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Consider a point, say $S(2,3)$. Now here $3$ indicate that $S$ is $3$ units away from x axis. Right?

Now consider what Wikipedia says:

The trigonometric functions cos and sin are defined, respectively, as the x- and y-coordinate values of point A.

This definition of $sin$ and $cos$ is based on unit circle. In this definition sin is defined as Y-coordinate of point $A$ on unit circle. But now what do we mean by Y-coordinate?

Y-coordinate is distance between point $A$ to $x$-axis (Right? ). How could sin or for that matter any trigonometric function can be a distance? Trigonometry functions, for acute angle, are defined as ratios of sides. How could they be "distance"(with unit) in one definition and "ratio" (unitless) in other?

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    $\begingroup$ They're both, if we assume a unit circle radius of 1. Otherwise and in general, they're always ratios. $\endgroup$ – smci Nov 23 '19 at 23:04
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This is a good question. I think it best always to regard the values of $\sin$, $\cos$, and so on as ratios. What allows these values seemingly to be defined as distances in your quotation from Wikipedia is that that definition refers to the unit circle—a circle whose radius is $1$. A similar definition that works for circles of arbitrary radius would be

The trigonometric functions $\cos$ and $\sin$ are defined, respectively, as the $x$- and $y$-coordinate values of point $A$ divided by the radius of the circle.

In this definition, the values are again ratios.

Added: There is a sense in which lengths stated within some system of measurement are ratios too. To say that a tree is $3$ meters high is to say that the ratio of its height to that of the fundamental unit of measurement—whether that's defined by an actual meter stick somewhere, or something else—is $3$. When we quote units with our lengths, we are implicitly carrying along a physical length to be used for comparison. So mathematically, lengths quoted with units are pairs, where the two elements of the pair are

  • the ratio of that which is being measured to the fundamental measuring unit, and
  • the fundamental measuring unit itself.

When we talk about the unit circle, we are abstracting away from that a bit by using the circle's radius itself as a measure, rather than something external. All quoted lengths are now really ratios of lengths defined within the figure itself since the unit of measure is within the figure.

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    $\begingroup$ I've added something to my post, which I hope somewhat addresses this. I would say that Wikipedia's definition is a manner of speaking, not something that is actually wrong. For most readers it would complicate things to make these subtle points explicit. What I'm trying to express is that whenever you talk about the unit circle you are taking about ratios the whole time, without actually saying so. $\endgroup$ – Will Orrick Nov 23 '19 at 12:16
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    $\begingroup$ Wikipedia's definition is a correct definition of the sine and cosine functions over the domain of real numbers. (There are other equally correct definitions.) It is possible to over-interpret that definition and thus be led down a wrong path, which I think is what has happened in the question. $\endgroup$ – David K Nov 23 '19 at 21:25
  • $\begingroup$ @DavidK I agree that one can be flexible about interpretations, as you state in your nice answer. The OP seems to be a high school student with an interest in physics and is therefore probably used to thinking of lengths as dimensionful quantities. I'm guessing this is what led to the question. What I wanted to get across is that it's not necessary to think of lengths that way and that there is no incompatibility in the two pictures Wikipedia gives. $\endgroup$ – Will Orrick Nov 24 '19 at 1:10
  • $\begingroup$ "No incompatibility" is indeed the main point. My comment actually was really a response to OP's comment, and I given the addendum you had already posted, possibly redundant. The more I read this answer, the more satisfied I am that OP selected it to be at the top. $\endgroup$ – David K Nov 24 '19 at 3:04
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    $\begingroup$ @Shekhar Wikipedia is correct, because it says that this is a unit circle (radius 1). It even shows "X=1" and "Y=1" on the graph. Multiplying or dividing by 1 gives the original value, so this value is the same as the ratio. $\endgroup$ – Graham Nov 24 '19 at 10:27
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The output values of the sine and cosine functions are numbers.

A number may be interpreted as a distance or as a ratio, depending on how you use it.

