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I realise that teachers tend to focus on right-angled triangles when introducing trig functions, and for those I can see that the most intuitive approach seems to be starting with the opposite and adjacent sides of a triangle matching the right and bottom of a rectangle. But otherwise, it's always struck me as odd that we go anti-clockwise from the right, rather than clockwise from the top - which gives the same results, but with sine corresponding to the horizontal direction, and cosine on the vertical. This is how bearings are conventionally given, after all, and even in the digital age the analogue clock-face is something almost everyone's familiar with. And it's not so hard to draw a triangle with its point facing downwards, right?

I'm just wondering if I'm missing something here. Would I be doing my students a terrible disservice if I introduced them to trig functions treating them as going clockwise from 12 o'clock?

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    $\begingroup$ How would they react if you drew Cartesian coordinate planes with the positive x-axis going vertically and the positive y-axis going to the right? And then graph a function y=f(x)? You can buck conventional notation, but at some point your students will have to deal with conventions used in other contexts. Point out the differences. Gerhard "If You Dare To Differ" Paseman, 2015.10.29 $\endgroup$ – Gerhard Paseman Oct 29 '15 at 21:28
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    $\begingroup$ Really asking two questions here. First is why the tradition originated and second is whether you should teach it differently. I also wonder about the origin, but I would never let that wonder dictate how I taught students unless it was some type of advanced graduate level proofs or theory course. $\endgroup$ – oemb1905 Oct 30 '15 at 1:22
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    $\begingroup$ In addition, compass directions are also (not surprisingly) measured in a clockwise direction. $\endgroup$ – copper.hat Oct 30 '15 at 2:41
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    $\begingroup$ Might have some luck on MSE. I added a question: math.stackexchange.com/questions/1504483/…. $\endgroup$ – copper.hat Oct 30 '15 at 3:07
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    $\begingroup$ From Wikipedia: "it is known that the systematic introduction of the 360° circle came a little after Aristarchus of Samos composed On the Sizes and Distances of the Sun and Moon (ca. 260 BC)". Note the link between astronomy and using a circle for trigonometry. I mention that because when the solar system is viewed from above the Earth's north pole, the planets revolve anti-clockwise. That is a possible source for the direction of rotation. $\endgroup$ – Todd Wilcox Oct 30 '15 at 14:38

12 Answers 12

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There is no such thing as a natural sign/direction convention until long after the fact. Consider another answerer's comment that anyone using the left hand rule to construct the cross product of two vectors would be mistreated. Of course, the left hand rule is exactly the correct rule to use for the path of electrons in a magnetic field? Why? Because the convention of positive and negative charge was set before anyone knew that the mobile charge carriers had been called "negative". Thanks, Ben Franklin.

Consequently, conventions are made and the ones that achieve critical mass succeed and overwhelm the alternatives. Frequently, it is the ones that can be described as somehow natural. In this context:

  • I teach my trigonometry students that all three pairs of (cos, sin), (adjacent, opposite), and ($x$,$y$) are in alphabetical order.
  • I teach them that angles are measured from the positive half of the first ($x$-) axis in the direction of the positive half of the second ($y$-) axis.
  • Reference triangles have two points on the $x$-axis, in fact, their first two points. That is, start at the origin with the first point, go along the first axis to the second point, then turn at right angles and go parallel to the second axis to the third point. For some students, it even helps to see this as a sequence: "$(0,0), \xrightarrow{x}, 1, \xrightarrow{y} 2$".

Later, I explain to them that bearings are different and for a very practical reason: bearings are measured from the direction compasses point (North). I also warn them that there are many, many mismatched conventions. (Nearly every "reasonable" reference vector has been used in a navigation system at some point in the past 600 years). Although North is a common reference vector now, for astronomy and stellar navigation, it was common to use South as the reference vector, as it was then zero at the same time that the hour angle of a star was zero. Then, once one is in this mode of comparing bearings to times and star positions, it is only sane to measure angles clockwise so that positive offset times correspond to positive offset angles. That is, the conventions of navigation in the past were based on the need to have an easily interpretable table of stars and times, but the conventions of navigation in the present are based on the underlying convention that North is the basis of bearing measurements (which is largely a consequence of gradually narrowing cartographic convention in the last half of the second millenium).

