How can the circular function $\tan(\theta)$ be both a length and a ratio of lengths?

Question

The circular function $\tan(\theta)$ is defined as $\tan (\theta)=\frac{\sin (\theta)}{\cos (\theta)}$. If we look at this in the context of the Unit Circle:

From this picture it can be seen that $\tan(\theta)$ is the $y$-coordinate of the $Q(1,y_1)$, the point on the terminal side of $\theta$ that lies on the vertical line $x=1$. So like $\sin(\theta)$ and $\cos(\theta)$ it signifies a length on the Unit Circle.

My question is: How can $\tan(\theta)$ be a length if it is a ratio of two lengths? The ratio of two lengths is dimensionless, so how does $\tan(\theta)$ signify a length?

EDIT: I guess this goes for $\sin(\theta)$ and $\cos(\theta)$ as well; as they are ratios of two lengths.

EDIT 2: mislabeled sine and cosine on the unit circle

Answer 1

$\sin \theta$ and $\cos\theta$ are themselves dimensionless (since they're ratios of length) but their value represent the magnitude of a length in the unit circle.

Answer 2

Note that all quantities involved: $\theta$, $\cos\theta$, $\sin\theta$, and $\tan\theta$ are just real numbers. They appear as lengths only after you draw a picture: Length of a certain arc on the unit circle (whatever that means), lengths of certain segments in the plane.

It is the introduction of a coordinate system that make this possible: Marking the points $0$ and $1$ on each of the axes allows to interpret not only real numbers $x$, but also expressions like $x^2$, $x^3$, $e^x$, etc. for given $x\in{\mathbb R}$ as lengths. The ancient Greeks could view quantities like $a\cdot b$ or $x^2$ only as pieces of area.

Answer 3

$\tan\theta$ is the vertical offset of the point relative to its horizontal offset (from the origin). In this respect, one can only considered $\tan\theta$ as a length if the horizontal offset is equal to 1 (which is in fact the case with the Unit Circle).

Answer 4

Both $\sin \theta$ and $\cos \theta$ as well as $\tan \theta$ are ratios of lengths and do not specify lengths in themselves. In your diagram, you have interchanged the places where $\sin \theta$ and $\cos \theta$ are written.

You are able to write the lengths of the base and height of the right triangle as $\cos \theta$ and $\sin \theta$ only because the length of the hypotenuse is taken to be $1$. Hence, don't confuse the ratios in general with the length.

Answer 5

All trigonometric functions like $sin\theta ,cos\theta ,tan\theta$ are dimensionless just like $\theta$.They are defined as lengths on the unit circle just the way 1 N(newton) is defined as force on 1 kg object moving at 1 m/s^2

Answer 6

I personally prefer this rendering of the trig segments:

(My avatar ---the logo of my company--- is based on it, and it's the inspiration for my iPhone app, Trigger. My attachment to this figure is significant ... but I digress ...)

I look at this picture as definitional. To me, $\tan\theta$ is defined as the length of a segment that's tangent to the unit circle. (So, the name makes sense!) The ratio that most textbooks give as a(n entirely-unmotivated) definition is actually a theorem. It expresses a proportion based on the similarity of the $\tan\theta$-$1$-$\sec\theta$ and the $\sin\theta$-$\cos\theta$-$1$ right triangles in the figure:

$$\frac{\tan\theta}{1} = \frac{\sin\theta}{\cos\theta}$$

Likewise, similarity with other triangles gives these proportions $$\frac{\tan\theta}{1} = \frac{1}{\cot\theta} \qquad\text{and}\qquad \frac{\tan\theta}{1} = \frac{\sec\theta}{\csc\theta}$$ but I think I'm digressing again. :)

In any case, we resolve the length-vs-ratio-of-lengths conflict by realizing that

Ratio $\tan\theta$ represents length $\tan\theta$, over length $1$.

Conversely, length $\tan\theta$ represents length $1$, scaled by ratio $\tan\theta$.

The same goes for $\sin\theta$, $\cos\theta$, and the rest. They're lengths, and they're ratios-of-lengths.

In practice, you'll rarely think of trig values as lengths, just as you rarely think of the product $xy$ in some formula as the area of an $x$-by-$y$ rectangle. Nevertheless, the duality can be instructive; seeing the trig functions as lengths in the figure above explains a great deal about their properties and relations.

I'll end with a pointer to this answer of mine which describes how the power series for sine and cosine, and for tangent and secant, have geometric interpretations. Not only are the trig values presented at lengths, so is $\theta$ ---naturally, the length of a circular arc--- as well as each power of $\theta$, despite all of these things ostensibly being "dimensionless".