A coordinate also is a number. It's convenient to interpret that number as a distance from an axis while it is still in the context of the list of coordinates of a point; but once you abstract it from that context (as the trig functions do in the unit-circle definition), it is no longer necessarily a distance.

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  • $\begingroup$ So you mean x and y coordinates are not really distances, but they are just numbers? If our purpose is to locate points then we take coordinates as distances but in different context such as in trigonometry we use coordinates in different context. $\endgroup$ – HiterDean Nov 24 '19 at 5:11
  • $\begingroup$ That's one way you can think about it. I find this way helps me sometimes. If you can think of a number as simultaneously a distance and a ratio, that's OK too, as long as it doesn't stop you from using the formulas and methods of trigonometry. $\endgroup$ – David K Nov 24 '19 at 16:05
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That Wikipedia article clearly states that it is dealing with distinct definitions. The definition of $\sin$ and $\cos$ as ratios, appears in the section Right-angled triangle definitions, whereas the definition of $\sin$ and $\cos$ as coordinates, which appears in the section Unit-circle definitions. These are two distinct approaches.

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One way to look at this is that neither are really definitions of $\sin$ and $\cos$, but merely different geometric interpretations. The functions $\sin$ and $\cos$ are abstractly defined as functions $\mathbb R\to\mathbb R$ which happen to satisfy the geometric conditions that a point $A$ of signed angle $\theta$ to the positive $x$ axis has coordinates $(\cos\theta,\sin\theta)$. In particular, the functions themselves are not defined as ratios of anything, nor are they measures of lengths of anything, but are abstractly considered functions where notions of units don't come into play.

So your question

How could they be "distance"(with unit) in one definition and "ratio" (unitless) in other?

Is a non-issue. The trigonometric functions, whichever (equivalent) geometric interpretations you might want to endow onto them, are fundamentally abstract mathematical functions, not physical devices of measure. So questions about units can and should be disregarded in this context.

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Given a circle centered at the origin with radius $R$ we define the the $x$ and $y$ coordinate values of point $A$ as

  • $x=R\cos \theta \implies \cos \theta=\frac x R$

  • $y=R\sin \theta\implies \sin \theta=\frac y R$

therefore both $\cos \theta$ and $\sin \theta$ are ratio, that is dimensionless values, even if in the definition we refer to a unit circle ($R=1$).

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A point on the unit circle can be represented as $(\cos \theta, \sin \theta)$, because $\cos^2 \theta + \sin^2 \theta = 1$.

This means that the horizontal distance from the origin is $\cos \theta$, and the vertical distance is $\sin \theta$.

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The two approaches are equivalent for acute angles. Let $A$ be a point in the first quadrant on the unit circle. Then we can form a right-angled triangle $OAB$, where $B$ is the point on the $x$-axis vertically below $A$. Then the hypotenuse $OA=1$, and the angle $AOB=\theta$, so using the ratio definition, we can show $OB=\cos\theta$ and $BA=\sin\theta$. The converse is even easier to prove.

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  • $\begingroup$ Actually my question is on dimensional nature of trigonometric function. In right triangle definition trig... untions are unit less but in unit circle definition they are considering as distance which have unit. How could this be? $\endgroup$ – HiterDean Nov 23 '19 at 11:48
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    $\begingroup$ No, the unit circle definition does not specify units. $\endgroup$ – Toby Mak Nov 23 '19 at 11:50
  • $\begingroup$ @TobyMak "the unit circle definition does not specify units" How? Please explain. $\endgroup$ – HiterDean Nov 23 '19 at 12:38
  • $\begingroup$ Why do you think the unit circle definition specifies units? $\endgroup$ – Toby Mak Nov 23 '19 at 12:46
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    $\begingroup$ @Shekhar : Generally, $x$ and $y$ coordinates may represent quantities (that is, a value and a unit). They can also be unitless values. In the Wikipedia's phrase "x- and y-coordinate values of point A", if the coordinates are values, those values are being directly referenced; if the coordinates are quantities, their values are being directly referenced. In either case, nothing with a unit is being referenced. $\endgroup$ – Eric Towers Nov 24 '19 at 5:16

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