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I see an image like this:

enter image description here

In quadrant 2, even though the cosine is negative, because the x coordinate goes to the left, the hypothenuse is still positive. Why is this? It seems like the direction of x matters when determining if cos is positive or negative, why doesn't this hold for the hypothenuse too? Why is the hypotenuse always positive?

Why do some people talk about trig using triangles vs unit circles? Is one more correct than the other?

enter image description here

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    $\begingroup$ There's not really any deep reason, it is just a matter of convenience. The easiest way to represent the hypotenuse in terms of coordinates is as the norm of the vector from $(0,0)$ to $(x,y)$, that is, $\sqrt{x^2 + y^2}$, which is always positive. You could also think of the hypotenuse as the vector itself, but there is no particularly natural way to give a sign to this vector. $\endgroup$ – Jair Taylor Dec 14 '17 at 20:28
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    $\begingroup$ Hypothenuse, as any other sides of a triangle, has a length. Length is by definition/convention a positive value. $\endgroup$ – Vladislavs Dovgalecs Dec 15 '17 at 1:26
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    $\begingroup$ The real answer is that the book does not give a good explanation of trigonometric functions. Sadly, I don't have a recommendation for a better book. Many online notes are equally flawed. The bottom-line is that it is meaningless to define trigonometric functions in terms of arbitrary right-angle triangles with the specified angles unless you have already proven that the ratios do not depend on the choice of triangle. It is far simpler to use the $x,y$-coordinates of the point on the unit circle to define the functions. $\endgroup$ – user21820 Dec 15 '17 at 5:54
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    $\begingroup$ Well, ultimately the thing most odd to me is that even though in quad II, the x coordinate is negative, the hypotenuse IN THE SAME DIRECTION is still positive. What's the main intuition behind this? Should hypotenuse not be treated as something with direction? $\endgroup$ – Jwan622 Dec 15 '17 at 10:18
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    $\begingroup$ First of all, a "hypotenuse" is a line segment, not a number, so it makes no sense to talk about it being "positive" or "negative". The length of the hypotenuse is a number and length is always positive. The coordinates of the vertices of a triangle, in a given coordinate system may be positive or negative but the lengths of the legs of the right triangle are also positive. $\endgroup$ – user247327 Dec 15 '17 at 15:15
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The hypotenuse is a number that represents a distance, i.e. the norm of a vector. Distances are always positive.

That said, the figures are a bit misleading, because the green numbers have two different meanings: for the legs of the triangle, they represent the x- or y-coordinate, while for the longest side, they represent its actual length a.k.a. the hypotenuse. All green numbers should be positive if you want to interpret them as distances. I think the figures are just not mathematically sound.

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In many other textbooks, the trigonometric functions of angles outside the range of $0$ to $90$ degrees would be defined with the help of a unit circle, the circle of radius $1$ centered at the origin. Starting at the point $(1,0)$ on the circumference of that circle, we travel some distance counterclockwise around the circle until we reach a point. The path to that point determines an angle (where $360$ degrees equals one full trip around the circle), and the $x$ and $y$ coordinates of that point are the cosine and sine of that angle. The tangent of that angle is the slope of the line through that point and through the origin.

For some reason this textbook took a different approach; although the words "unit circle" appear in the right-hand margin of the excerpt you showed, no unit circle is shown. Instead we have triangles that are too large to relate easily to the unit circle; there is a circle on which the other end of the hypotenuse always lies, but it has radius $2.$

Despite this different way of trying to explain the trigonometric functions of large angles, the book calculates those functions in a way that is consistent with the unit-circle definition. Cosine is always positive when the direction of the given angle points to the right of the $y$ axis; the tangent is positive when the angle points into quadrants I or III, since then the line in that direction has positive slope; and so forth.

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  • $\begingroup$ but why is the hypotenuse for everything positive, even the ones in quadrant 2? $\endgroup$ – Jwan622 Dec 14 '17 at 21:07
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    $\begingroup$ Because if you made any hypotenuse negative in those figures, you would get results that do not agree with everyone else's definitions of sine, cosine, and tangent. What you are looking at is just some textbook writer's attempt to organize the trig functions of large angles; it is not how mathematicians define those functions. $\endgroup$ – David K Dec 14 '17 at 22:37
  • $\begingroup$ Is it wrong to define trig functions using triangles and not the unit circle? $\endgroup$ – Jwan622 Dec 15 '17 at 20:38
  • $\begingroup$ It is only wrong if you come up with answers that are different from the ones on which millions of pages of mathematical, scientific, and engineering work were based. If you do want the answers to be different, please call your functions something else so people won't be misled. Or if you really prefer the definition by triangles, everyone is happy as long as you accept "hypotenuse is always positive because definition says so." $\endgroup$ – David K Dec 16 '17 at 14:04
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The diagram is a little misleading. It uses double-headed arrows everywhere, but some of those arrows are labeled with a negative number, and others aren't.

