divided by the length of one of the straight sides. "Radians" are not actually a unit at all because length divided by length is dimensionless, but it is convenient to treat radians as a unit. With this definition it is easy to see why an angle of pi radians is a angle of 180 degrees. Also, one radian is just about the right amount of pie to eat, which is a nice property. :-) – Eric Lippert Feb 16 '15 at 23:16
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    @smci But without the units on one side. Is 1 pound = 454? – user253751 Feb 17 '15 at 01:20
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    @immibis: ok, but radians are unitless. Hence a number used as angle measure (without units) is understood to be radians. – smci Feb 17 '15 at 05:55
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    @smci Or even closer: 1 pound = 454 g. Is 1=454? – mrf Feb 17 '15 at 09:45
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    @mrf: that's exactly what I asked 6 posts above ^^^^^^ (The answer of course is that units introduce a conversion factor) – smci Feb 17 '15 at 12:07
  • Size or magnitude of any physical quantity can be measured in different ways using different units of measurement. The size of the very same turning around amount can be $\pi$ in radian measure, 200 in grad measure and $180^0$ in degree measure or $ 180 \cdot 60 \cdot 60$..in seconds of arc measure. – Narasimham Feb 17 '15 at 12:59
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    The way I usually think about it is like this: π is a real number, and whenever one writes 2π that is understood as a product of two real numbers. Similarly I see ° as denoting a different real number, and 360° meaning the product of two real numbers. Then considering that 2π = 360°, one will find that ° would be the number π/180. This might not be the standard interpretation of the ° symbol, but so far I haven't noticed any scenario were it produced any inconsistencies. – kasperd Feb 17 '15 at 22:31
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    Does it make sense to say that degrees and radians are both units, even though angle is a dimensionless quantity? A distinction between units and dimensions appears to be needed here. You need a conversion factor when comparing measures of different units. Dimensionless is just a special case of dimensions, and not relevant. (The pound-mass vs grams example is the same as the degrees vs. radians case.) I think the main source of confusion is that radians are implicit for measuring angles. That's just a typographic / notation convention. – Peter Cordes Feb 18 '15 at 07:32
  • @PeterCordes - you are absolutely right. The only difference between radians and degrees is that radinas are implied if the "unit" is left out, and degrees must be explicitly specified. This is purely notational convention. There is absolutely nothing intrinsically different about them, they both express the exactly same coefficient. – Davor Feb 18 '15 at 08:07
  • @kasperd I like that view. It also makes it easy to see and remember that for solid angles, a so-called square degree means the number $\left( \frac{\pi}{180} \right) ^2$ (and a steradian means unity, the number one). – Jeppe Stig Nielsen Feb 18 '15 at 21:59
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    "radian" is just a silly way of writing 1. Education systems should really stop with radians, they are confusing the students... after all, even when you say "angle is in radians" you don't normally write "rad" after the number. On the other hand, I might as well say I have eaten 8 radians of apples today, it's completely ok, although unconventional. – orion Feb 18 '15 at 22:08
  • I'm not comfortable with the ideas that radians are dimentionless. Angle is not an arc length; it is a change in direction. The way that we measure that change in direction is by measuring the arc length subtending the rays of direction on a standard circle. For radians, the standard circle has a radius of 1; for degrees, the standard circle has radius of $180/\pi$. – John Joy Mar 02 '15 at 15:23
  • @WarrenHill notes that radians are 'special', and are a strong indicator that the value is an Angle on the number circle, rather than a value on the number line. From a mathematics perspective, all notions of Dimenions are abstracted away, leaving a vary flexible notion of dimenions that is context related. For SI units, the issue is that Length is both a single Dimenions and 3 dimenions - it's the norm of the space. Torque should be in Newton.Metres/Angle(rad), but unfortunately it was abstracted away, so it now has the same units as Work - Newton.Metres. Tres unfortunate. – Philip Oakley Nov 28 '16 at 10:52

12 Answers12

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Not $\pi$ but $\pi$ radians equal $180°$