However, none of that explains why angles are measured up from right. Early applications of trigonometry were to astronomy and navigation (see prior paragraph) and architecture. In architecture, the problem is to pile up building materials (either in preparation to construction or placed in the constructing itself). Consequently, one is measuring grades of piled up or placed materials. Then one measures angles in quadrants I or II up from the $x$-axis. One always puts the angles at the origin so that the sine and cosine actually correspond to the coordinates of the third point (projected onto the unit circle) because the practical historical problem is to measure an angle up from the horizon.

One can see this in Euclid's Elements, Prop. 1. The construction of an equilateral triangle from a given line segment AB. The point $A$ is drawn at the lower left, the segment AB proceeds to the right, and the third point is constructed above the segment AB (even though constructing it below is just as easy). That is, the author finds it natural to place the starting segment horizontally with the starting point at the left. (The next few triangle propositions are about isosceles triangles, which are drawn with the first point on a vertical symmetry axis -- i.e., modeling a pile of stuff. I recall reading that this was intended, but I have no hope of dredging up a reference.)

The final reason to do this is so that derivatives work in the expected way in Calculus: a slope of zero does not correspond to a vertical tangent line. Having to unlearn this will be a significant disservice to your students.

Edit: (More on the Calculus reason) Pick a point on a function at which the function is continuous and translate that point to the origin (just to make the rest of this clearer). Positive derivatives at that point correspond to tangent lines lying in quadrants I and III. Negative derivatives correspond to tangent lines lying in quadrants II and IV. Further, a derivative of zero corresponds to a horizontal tangent line. If we arrange angles to be measured in the right-handed orientation from the horizontal axis, then a zero angle coincides with a zero derivative, a positive angle with a positive derivative, and a negative angle with a negative derivative. We can actually finish this correspondence by observing that the tangent of the angle (measured in the standard way) of a tangent line is the slope of that tangent line. That is, the slope is the $\frac{\text{rise}}{\text{run}} = \frac{\Delta y}{\Delta x} = \tan \theta$. And I've seen "the lightbulb" for students when they realize the tangent line's angle's tangent is the slope. (I don't know that this is a valid "Aha!". There's linguistic repetition, but the ideas are really unrelated. But some students seem to make progress with this association.)

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  • $\begingroup$ That's a very good answer, but I'm not sure I follow your last point - could you possibly elaborate a little? $\endgroup$ – Oolong Nov 1 '15 at 10:19
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    $\begingroup$ +1 This is a wonderful and comprehensive answer, well worth the wait. $\endgroup$ – mweiss Nov 1 '15 at 14:57
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    $\begingroup$ Also, I see this is your first post on MESE. Welcome to the site! $\endgroup$ – mweiss Nov 1 '15 at 14:58
  • $\begingroup$ @Oolong : I've elaborated the last paragraph. It's early here; please let me know if it's not quite coherent. (I haven't had my second cup of coffee yet.) $\endgroup$ – Eric Towers Nov 1 '15 at 16:03
  • $\begingroup$ Thanks Eric, I think that makes sense. I'm still a bit unsure on one point though - it seems to me that going clockwise from the top, the tangent actually represents the gradient of a line tangent to the circle. Going anticlockwise from the right, it's instead the gradient of the radial line (the hypotenuse) - which is also nice, but seems harder to relate to the name of the function. Am I confused? $\endgroup$ – Oolong Nov 5 '15 at 17:18
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Regarding the second part of your question:

Would I be doing my students a terrible disservice if I introduced them to trig functions treating them as going clockwise from 12 o'clock?

I think it's important to stress that the convention is just a convention, and there is no intrinsic reason why one convention is better than another. But at the same time, this particular convention is a nearly universal one, and teaching your students a nonstandard convention is roughly equivalent to a language teacher teaching his or her students a dialect that nobody else speaks. As soon as they leave your classroom they will find themselves in a mathematical world that does things differently, and you are doing them a disservice if they are not thoroughly used to the conventions that everybody uses.

