Are trigonometry functions Ratios or Distance?

Question

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?

Answer 1

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.

Answer 2

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 $\sin \theta$.

Answer 3

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.

Answer 4

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.

Answer 5

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 ($R=1$).

Answer 6

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.

Answer 7

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.