Leox
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    Are angles not a ratio of lengths, and thus dimensionless? – Keen Feb 16 '15 at 13:33
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    Maybe, but in mathematics one almost never writes "radians". I've never seen $\sin(\pi,\mathrm{radians})=0$ written, for instance, it's just $\sin(\pi)=0$. Or worse $\sin'(x)=\cos(x),\mathrm{radians}^{-1}$. For practical purposes the "radians" unit is just the unitless unit $1$. – Marc van Leeuwen Feb 16 '15 at 13:36
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    just a quick note about WHY $\pi$ rad was chosen to equal 180°, in case it's not obvious: the length of an arc (on the unit circle) of measure $x$ rad is $x$. This is convenient for all sorts of stuff in trigonometry. – MichaelChirico Feb 16 '15 at 13:45
  • @Cory : no, angles can have many dimensions. In this question only, radians and degrees have been mentioned. There are others. Where did you read that angles are a ratio of length? – njzk2 Feb 16 '15 at 14:46
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    @njzk2 The definition of angle as a ratio of lengths is fairly common, and angles are (often) considered dimensionless. On the other hand, dimensionless quantities can have units. – David K Feb 16 '15 at 15:25
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    @njzk2 The lengths in the ratio are generally an arc-length on a circle and the distance from that arc to the center of that circle. (We choose the circle and arc based on what angle we want to evaluate. This ratio turns out to be independent of the radius of the circle chosen--good since we leave that arbitrary.) – Keen Feb 16 '15 at 15:38
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    @DavidK I wonder what a unit means for a dimensionless quantity. It seems that such a unit must itself be a dimensionless constant, which would give a trivial proof for $\pi = \pi$ radians. I would like to know why this is not the case. – Keen Feb 16 '15 at 15:40
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    @Cory: So the angle is basically the length of the arc, normalized by the radius. That would be equivalent to the angle in radians, but I do see how it is in fact dimensionless. Thank you for the explanation. – njzk2 Feb 16 '15 at 15:42
  • @Cory: in what sense is 180 a ratio of lengths? – Martin Argerami Feb 16 '15 at 15:45
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    @Cory: apparently there are a few other examples of dimensionless quantities that have units : http://en.wikipedia.org/wiki/Dimensionless_quantity – njzk2 Feb 16 '15 at 15:45
  • @MartinArgerami $180°$ by itself is simply a quantity. If you obtain it as the result of measuring an angle, you obtain it by measuring an arc-length created by a circle defined to have its center at the point of intersection of the two line segments whose angle you're measuring, then dividing that arc-length by the radius of that circle. – Keen Feb 16 '15 at 15:53
  • I can see how an angle might have dimensions if we measure it in a space where different directions are measured with different units, but I haven't studied that area in depth. Does anyone actually use this approach, and is there any sensible way to talk about angles in such a space? (Maybe I should find/ask a new question if that goes too far outside the scope of this one.) – Keen Feb 16 '15 at 15:56
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    This answer is not correct: it would be correct if the part "not $\pi$ but" were deleted. In fact the number $2\pi$ is literally equal to $360^\circ$. This is the definition of $\circ$. – hunter Feb 16 '15 at 16:14
  • @Cory Units are not "constants", at least not in the sense anything else is a constant. If $\pi = \pi\mbox{ radians}$ has a "trivial" proof, then why isn't it equally true that $\pi = \pi^\circ$ and therefore $\pi\mbox{ radians} = \pi^\circ$? In fact none of these equations is true. – David K Feb 16 '15 at 16:14
  • @hunter Do you mean to say that the area of a circle of radius $\sqrt2$ is $360^\circ$? That is what you have implied. – David K Feb 16 '15 at 16:17
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    @David K that is in fact true. the symbol $\circ$ means $2\pi /360$. See the answer by goblin. – hunter Feb 16 '15 at 16:20
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    @DavidK It seems that 1 radian = 1, whereas $\pi° = \pi^2/180$. That is, the unit ° appears to be a dimensionless constant equal to $\pi/180$. This answer claims that this is not the case, but I would like to see that claim explained, since it appears to be the crux of the question. – Keen Feb 16 '15 at 16:39
  • @Cory OK, my explanation is I have never seen anyone use degrees to measure an area, therefore I find the interpretation of $^\circ$ as a numerical constant inconsistent with usage. What is the evidence/explanation for your claim? – David K Feb 16 '15 at 19:56
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    @DavidK I think the area of a circle with radius $\sqrt{2}$ is indeed $360^{\circ}$, but that this usage falls under the umbrella of "serious stylistic issue" not "incorrect". But I think it's a moot point anyways - in the contexts that degrees do get used, they get used in a sense consistent with $^{\circ}$ being a constant. This doesn't imply we'd expect to see them in every situation where a constant is appropriate. – Milo Brandt Feb 16 '15 at 20:17
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    @DavidK: The official SI brouchure says: "The radian and steradian are special names for the number one that may be used to convey information about the quantity concerned. In practice the symbols rad and sr are used where appropriate, but the symbol for the derived unit one is generally omitted in specifying the values of dimensionless quantities". Thus $\pi=\pi,\mathrm{rad}$ by the official definition of "radian". – hmakholm left over Monica Feb 16 '15 at 20:36
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    @HenningMakholm Elsewhere in the brochure, "... a special name is given to the unit one, in order to facilitate the identification of the quantity involved. This is the case for the radian ... ." So at least according to SI, "radians" does not express angular measure but merely suggests it. I give up. At least this does not actively conflict with actual usage. – David K Feb 16 '15 at 21:57
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    @Leox. If $\pi$ radians isn't dimensionless, how would you define $\text{sinc}$ function which is defined as $\text{sinc}x=\frac{\sin x}{x}$? –  Feb 17 '15 at 15:40
70