When calculating $\sin$ and $\cos$ and the inverse functions, the sign of the $x$ and $y$ coordinates matter, which is why those parts are labeled as $-2$, $-1$, etc.

The lengths of these segments, of course, are positive. What is being labeled, though, is the displacement from the origin, which is negative if it's left on the $x$-axis, or down on the $y$-axis.

There's no real need to talk about where the hypotenuse points, since we already know where it points from the $x$ and $y$ displacements. So, it's labeled with a positive value.

I do see your confusion, though. A single-headed arrow pointing away from the origin would have reduced this confusion.

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  • $\begingroup$ whats teh diff btw a double and single headed arrow? Why would a single headed arrow pointing away from the origin to the left help here? Can you clarify? $\endgroup$ – Jwan622 Dec 14 '17 at 21:06
  • $\begingroup$ Just taking a guess as to something that may have contributed to the confusion. A single-headed arrow would have indicated "$1.732$ units, this way" more clearly. Keeping the double-headed arrow on the hypotenuse would be fine. Anyway, if I'm confusing you more with the arrows, just ignore. Hopefully the rest makes sense. $\endgroup$ – John Dec 14 '17 at 22:24
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Some of these may seem like conventions, but here are the reasons I think might explain why the hypotenuse is always positive.

  1. It is a distance, and distances are given by non-negative real numbers.
  2. $r=\sqrt{x^2+y^2}$ and the square root function always returns a non-negative real number.
  3. moving along a given circle, the $x$-value and $y$-value move from positive to negative and back, but each vanishes as it passes from positive to negative and back. The hypotenuse of the triangle never vanishes; so to be continuous, its sign must remain positive.

People talk in terms of unit circles because it is simpler and the situation is general except for a scale (via similar triangles).

Using a unit circle allows us to say $x=\cos(\theta)$ and $y=\sin(\theta)$.

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I learned trigonometry in fourth grade of high school (it was considered an advanced topic, go figure) and, unfortunately, my textbook used the same conventions (without the fancy arrows to mark the lengths): segments parallel to the axes were to be considered positive or negative according to their “orientation”. Needless to say, several of my classmates understood very little about that definition.

Happily for them, the definition was soon dumped and one just had to remember the rigmarole

  • first quadrant: sine positive, cosine positive, tangent positive;
  • second quadrant: sine positive, cosine negative, tangent negative;
  • third quadrant: sine negative, cosine negative, tangent positive;
  • fourth quadrant: sine negative, cosine positive, tangent negative.

Only later, when discussing trigonometric inequalities, the unit circle was introduced, but just as a “graphic device”.

I don't know whether your book tells you how those triangles are built and should be thought of. You consider the ray forming the given angle with the positive $x$-axis, pick a nonzero distance (whatever you want), find the point $P$ on the ray having that distance from the origin and draw the perpendicular to the $x$-axis: in other words, you just consider the coordinates of $P$.

The distance from the origin is just a scale factor which becomes irrelevant because you always consider a ratio of two sides. You simply remove the possible negative signs, compute the ratio and then reinsert a minus sign if you removed just one. But, and here your doubt kicks in, the hypothenuse doesn't carry a sign because it is not generally parallel to one of the axes. Oh, well: what about the straight angle? Yes, there's a contradiction! How do they solve it? The $0^\circ$, $90^\circ$, $180^\circ$, $270^\circ$ and $360^\circ$ angles are special because the triangle is degenerate, so the definition must make particular cases for them.

Not the best way to define the trigonometric functions. Using $1$ as the arbitrary distance and defining cosine and sine as the coordinates of the intersection between the ray and the unit circle is much easier and avoids all doubts about which segments carry a sign and which ones don't. Then one can define the tangent as the ratio between sine and cosine. No special cases (except, of course, for the tangent of $90^\circ$ and $270^\circ$, which doesn't exist).