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    $\begingroup$ Actually I disagree with your answer based on the reasons I gave in my answer. A lot of mathematical objects that seem tied to the anti-clockwise direction are nowhere near arbitrary but have to do with the way we have chosen to position the coordinate axes on paper. For example signed area is positive for anti-clockwise traversal of a non-self-intersecting polygon in the plane. The relation between the anti-clockwise contour integral around a pole and the residue there as given by the Laurent series is another. Not coincidental at all. $\endgroup$ – user21820 Oct 30 '15 at 13:40
  • $\begingroup$ Seems like we could just as well write complex analysis based on CW-loops. It'd just introduce a few signs here and there. Maybe CCW is just what we're used to. $\endgroup$ – James S. Cook Oct 31 '15 at 1:21
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    $\begingroup$ To me, the most important takeaway from this question is this: "As soon as they leave your classroom they will find themselves in a mathematical world that does things differently, and you are doing them a disservice if they are not thoroughly used to the conventions that everybody uses." +1 $\endgroup$ – Todd Wilcox Oct 31 '15 at 8:53
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    $\begingroup$ I don't think we should be upvoting this so much, since it does not answer the main question. Am I wrong on protocol here? $\endgroup$ – oemb1905 Nov 2 '15 at 2:11
  • $\begingroup$ @oemb1905 If by "the main question" you mean "the question in the title", remember that the title is a just a headline; the question is what's in the body of the post. This particular post asks two questions; the second one asks about teaching, the first one about historical origins. It may be worth noting that it is only the second question that makes this on-topic for MESE; the first question by itself would be better for MSE or HSM. $\endgroup$ – mweiss Nov 2 '15 at 2:42
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My guess is that it comes from drawing complex numbers. Putting real numbers on the $x$-axis from left to right and imaginary numbers on the $y$-axis from bottom to top matches with the way we tend to think (in cultures based on Latin at least). Once you establish that, then the function $x\mapsto e^{2\pi ix}$ takes you anti-clockwise from the positive $x$-axis.

On the other hand, if I'm not missing something, a clock face is based on living in the northern hemisphere.

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    $\begingroup$ Interesting! I wonder if that is indeed how it started. I'm not sure what you mean about 'a clock face is based on living in the northern hemisphere'? $\endgroup$ – Oolong Oct 29 '15 at 21:54
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    $\begingroup$ @Oolong A clock face is based on the position of a shadow moving on a sundial during the day. $\endgroup$ – PVAL Oct 29 '15 at 23:30
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    $\begingroup$ The problem with this answer is that it is definitely false because Euler's identity came long after the definition and use of the trigonometric functions. As in my answer, I'm not saying that this isn't a good reason, but it certainly wasn't the original reason. =) $\endgroup$ – user21820 Oct 30 '15 at 13:36
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    $\begingroup$ @user21820 Is it more about when the trig functions were invented or when the unit circle was developed as a way to understand and represent them? The history of the use of the unit circle is something I've been looking for all morning and haven't yet really found. If it post-dates Euler, then this answer could very well be a very good one. $\endgroup$ – Todd Wilcox Oct 30 '15 at 14:27
  • $\begingroup$ My understanding was that the question was about why we now draw triangles oriented a particular way, not about why sin and cos are defined the way they are. Did those using trig before Euler always draw their triangles the same way? I would assume they spent their time on real-life questions, not school exercises. $\endgroup$ – Jessica B Oct 30 '15 at 15:30
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I would avoid teaching it only as clockwise. If you wanted to teach it that way, I would recommend showing both throughout the process, and pointing out that while clockwise might make more intuitive sense at first, the world will expect you to think of it anti-clockwise later.

The reason I point it out is not a mathematics one. I am an engineer, who deals with things like coordinate conventions every day of my life. Even after doing it for years, coordinate conventions still confuse engineers enough to make machines break and planes crash every year (hopefully simulated machines and simulated planes! We do like to fix these things before they get manufactured!) The topics we have to deal with are so exacting that we oft do not have the spare brain-power to consider what might happen if our coordinate system is wrong.