It would be reasonable to define that: $$x^\circ = \frac{2\pi}{360}\cdot x$$ in which case yes, $180^\circ$ literally equals $\pi$. Leox's answer is probably a little more correct though.

Addendum. After a bit of thought, I've changed my mind slightly; I no longer think that Leox's answer is correct anymore. To summarize my current beliefs about the issue: $\pi$ literally equals $180^\circ$, both are unitless (as others have argued), and neither degrees nor radians are really units at all (again, as others have argued.) In particular, I think that "radians" and "degrees" are basically systems of conventions, not units like meters or seconds.

Lets discuss this a little. In my opinion, what's really going on is that there is a function

$$\mathrm{AngleInRadians} : \mathbb{R}^2 \times \mathbb{R}^2 \rightarrow [0,\pi]$$

given by

$$\mathrm{AngleInRadians}(v,w) = \mathrm{arccos}\left(\frac{v \cdot w}{\|v\| \cdot \|w\|}\right)$$

and another function,

$$\mathrm{AngleInDegrees} : \mathbb{R}^2 \times \mathbb{R}^2 \rightarrow [0,180]$$

given by

$$\mathrm{AngleInDegrees}(v,w) = \frac{180}{\pi}\mathrm{arccos}\left(\frac{v \cdot w}{\|v\| \cdot \|w\|}\right)$$

Observe that both functions return unitless numbers. So really, degrees and radians aren't units at all; they're not like meters or seconds. They're more like consistent systems of conventions than anything.

If we want to formalize the relationship between these conventions, then $x^\circ$ should be defined as stated in my original answer, as the result of evaluating a function $(-)^\circ : \mathbb{R} \rightarrow \mathbb{R}$ at a (unitless) number $x.$ Explicitly:

$$(-)^\circ : \mathbb{R} \rightarrow \mathbb{R}$$

$$x^\circ = \frac{\pi}{180} \cdot x.$$

It follows that:

$$\mathrm{AngleInRadians}(v,w) = (\mathrm{AngleInDegrees}(v,w))^\circ.$$

Under this convention, statements like $\pi = 180^\circ$ and $\cos(\theta+180^\circ) = -\cos \theta$ are literally true, where $\cos$ is viewed as a mathematical function $\mathbb{R} \rightarrow \mathbb{R}$. So the inputs to $\cos$ are mere numbers; they have no units, and neither do its outputs.