Maybe some rough history can help. Sine and cosine were first defined using right angles, which limited them to be acute. However the sine and cosine laws were soon discovered and they opened the way for defining sine and cosine also for obtuse angles. The sine of an obtuse angle must be positive in order that the sine law still unchanged for obtuse triangles; to the contrary, the cosine of an obtuse angle must be negative in order for the cosine law to hold unchanged. At the time, negative numbers were just considered “computational tricks”, not having a meaning of their own.

However, describing circular motion in physics made evident that angles should be considered of any value, positive or negative. And then the unit circle become an important tool. It actually doesn't matter what radius you use, as soon as the coordinates are divided by the length of the radius (which is positive by definition): just consider similar triangles. Radius one avoids computing the ratio.

It should be remarked that in ancient times, sine and cosine were not defined through a ratio leg/hypothenuse, but as the length of the suitable leg with a quite large hypothenuse, say $10\,000$ units, so decimal numbers were not needed. However, this length had the bad habit of popping out in formulas such as the addition formulas. At last, mathematicians realized that normalizing the hypothenuse to $1$ would simplify the work with the small drawback of having to use decimal numbers.

I'm not saying that the historical development was precisely along these lines: mathematics doesn't evolve in a linear way. It took long to arrive at easy definitions like the one with the unit circle. But textbook writers often think different.

The “oriented segments” approach is a mixture of the classical method with triangles outlined above and analytic geometry in order to cope with angles larger than the straight angle. Apparently, defining cosine and sine simply as coordinates of a point is considered “difficult” by the author of that textbook.

Just forget it: the hypothenuse is simply a scale factor and, as such, it doesn't carry a sign.

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The issue is that the "hypotenuse" is the distance you travel in the direction specified by the angle to get to the point which defines your triangle. Similarly, the "adjacent" is how far you travel in the positive $x$ direction, and the "opposite" is how far you travel in the positive $y$ direction. A negative value for the adjacent means travelling in the opposite direction, i.e. left of the origin.

So it would make perfect sense to have a negative hypotenuse, but this would mean the angle is not where you expect it to be: a hypotenuse of $-2$ means going $2$ units in the opposite direction to that given by the angle.

Here's what that might look like: $\sin 135^{\mathrm o}=\frac{-\sqrt 2}{-2}=\frac1{\sqrt 2}$.

enter image description here

But you never need a negative hypotenuse in the way that you need negative adjacents or opposites. The positive $x$ and $y$ directions are fixed, but by choosing the correct angle you can always make sure the hypotenuse is going in the positive direction.

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  • $\begingroup$ nice, where is this image from? do you use any math software? $\endgroup$ – RollRoll Dec 15 '17 at 19:31
  • $\begingroup$ What is the 'adjacent'? $\endgroup$ – Jwan622 Dec 15 '17 at 20:50
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    $\begingroup$ @ThePoet Looking at the angle denoted by green circular sector, I guess it's Geogebra. / Jwan622: Adjacent side in right triangle. $\endgroup$ – user202729 Dec 16 '17 at 5:59
  • $\begingroup$ nice, can this be done in R? $\endgroup$ – RollRoll Dec 16 '17 at 16:42
  • $\begingroup$ Yes, it's GeoGebra. I'm sure you can do it in R, but R is still on my "to learn" list :) $\endgroup$ – Especially Lime Dec 17 '17 at 22:28
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Except for the reference angles (which are always non-negative, acute, and measured from the nearest part of the $x$-axis), all the values in these diagrams are signed values. In particular,

  • The labels for the standard position angles start at the positive $x$-axis and are positive for anticlockwise and (although there are no examples of this in the given diagrams) negative for clockwise.
  • The labels for the legs running along the $x$-axis are signed lengths, positive to the right and negative to the left.
  • The labels for the vertical legs are signed lengths, positive for up and negative for down.
  • The labels for the hypotenuses are signed distances from the origin, positive for radially outward.

Of course, since there's no other direction radially than "outward", all hypotenuses are non-negative.