How bad is it? I work on a simulation where we support multiple frames / coordinate systems because the work done in the simulation is too complicated to do them in the wrong frame. We support, at any time, at least 5 frames and another dozen or so coordinate systems associated to those frames. We handle Euler angles and Tait-Bryan angles, all 1000+ combinations thereof (did you know there were that many? Most people I work with are aware of 2 or 4!). I admit I get a sadistic gleam in my eye whenever anyone asks me to read in some Euler angle data, and I ask "which convention are you using?" It's especially satisfying when I make them go back to the vendors to make sure both vendors supplying data are using the same definitions! (usually they do. Most of our vendors follow the DIS convention, thank goodness)

You can call it a very powerful group-think, which it is, but its important for young engineers to learn the conventions we use, because we will use them. Circular, but true. Initially teach it however your instincts recommend, but please make sure your students are comfortable with the conventional way of doing it before they leave your class. In the best of worlds, a new engineer who is totally comfortable switching conventions is the best of all. However, a new engineer who has an alien convention is a liability until they can be trained to group-think like the rest of us.

And that's where I leave it to you. I'm no educator. I'm just a guy in the field. If you think you can train students to switch conventions in their mind without sweating it, please do so. Minds that can switch on a whim are tremendously valuable. If you think some of the students might have great trouble switching later on, I'd think twice (especially if you think any of those students might enter STEM fields).

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    $\begingroup$ As an amusing aside, we do have one convention we all agree upon in engineering: right handed coordinate systems. Thank goodness for that. I oft joke that we have the "engineering gang sign," the thumb index and middle fingers of the right hand splayed out into a right handed coordinate system. If anyone flipped us the left-handed gang sign, we might murder them. $\endgroup$ – Cort Ammon Oct 30 '15 at 19:45
  • $\begingroup$ You're probably right, but I'm still wondering how much difference it actually makes when that convention does come along. So many things come out the same if you mirror the whole thing around the a line from the top-right to the top left... $\endgroup$ – Oolong Oct 31 '15 at 15:23
  • $\begingroup$ I am truly torn between saying "try it," because that's what learning is all about, and we need to try new things... and knowing that we've pelted the surface of Mars with slugs of what-was-a-rover-at-one-point for smaller convention mistakes ;-) $\endgroup$ – Cort Ammon Oct 31 '15 at 18:17
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It's starting on the positive branch of the real number line; and passing through the positive quadrant first.

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  • $\begingroup$ Agreed - the convention predates complex numbers by many years. $\endgroup$ – oemb1905 Oct 30 '15 at 1:24
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    $\begingroup$ @oemb1905 This answer doesn't really make sense without complex numbers, since it views the 'real number line' as significant. $\endgroup$ – Jessica B Oct 30 '15 at 8:00
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    $\begingroup$ @JessicaB Complex numbers were discovered nearly five thousand years, possibly ten thousand years after the usage of the unit circle, degree measure, arc length, etc., for astronomy. If the question is about the origins of the convention, then I am not sure the work of Hamilton and Gibbs will inform the historical basis for that convention. $\endgroup$ – oemb1905 Oct 30 '15 at 15:56
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Perhaps the convention is rooted in how the big dipper rotates around the north star--in a counterclockwise direction. Also think about how the earth rotates around its axis--in a counterclockwise direction.

Since the origins of trigonometry are inextricably tied to navigation, you would think our modern convention might descend from this.

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    $\begingroup$ Of course, technically the Earth only rotates anticlockwise viewed from above the North pole... $\endgroup$ – DavidButlerUofA Oct 30 '15 at 3:19
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    $\begingroup$ And yet, navigators use bearings clockwise from North instead... $\endgroup$ – Oolong Oct 30 '15 at 8:19
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    $\begingroup$ Longitude on the celestial sphere is measured as eastward along the ecliptic plane from the vernal equinox, which serves as 0. So in the northern hemisphere this is a counterclockwise orientation. $\endgroup$ – user52817 Oct 30 '15 at 18:03
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Question 1 - Why: it has to do with the right-handed coordinate system (Ampere's invention). X and Y are not the only axes, there is a third one- Z. In a right handed CS, if X points to the right and Y points up, then Z must point out of the wall (towards observer). Almost everything in physics, engineering and math relies on the right handed CS. The Cross-product X(cross)Y = +Z, Y(cross)Z = +X, Z(cross)X = +Y, whereas Y(cross)X = -Z. Positive moments are defined following the cross product notation: positive moment (or rotation) about Z axis goes from X towards Y axis. Take a screw for example - rotate it counterclockwise and it goes out of the wall towards the observer (positive Z direction), rotate it clockwise - it goes into the wall (negative Z direction). Question 2 - Since this convention is ubiquitous across all the scientific fields, teaching in a left handed CS will do a disservice to students, since re-learning is more difficult than learning from scratch.