goblin GONE
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    Does π only equal to 180∘ when radian=1? Thank you – Chin Huan Feb 16 '15 at 10:55
  • @Al3, that's one way of looking at it. – goblin GONE Feb 16 '15 at 10:57
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    I agree: the best way to look at $^\circ$ is to pretend it's the factor $2\pi/360$. It's like the symbol for “meter”, which is just an unspecified factor; in the case of degrees we even know the factor. – egreg Feb 16 '15 at 11:38
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    Or another way to look at it is that "sin of an angle in degrees" and "sin of an angle in radians" are two different functions. Then we don't have to worry about whether $180^\circ$ is equal to $\pi$ (or $\pi$ radians) or not. We know they're different ways to represent the same angle, is all. The trouble with using the units alone to distinguish is just that some day someone might write sin(sin(x)). This may not be geometrically significant, but suddenly the units become important. – Steve Jessop Feb 16 '15 at 12:57
  • Your discussion certainly has merit, but I think you should note, that your conclusions stem from the fact, that in mathematics we choose “radians” as the default way to measure the size of an angle and then define the trigonometric functions in terms of it. – bodo Feb 17 '15 at 13:23
  • @canaaerus, actually, I think the conclusions are independent of that fact. Really, there are two different versions of $\cos$, namely $\mathrm{CosInRadians}$ and $\mathrm{CosInDegrees}$, and correspondingly two versions of $\mathrm{arccos}$. Importantly, the inputs to all these functions are pure numbers, not "unit-ed" numbers. – goblin GONE Feb 17 '15 at 13:25
  • Sure, there are these different functions, depending on which convention is chosen. But usually the radians-version is used. You yourself acknowledge this by defining the $(-)°$-function which maps a value in degrees to its counterpart in radians. After all one writes $\sin(π)$ and $\sin(180°)$ instead of $\sin(π\mathrm{rad})$ and $\sin(180)$. – bodo Feb 17 '15 at 13:48
  • @canaaerus, sure. I don't think we're disagreeing. – goblin GONE Feb 17 '15 at 13:56
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    By your reasoning, Leox's answer is still correct. Neither degrees nor radians are "units" in the sense of dimensions. But rather they're annotations/suggestions that the number is used to measure a parameter (in this case, angles). Which incidentally is the reason we invented "units" in the first place. So π radians simply means that the π represents an angle (dimensionless though it may be) – slebetman Feb 18 '15 at 03:50
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    You say π = 180, but π can't = 180 and 3.1415926535897..... I don't think I fully understand your argument. – Travis Feb 18 '15 at 04:43
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    @Wyatt, you may have a browser that isn't displaying the math correctly. There's a degree symbol; it says "180 degrees literally equals pi" not "180 literally equals pi" – goblin GONE Feb 18 '15 at 06:00
  • @goblin that's probably it, can't see the degree symbol. – Travis Feb 18 '15 at 06:03
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    @canaaerus, I think the point I'm trying to argue is that $\sin(\pi \mathrm{rad})$ doesn't mean "$\sin$ evaluated at $\pi$ radians"; rather, it is sloppy notation for "the radians-version-of-$\sin$ evaluated at $\pi$." Angles themselves have no units. To be honest, I think the argument needs a good rewrite. – goblin GONE Feb 18 '15 at 06:05
  • @Wyatt, haha okay. – goblin GONE Feb 18 '15 at 06:05
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    From a physics perspective, I think there's a difference between dimensions and units. You can have units for dimensionless quantities, just like you can have multiple units with different scales for quantities with the same dimension. $E=mc^2$ only gives the right numerical answer if you have your variables in consistent units. I see 1 pound-force = 454g as exactly the same kind of thing as $180^\circ = \pi$ rad. I think there's some confusion due to rad being implicit for angles. You'd have to go out of your way to say "$\pi$ the number", instead of just omitting rad or deg – Peter Cordes Feb 18 '15 at 07:44
  • @goblin would it be possible for you to remove all the fluff from the first part of your otherwise excellent answer? some statements are redundant, all the ad personam there are IMO completely unnecessary - and, although I easily followed the part after the horizontal bar, I still don't get the first part at all; my edit proposal got reject as "too drastic", yet I hope you don't see the idea I propose in the same way. –  Feb 18 '15 at 10:13
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    @vaxquis, the reason for the whole "addendum" thing was that by the time I had actually thought carefully about the issue, my answer had already accrued many upvotes, and I didn't want to misrepresent what those upvotes meant; many of them indicate agreement with my original answer, and not necessarily agreement with the edit. I agree that the answer needs a pretty serious edit; it could be a lot clearer. If I find the time later today, I will fix it. – goblin GONE Feb 18 '15 at 11:04
  • Is radians a "dimension" or a "direction"? If something is rotating once per second, a point one inch from the center will be moving at 2π inches per second in a direction perpendicular to the radius; likewise a point one cubit from the center would be moving at 2π cubit per second in a direction perpendicular to the radius, etc. The difference between torque and energy is that in the former force and displacement are perpendicular, while in the latter they are parallel; the magnitudes of the units, however, are the same. – supercat Feb 19 '15 at 01:08
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    @egreg There is a much, much better and more common comparison to make than meters: the $%$. $ x % = 100 x $. In the same way as degrees and radian, the number remains unitless even in the percentage form. – jpmc26 Feb 20 '15 at 02:03
  • @jpmc26 Really! Although it's $x/100$. – egreg Feb 20 '15 at 07:30
  • @egreg Oh, shoot. Had it the wrong way around. – jpmc26 Feb 20 '15 at 14:43
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    @goblin I have just now stumbled upon this question and your answer, and I thought this was very interesting. About when you say that $\sin\left(\pi\text{ rad}\right)$ is just a way to say "the radians-version-of-$\sin$," Michael Spivak takes the same approach in his textbook, "Calculus" in Chapter 15. He defines a function $\sin^\circ$ to be the "degrees version" of $\sin$ and $\sin^r$ to be the "radians version" and related them by $\sin^\circ\left(x\right)=\sin^r\left(\frac{\pi x}{180}\right)$. He then goes on the using just $\sin=\sin^r$. – Brian Apr 28 '15 at 17:07
28

