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In a calculus course I took a while ago, we defined an angle between two rays in $\mathbb{R}^2$ sharing a common endpoint as the length of the circle with radius 1, centered at the point of intersection.

Going back to my old notes, I was wondering: Is this definition even well-defined ? Because one has to give additional information at which of the rays forming the angle to start measuring, if I'm not mistaken.

And then there's also the problem of measuring an angle - Why does one use dimension-sounding phraseology, like "degree" or "radian", if at the core of the definition, the measure of an angle, is nothing more than a function, associating every pair of rays sharing a common endpoint a number, by a (strictly mathematically speaking) rather complicated process, of computing the length of a curve ?

Do you know of other definitions of an angle ? It seems to me, like the mathematical object "angle" is getting too little attention - in almost no textbook during the first years of university does it come up (although one needs it for example in complex analysis, for the multiplication of complex numbers), and when it is used, just an intuitive definition of it is assumed.

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    "Because one has to give additional information at which of the rays forming the angle to start measuring" - hence the usual convention of measuring anticlockwise. Of course, one could set up a system where angles are measured clockwise...2011-11-20
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    Many calculus books will mention angles w. r. t. the dot product of two vectors.2011-11-20
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    On the other hand... in the plane, if you can define slopes, then angles easily follow.2011-11-20
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    I hope this will get some good answers. Students of mathematics are cheated on this matter. They are told the analytic definitions of sin and cos without proof that they coincide with what they learned in school. Once they have forgotten about the difference, angles are defined via the cosine and it seems perfectly natural. Only it isn't, because actually cos is just some power series. I have often searched the literature for a definition of angle that is both intuitive and rigorous. Nothing was really satisfying, but closest came the books about Euclidean Geometry, e.g. Geometry by Marcel ...2011-11-20
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    Berger, Geometry by Audin, "Linear algebra and geometry" by Dieudonné and some parts of Bourbaki's General Topology that are quite similar. Their approach is via the orthogonal group of the plane. I am not competent enough to give a summary of that approach, but maybe you find something of worth to you in these books.2011-11-20
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    @Stefan, where do you assert that such cheating happens? I notice that you mention neither any particular schools nor any particular books that commit such an omission. Are you just extrapolating from your personal experience to "students of mathematics" in general?2011-11-20
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    @HenningMakholm: Actually, my personal experience was quite good. I was told that it is not obvious and will not be proven in the lecture that the analytical definition corresponds to the intuition that is acquired at (German) schools. I should have put "cheated" in quotation marks, as I certainly do not mean to complain about books or instructors. As to particular schools or books, I think that no introductory book on calculus gives a satisfying definition of "angle". We can take Baby Rudin, if you like. If you can convince me of the contrary, I would be most grateful.2011-11-20
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    @Stefan, not getting to see a proof that the power series work sounds pretty substandard. Again, on what basis to you claim that such an omission can qualify as a "quite good" experience? I don't know Rudin's book, but my formal introduction to power series was _Complex Analysis_ by Ian Stewart and David Tall, which (of course!) meticulously proves that the power series definition of sine and cosine agrees with the usual geometric notions. -- I will refrain from trying to guess what you would consider a "satisfying definition of angle".2011-11-20
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    @Henning: I detect a certain sharpness in your tone that I don't think I have given cause to. Anyway, what I would consider satisfying is: a rigorous definition of "angle measurement" (by that I mean assigning a real number to every pair of rays starting in the same point) that corresponds to our intuition. A definition of sine and cosine in terms of leg and hypotenuse (or something similar) and a derivation of the analytical properties of these functions starting from that definition. Textbooks don't define the concept of angle measurement at all, so how can they prove anything about it? ...2011-11-20
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    @Henning: in other words, textbooks (and from what I can see over google books the book you mentioned is no exception) may deliver some heuristics to convince the reader that "all is well", which is a good thing no doubt, but they don't even clearly state the geometric notions you claim they prove to coincide with the analytical ones.2011-11-20
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    @StefanWalter You perfectly put in words, in the comments above, the problem I have. Sadly, it seems that the mathematical community has no wish to fix this "gap" of undergraduate education (not even in this forum, since there haven't been any responses so far). Nonetheless it is good to know, that I'm not the only one, who is bothered. If you put your comments in an answer, I would accept it.2011-11-24
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    @HenningMakholm I can only confirm what Stefan Walter said - on page 121, there is indeed, lets say, an argument (calling it a proof would be too much, because the book doesn't say anywhere, what an angle or a radiant it) that the analytical definition of the measurement of an angle coincides with the geometrical one - but calling that "meticulously" would be, at least in my opinion, rather far-fetched.2011-11-24
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    I'd rather leave the question open and wait for someone who can give a real answer. Maybe I will be the one, but not for some time.2011-11-24
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    @StefanWalter: There's a nice geometric interpretation of the power series of sine and cosine (and also tangent and secant; cotangent and cosecant are currently a mystery). http://dlnds.com/bloog/the-geometry-of-the-power-series-for-trig-functions/ . Does this help?2013-02-23

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As soon as we leave the plane into 3D space, any thoughts about clockwise vs. anticlockwise measureing lose their meaning so that we might justify speaking only of angles from $0^\circ$ to $180^\circ$. But let us stay in the planbe for a while.