PS I know at least one public clock that goes the other way around.

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I did a google search for "right triangle". The first image I got is

a right triangle

Suitable for viewing angles counterclockwise from East or clockwise from South.

*: I use the compass points for direction, since I find phrases like "clockwise from down" awkward, and did not want to use "counterclockwise from the positive $x$ axis", since sometimes people do have the $y$ axis pointing East and the $x$ axis pointing either North or South

The second image I got was

enter image description here

suitable for viewing angles clockwise from West, or counterclockwise from South.

The twenty-fourth image was

enter image description here

and is the first triangle suitable for having an angle measured clockwise from North.

This is by no means conclusive, but if we were to believe the idea of measuring angles from coordinate axes and extending the trig functions to the whole circle came about as an extension of measuring right triangles, it's plausible that the way we tend to write triangles had some influence.

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  • $\begingroup$ It is, however, also plausible that the way we draw triangles is influenced by the convention that angles should run anti-clockwise from the right... $\endgroup$ – Oolong Oct 31 '15 at 15:26
  • $\begingroup$ @Oolong: The images seem to be mostly split evenly between the first two types: I think the relevant bias is more likely to be that the base of the triangle should be on the bottom than anything else. $\endgroup$ – Hurkyl Nov 1 '15 at 23:40
  • $\begingroup$ I'm not seeing the connection between this answer and the question. What did I miss? You say it's "plausible" but how? Every image is the same when viewed from somewhere else ... $\endgroup$ – oemb1905 Nov 2 '15 at 2:10
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A large source of problems that trigonometry solved was astronomy. Our solar system is "right handed". If you point your thumb "north", your fingers will curl in the direction that the planets revolve around the sun. The orbits of most (not all) of the planets in our solar system are also right handed. It makes sense then to use the right hand rule to describe the direction in which angles increase.

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    $\begingroup$ I see no natural reason to prefer north over south. If you look at the solar system from the opposite side, the planets revolve the other way. It is true that the conventions of preferring north and that of preferring counter-clockwise direction are compatible with each other, but neither choice seems obvious. $\endgroup$ – Joonas Ilmavirta Oct 31 '15 at 23:13
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    $\begingroup$ There is no natural reason, just historical. Astronomy from the days of yore was mostly developed by people in the northern hemisphere who centered things around Polaris, or the "North Star." $\endgroup$ – user52817 Nov 1 '15 at 2:17
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There are many possible reasons why it should be defined that way, but I'm not sure what were the initial motivating reasons. $\def\nn{\mathbb{N}}$ $\def\rr{\mathbb{R}}$

Firstly, $(\cos,\sin)$ is a basis for all the real-valued functions satisfying $f''=-f$. It is then natural to choose the basis elements so that $\cos(0) = 1$ and $\cos'(0) = 0$ and $\sin(0) = 0$ and $\sin'(0) = 1$. Another viewpoint is that this differential equation immediately implies that if they have a Taylor series $x \mapsto \sum_{k\in\nn} a_k \frac{x^k}{k!}$, then the coefficients $(a_n)_{n\in\nn}$ would have to satisfy $a_{n+2}=a_n$ for any $n\in\nn$. If we do not want to invoke the Taylor theorem we could also have defined them by the infinite series, prove that it converges, and then prove directly that it satisfies the differential equation. Either way, we obviously want the basis elements to be $(1,0,-1,0,\cdots)$ and $(0,1,0,-1,\cdots)$.