Strictly speaking, there are two functions that are commonly denoted by $\sin(\cdot)$. The mathematical sine function, $\sin: \mathbb R \to \mathbb R,$ has the real numbers as its domain. That is, the function does not take angles to numbers; it takes numbers to numbers.

The domain of the other sine function is angle measurements; an angle measurement consists of a number and the units of measurement of the angle. Just as a single interval of times or a single linear distances can be written in multiple different ways using various numbers with various units, a single angle can be written in multiple different ways with different units. Hence it is correct to write

$$ 180^\circ = \pi\mbox{ rad},$$

where the symbol $^\circ$ indicates units of degrees and rad is the symbol for radians, or (if you are in a context where it is appropriate to use the "other" function named $\sin(\cdot)$) to write

$$ \sin(30^\circ) = \sin\left(\frac\pi6 \mbox{ rad}\right).$$

On the other hand, in a more "pure" mathematical context using the function $\sin: \mathbb R \to \mathbb R,$ strictly speaking we should write

$$\sin\left(\frac\pi6\right) = \frac 12 \neq \sin(30) \approx -0.988.$$

In practice, the tendency to interpret the notation $\sin(30)$ as $\sin(30^\circ)$ is so strong that if you type sin(30) as input to Wolfram Alpha (for example) it will return $0.5$ as the answer. On the other hand if you put =sin(30) in a cell in some widely-used spreadsheet programs you may be in for a surprise. One just has to be aware of this potential source of confusion (identical names for two different functions) and make sure one uses the correct function in the given context.

David K
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    For example in Matlab, sin(pi/6) = sind(30), so one has to distinguish between radian and decimal representation of the input oneself. – Thomas Feb 19 '15 at 11:12
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    I would say that interpreting sin(30) as $\sin(30^\circ)$ is generally just plain wrong. Wolfram|Alpha does this, but it's only ok because it's nothing but a web interface and shows a clear warning about the degrees-assumption; Mathematica doesn't, nor any other properly designed programming language I'm aware of. Heck, if even Matlab gets it right... – leftaroundabout Feb 19 '15 at 12:51
15

This image below shows how to interpret Degrees and Radians. A full-circle= $360^\circ$.