Angles are simply pairs of rays with a common endpoint, and so far the definitoin poses no problems. But we want to assign a measure to angles that is invariant under Euclidean movements (or preferably only under orientation preserving movements, i.e. translations and rotations), and that is where problems come into play. We have a few special angles:

  • If the rays are the same, we have the zero angle (or the full angle?)
  • If the rays together form a line, we have a straight angle
  • If the rays are orthogonal, we have a right angle (this definition is not circular: two lines $g,h$ are orthogonal if $(g,h)$ is congruent to $(h,g)$)

Clearly, the straight angle is in some sense twice as big as the right angle because we can juxtapose two right angles to obtain a straight angle. We could easily add $\frac1n$ of the full angle to the above list by similar definitions as for the right angle, but these do not belong to the Greek tradition, especially as a method to contruct a given angle with ruler and compasses cannot be given for all $n$, which may be disturbing. ;) Especially, the $1^\circ$ angle cannot be constructed exactly with ruler and compasses so apparently the Babylonians were maybe a bit more inclined to practice rather than theory. And it is their "fault" why we measure in degrees. Why did they not invent radians or simply use a dimensionless proportion? We can guess the answer to the first question: All numerical approximations to $\pi$ are inexact; moreover it is awfully impractical when constructing a house to erect walls at angles like $\approx 3+\frac8{60}+\frac{30}{3600}$ instead of plain $90^\circ$. They could have used the right angle as unit, but you know those babylonians: They simply like to divide by $60$. So probably they took the angle belonging to the hexagon (readily constructed!) and divided it into $60$ equal parts (this would need a rope or the like) to define the degree and were happy that the right angle is an integer multiple thereof. It is only since the time we tend to define the trigonometric functions non-geometrically (by their series or differential equations, for example) that we get rid of units (or note that the radian is the "better" unit and yet redundant).

To measure general angles we need an invariant under movements. It suggests itself to draw a "unit" circle around the vertex $O$ that intersects the rays in $A$ and $B$, say, and then measure the line segment $AB$. However, we frown upon the "unit" and remembering what similarity is, we simply take any circle around $O$ and measure the proportion $AB:OA$. This would give $2$ for the straight angle and $\sqrt 2$ for the right angle. This allows us to impose an order relation among angles and indeed to "classifiy" them: Two angles give the same proportion iff they are congruent. However, we'd prefer angle measures to be additive (as far as possible), that is for three rays originating at $O$, the smaller angles should (usually) add up to the biggest angle. This brings us to the idea (and the problem) of measuring an arc instead of a line segment. This necessarily involves limit processes in a theory that can really only set straight line segments into proportion (and even there has some difficulties with incommensurable lengths). Without limit processes, we can only define the measures of angles which happen to be rational multiples of the full circle, but at least for these the desired additivity of angles (modulo full angle) is given - more or less by definition: $2^\circ + 3^\circ=5^\circ$ because each of these three angles is by definition the corresponding multiple of a $1^\circ$ ur-angle.

In summary, measuring angles is governed by

  • classical Greek aspects, where only constructible angles deserve a specific measure (or simply a name). You won't find measurements of angles in Euclid. And even when he constructs a pentagon, there won't appear a $72^\circ$ angle, but at most "the angle subtending the side of a pentagon".
  • real world practical aspects, where it is important that widespread angles can be easily named with numbers of reasonable size and precisoin. This is the case with the degree, minute, seconds system inherited from the Babylonians. This is at least as cumbersome as the hours, minutes, and seconds of time units (for us decimal people), but then again I've heard about people still using inches, feet and miles for distances. Attempts to replace $90^\circ$ with $100\,\text{gon}$ were fruitless in spite of possible advantages (e.g. $1\,\text{km}$ originally corresponds to $0.01\,\text{gon}$ along the meridian of Paris, which is somewhat similar to one nautical mile corresponding to roughly $1'$ and the utility of this relation for mental calculations is well-known for practical nautics)
  • modern theoretical aspects, where angles are dimensionless and indeed (their measure) not defined before suitable tools like dot product and the trigonometric functions are available.

Fortunately, the main aspect needed e.g in multiplication of complex numbers in polar coordinates (which you mention), does not depend on any specific unit of measuring angles, but only on the additive structure (modulo full angle); also only the addition theorems of $\sin$ and $\cos$ are needed, not their Taylor expansion or the like.