As a side note, harmonic motion is often started off with nonzero displacement but zero velocity, corresponding to a multiple of $\cos$, whereas $\sin$ corresponds to zero initial displacement but nonzero initial velocity. In some sense, rest is simpler than motion, just as first-order terms in a Taylor series are 'more' important than second-order terms.

Secondly, $\exp(it)$ goes round the unit circle for $t\in\rr$. We can see this from the properties of $\exp$ that follows from the infinite series definition, which is motivated by the differential equation $f'=f$. The other definitions do not reveal the underlying structure. We can then choose $\cos,\sin$ as the $x,y$ coordinates.

In both reasons above, they do not fix a starting point or direction for the path traced by $(\cos,\sin)$ in the plane. What fixes it is the choice of our coordinate system where the $y$-axis is $90^\circ$ anti-clockwise from the $x$-axis, and we plot a pair with the first element as $x$-coordinate and the second element as $y$-coordinate.

In fact, a lot of mathematical objects that seem tied to the anti-clockwise direction are nowhere near arbitrary but have to do with the way we have chosen to position the coordinate axes on paper. For example signed area is positive for anti-clockwise traversal of a non-self-intersecting polygon in the plane. The relation between the anti-clockwise contour integral around a pole and the residue there as given by the Laurent series is another. Not coincidental at all. (I decided to put this paragraph in because amazingly people don't seem to get that my answer explains all these seemingly arbitrary conventions at one go.)

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    $\begingroup$ But why give the basis as $(\cos, \sin)$ rather than $(\sin, \cos)$? The latter seems (to beginners) much more "obvious". $\endgroup$ – mweiss Oct 30 '15 at 13:23
  • $\begingroup$ @mweiss: Simply because we write the first coordinate first. So the question that should be asked is not about $\cos,\sin$ but about the choice of depicting the cartesian plane on paper. $\endgroup$ – user21820 Oct 30 '15 at 13:31
  • $\begingroup$ @mweiss: It's the same reason we list the standard basis elements of $\mathbb{R}^2$ as $(1,0),(0,1)$ just as in the identity matrix rather than the other way around. It won't be the identity matrix if we switch the order but want to keep the same meaning of the matrix action on vectors. $\endgroup$ – user21820 Oct 30 '15 at 13:34
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    $\begingroup$ No, I'm not asking "Why not list $y$ before $x$?". My question is Why does $\cos$ come first? Why not choose $\sin, \cos$ as the $x,y$ coordinates? Then with our current conventions about how we represent the coordinate system on paper, we would follow the proposal in the OP -- i.e. to start at $(0,1)$ and move clockwise. $\endgroup$ – mweiss Oct 30 '15 at 13:44
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    $\begingroup$ But I think your second reason is the real one. The fact that $\exp(it) = \cos(t) + i sin(t)$, together with our conventions about writing complex numbers in the order "real part + imaginary part", argues for thinking about $\cos$ as the "first" function and $\sin$ as the "second". $\endgroup$ – mweiss Oct 30 '15 at 13:45
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I am so interested in finding an answer for the first part of the question that I copy (as community wiki) my "answer" to this somehow the same MSE question, hoping that it helps us to understand the roots of this particular "convention".

This is by no means answering the question as it is. However, It is just to give a historical piece (taken from "The history of Stokes' Theorem" written by Katz) that might come handy when thinking of the first part of the question (CW vs. CCW or ACW).

enter image description here

As it can be seen, Green's theorem as introduced by Cauchy (1846) is undecided about the orientation of the curve.

I repeat that I didn't attempt to answer the question as asked. I just tried to extend the question a bit hoping that it helps us to find the answer (if any) to the original question.

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I'm looking at the unit circle -

enter image description here

and it seems to me that if we deemed Y to be cosine, and X, sine, i.e. flip the graph about the Y=X line, we would achieve your goal. Right?

But, as Hurkyl indicates, the orientation of a 45 degree right triangle results in

enter image description here

and that would appear to be a bit disorienting to students. In the end, it's the combination of Cartesian coordinates being established as well as the 'comfortable' orientation for the 0-90 degree right angle triangles that dictated the placement.

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