Quoting directly from wikipedia:

Radian describes the plane angle subtended by a circular arc as the length of the arc divided by the radius of the arc. One radian is the angle subtended at the center of a circle by an arc that is equal in length to the radius of the circle. More generally, the magnitude in radians of such a subtended angle is equal to the ratio of the arc length to the radius of the circle; that is, θ = s /r, where θ is the subtended angle in radians, s is arc length, and r is radius. Conversely, the length of the enclosed arc is equal to the radius multiplied by the magnitude of the angle in radians; that is, s = rθ.

As the ratio of two lengths, the radian is a "pure number" that needs no unit symbol, and in mathematical writing the symbol "rad" is almost always omitted. When quantifying an angle in the absence of any symbol, radians are assumed, and when degrees are meant the symbol ° is used.

Now, if you were to equate Radians and Degrees,then:

$$2\pi \text { radians}=360^\circ$$ $$ 1\text { radian}=\frac{180^\circ}{\pi}\approx 57.3^\circ$$

enter image description here

Source : Lucas V. Barbosa on Wikimedia Commons and Tumblr

MonK
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    +1 for 9gag link. First time I see on of these on this site! – Crazy Yoghurt Feb 17 '15 at 14:23
  • The gif doesn't show degrees. – Potatoswatter Feb 18 '15 at 00:17
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    -1 for 9gag link. It being 9gag is not particularly impressive, despite Crazy's fanboyism. And you've already admitted that link-only answers are not acceptable, so why did you post it? o.O – Lightness Races in Orbit Feb 18 '15 at 00:49
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    Can we insert the image in this answer? I don't know if 9gag allows this, but I'm pretty sure it was uploaded on 9gag from a source without citing it. So it's like is it correct to steal a stealer?. – A.L Feb 18 '15 at 16:16
  • Its not an image its a gif file. I don't know how to add gifs here in SE. I guess our objective should be to "know". As long as the information is correct, I think source like 9gag.com shouldn't be a problem. I said link-only answers are penalized like you did..i never said they are unacceptable. – MonK Feb 18 '15 at 18:43
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    9gag links quickly go stale. This answer will soon be useless. Answers should be self-contained, links are for additional reference and detail. – Matthew Read Feb 18 '15 at 23:10
  • @MatthewRead : I guess this should fix the answer :) – MonK Feb 19 '15 at 04:16
  • I took the liberty of improving the gif quality and crediting the author, LucasVB. I made sure not to edit your description, since this is your answer, after all. :) But you can read Lucas' thoughts on the animation at the linked Tumblr page. – Chris Culter Feb 19 '15 at 04:31
  • @ChrisCulter Thanks for that mate! much appreciated. – MonK Feb 19 '15 at 04:32
9

YES. They are the same.

The are the same in the sense that $12 = 1\ \textrm{dozen}$.

Remember, that an angle is a ratio. Is is the ratio of the length of the arc to the radius. For a circle, that ratio is $2\pi$. For a half circle, that ratio is $\pi$.

If I said that the angle (the ratio) was $0.75$, or $3/4$, or $\textrm{three quarters}$, those are exactly the same.

"Radians" doesn't mean anything, other than the number is being applied to a angle, rather than some other ratio or number. It can always be left off and be perfectly valid.

"Degrees" is a quantity of units, similar to $\textrm{quarter}$ or $\textrm{dozen}$. The only difference is that one degree is not a rational number of units.

Just as you could say 5 dozen mph, you could say 3437.7... degrees mph.

Of course, no one measures in dozens mph, just as no one measures in degrees mph.

But they could.

Paul Draper
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For most branches of mathematics, the most useful definition of angle is the ratio between arc length and radius. As such, the angle is a quotient of lengths and hence dimensionless. To indicate that a certain number is meant as an angle in this sense, you can affix the unit “radian”, but that's just for clarification and therefore often not written down.