(If you go on to study polar coordinates, some sense of negative radii will be introduced. The method places a copy of the real axis through the origin along the ray of the given angle, with the non-negative half of the axis coincident with the angle's ray. In some ways, replacing "distance from origin" with "signed distance from origin" is analogous to extending trigonometry out of quadrant one by replacing "length of leg" by "signed length of leg". So in addition to all the coterminal aliases for points on the plane, all of these get duplicated to include new aliases for negative radii. For a point having coordinates $(r,\theta)$, the coterminal aliases are $(r, \theta + 2 \pi k)$, for any integer $k$, and the new aliases are $(-r, \theta + \pi + 2 \pi k)$, for any integer $k$.)

It really isn't proper to discuss trigonometry using triangles or the unit circle. One starts with geometric triangles, all lengths are positive. But since you start with right triangles, the other two angles are always acute. If you ever want to talk about an angle of measure greater than $\pi/2$, you will have to generalize somehow. (And talking about angle measures greater than $\pi/2$ is necessary. Not all angles are acute.) We introduce the trigonometric ratios while talking about geometric triangles and realize that these ratios are scale independent -- we could throw any geometric triangle on a photocopier, scale it up or down, and the ratios would be unchanged. This means we can pick some length to be constant in all our triangles and study a normalized subset of all the triangles, but still understand all of them by rescaling. In preparation for the next step, it is convenient to scale all the hypotenuses to be length one.

When we realize that every right triangle can be scaled so that its hypotenuse has length one, we take all the geometric triangles and do this to them and draw them together, with the "angle of interest" (the one we keep putting into sine, cosine, and tangent) in standard position at the origin. We get the unit quarter circle in the first quadrant. This is a direct consequence of Pythagoras's theorem/Euclidean distance. Well, we're certainly wily enough to continue that circle into quadrants two through four. When we do this, all the hypotenuses have length one; we don't get any negative hypotenuse lengths. But the legs point left and right, up and down. They are actually the coordinates of the point on the unit circle through which passes the terminal ray of the angle in standard position. If we extend our definition of "length of side adjacent to the angle" and "length of side opposite the angle" to allow positive and negative values, we get an extension of cosine and sine, respectively, from the first quadrant to the other three.

What good is this? Now our trigonometry functions tell us about all the points on a circle, not just the ones in the first quadrant. We can tell you where the attachment point of the coupling rods is all the way around the steam engine's wheel's rotation, without having to set up a new coordinate system in each quadrant. When the connection point is above the wheel's axle, the vertical coordinate, given by sine, is positive, and when below, negative. A similar observation applies for cosine. There are two basic mechanical motions -- linear and circular. Modeling linear motion with math has been in our wheelhouse since we studied parametrized linear equations in algebra. Now that we have a parametrization (by angle) for circles, we are in a position to model circular motion.

But we're not done. We need a further generalization. We started with trigonometric functions that could only accept acute angles. We generalized to trigonometric functions that could accept any angle from $0$ to $2\pi$ radians. But wheels don't roll around once and then stop. They keep going. And they go backwards. (Most likely, the idea of negative angles, large positive angles, and how each angle has infinitely many coterminal angles was explained when the term "angle" was defined.) We need to extend our trigonometric functions to handle all of these angles. How do we do it? By extending the unit circle to go around and around in both anticlockwise and clockwise directions. It's as if we have an infinite unit radius helix with axial period $2\pi$, but we only view it in parallel projection along its axis onto the plane (so what we "see" is a point doing infinitely many laps around the unit circle).

With that, we finish extending our trigonometric ratios from geometric triangles (acute angles) to an encyclopedia of values of sine and cosine (the unit circle, for angles in $[0,2\pi)$), and finally, to all possible real angles.

So it's not that you explain trigonometric functions in terms of triangles or unit circles. To initially define and then subsequently generalize, you need both. But then you have both tools in your toolbox going forward. If a particular problem is best approached with a geometric triangle, you are able to do so. If it is better approached with a unit circle, you can do that too. And in both cases, you know how the two approaches are related, so you can swap back and forth as needed.

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Maybe it is simply a choice consistent with the usual convention in polar coordinates $$x=\rho \cos \theta$$ $$y=\rho \sin \theta$$ with $$\rho \geq 0$$ and $$\theta \in [0,2\pi)$$

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The hypotenuse is a length (by definition it has a minimum value of 0). While the values of the other 2 sides may be signed (to represent extra information such as traveling South or West for example) a negative value for the overall distance does not add meaningful information. Beside that it would involve choosing the negative square-root over the positive which seems entirely arbitrary.

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