On the other hand, for most engineering purposes, the radian is a terrible unit of angle, and the degree is a much more useful one. So you might want to convert between these units. To do so you'd say “$180° = \pi\,\text{rad}$”. But that's the engineering point of view, where you treat “rad” as a unit. If you take the mathematical view, you read that unit simply as one and indeed end up with $180°=\pi$.

You might even go one step further, interpret the $180°$ as a product between the number $180$ and the constant (or dimensionless unit) $°=\pi/180$. In this fashion, things like $x°$ meaning $x\times\frac{\pi}{180}$ have a well-defined meaning as well. You can even plot the graph of some function (e.g. $\sin(x)$), label the ticks on one axis using plain numbers $30,60,90,\dots$ and label that whole axis to represent $x/°$. Which can correctly interpreted as “$x$ divided by that constant” but is usually better read as “$x$ measured in degrees”.

MvG
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  • Even within the realm of mathematics, I would posit that there are many cases where angles would be most usefully measured in units of whole circles than in units of radians, but multiplying by 2π can be done concisely enough that it's no big deal. It's a shame that programming languages never standardized on separate sets of trig functions for radian-based and whole-circle-based angles; even if hardware only supports radian-based angles, a function to compute sin2pi(x) could be faster and more accurate than code which multiplies x by 2π and then computes the sine. – supercat Feb 17 '15 at 20:25
5

This started as a comment, but got rather long.

Degrees and radians are two ways of relating the size of angles. Angles are dimensionless - geometrically similar figures have the same angles regardless of size. However the units with which an angle is measured do make a difference - essentially it comes down to what fraction of the whole circle is represented by the number $1$.

A division into $360$ pieces goes with measuring time in minutes and seconds. The scale factor $2\pi$ relates to the intrinsic geometry of the circle (the circumference is $2\pi$ times the radius), and also turns out to avoid some inconvenient constants in differential and integral calculus of the trigonometric functions.

Though dimensionless, the scale of measurement of angles also comes into particular focus when analysing circular motion, or motion in polar co-ordinates. Again it turns out that the radian scaling leads to formulae without inconvenient constants.

Mark Bennet
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    The important point is that dimensionless isn't the same thing as unitless. – Peter Cordes Feb 18 '15 at 07:35
  • Where degrees break down is when you start to enrich your function space with functions that aren't the classic/periodic trigonometric functions. In particular, $\frac{\sin(x)}{x}$, which occurs naturally enough, e.g. in Fourier analysis, would more cumbersome if you only had (in your function space) a $\mathrm{sind}$ function that took its argument in degrees; you'd have to write $\frac{\mathrm{sind}(\frac{180}{\pi}x)}{x}$. – the gods from engineering Feb 21 '15 at 14:58
  • I suppose $\frac{\sin(\pi x)}{\pi x}$, which is perhaps the more common way the function appears in Fourier analysis, would be slightly less painful to write in terms of a degree-argument sin, namely just $\frac{\mathrm{sind}(180 x)}{\pi x}$, but working at different scales still means different constants show up. – the gods from engineering Feb 21 '15 at 15:27
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Your $\tan$, $\cos$ and $\sin$ examples do not prove anything, even if you precede them with $\forall \theta$.

No one really uses $^\circ$ in any professional mathematics. If you were to define it in some reasonable terms then you need to follow what goblin wrote. And this definition explicitly uses $\pi$. Therefore, yes $180^{\circ} = \pi$.

Lisa
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Met
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$2\pi$ is equal to one turn, but since that is an impractical irrational number, what we do is to take a nice number with a lot of divisors like $360^\circ$ and use it to measure one turn. In this way, instead of using crazy fractions of $\pi$, we can call many common angles with nice integer numbers, but do not forget the $^\circ$!

Basically: $\pi\equiv180^\circ \neq180$.

Note that in the very same way you can define: $2\pi\equiv400^g$, this unit actually exists and is called gradian.

DarioP
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Obviously, $\pi$ is treated as a real number. While in trigonometric functions, it is treated as an angle in radian which is exactly equal to $180^o$.

Note: $\sin(\pi-\theta)=sin\theta$ Here, $\pi (\text{radian})=180^o$

Also note in simple calculations, $\pi-180\approx -176.8584073...$ Here, $\pi$ is treated as a real number ($\pi\approx 3.141592654$)

Harish Chandra Rajpoot
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There are two different ways of measuring angles, both using a circle centered on the vertex of the angle:

  • The fraction of the circle it cuts off. This is traditionally measured in degrees, where $1^\circ$ is $1/360$th of a circle. So e.g. a right angle is $90^\circ$.
  • The distance to travel along the unit circle to get from one edge to the other. So e.g. a right angle is $1/4 \times \text{the circumference of the unit circle}$.

The key point is that we almost always unify these, by defining$ \newcommand{\calc}{\begin{align} \quad &} \newcommand{\op}[1]{\\ #1 \quad & \quad \unicode{x201c}} \newcommand{\hints}[1]{\mbox{#1} \\ \quad & \quad \phantom{\unicode{x201c}} } \newcommand{\hint}[1]{\mbox{#1} \unicode{x201d} \\ \quad & } \newcommand{\endcalc}{\end{align}} \newcommand{\ref}[1]{\text{(#1)}} $ $$ \tag{0} 1^\circ \;=\; 1/360 \times \text{the circumference of the unit circle} $$ And since $$ \tag{1} \pi \;=\; 1/2 \times \text{the circumference of the unit circle} $$ we trivially have $$\calc 180^\circ \op=\hint{by $\ref{0}$} 180/360 \times \text{the circumference of the unit circle} \op=\hint{by $\ref{1}$} \pi \endcalc$$

(Note. Since $\;\text{the circumference of the unit circle}\;$ plays such an important role in trigonometry, the symbol $\;\tau\;$ is used as an abbreviation. More traditionally, textbooks still use $\;2\pi\;$.)

On radians (which the OP did not ask about, but everyone apparently wants to talk about): "2 radians" means "a distance of 2 traveled around the unit circle, to measure an angle". So the word 'radians' is there to describe the intent of the number, viz. to measure an angle.

As an analogy: I can say, "From my home to the office is 10 miles as the crow flies." The phrase "as the crow flies" does not change the distance of 10 miles, nor does it change the unit: it describes the intent, to make sure the reader is not confused.

MarnixKlooster ReinstateMonica
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If I compare them, I will get $\pi = 180^{\circ}$. Why?

Beyond the obvious fact that:

It takes $3.14\dots$ or $\pi$ lengths of a radius of a circle1 to rotate a point on it on $180^{\circ}$, i.e. the angle subtended at the centre of a circle is expressed as a length of an arc with units of measurement the radius of that same circle.

I will address a side that is not covered a lot, namely, why exactly $180^{\circ}$ an not any other number (conveniently skipping the irrational to rational "convertion" part)?

The use of the scale1 from $0$ to $360$ for measuring degrees of angles is probably related to the number system2 of the people that first started describing angles, viz. partly for historical reasons, partly for convenience reasons.

My guess is that in the used scale $[0,360]$, $180$ should be thought of merely as an approximation of $\pi$.

1. Which is a interval numeric scale.

2. Obviously not decimal but more likely the Babylonian Sexagesimal,i.e. with base 60 as it is easier to calculate fractions like 1/2, 1/4 etc.

Ziezi
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  • $180$ degrees is not just approximately $\pi$ radians, it is exactly $\pi$ radians. And degrees are not an interval scale; $60$ degrees of arc is twice as large an angle as $30$ degrees of arc (unlike degrees Celsius). – David K Sep 12 '19 at 01:19
  • My comment addresses the fact that a rational number, $180$, is equal exactly to irrational number, $\pi$. Why and how? – Ziezi Sep 12 '19 at 19:00
  • Of course the numbers are not equal. They are not even approximately equal in any meaningful sense. One number is 180, the other is less than 4. But in the only relevant sense (in this context) in which the two numbers relate to each other is when 180 refers to a number of degrees and $\pi$ refers to a number of radians. And the conversion factor from degrees to radians is exactly $\pi/180,$ which is an irrational number, which is how a rational number of degrees can be exactly equal to an irrational number of radians. – David K Sep 12 '19 at 19:34