tangents/cotangents on unit circles

Question

I was reading here http://www2.clarku.edu/~djoyce/trig/right.html about tangents and cotangents, and if we refer to the second figure, that is the green and yellow one, it states that the tangent for the angle FOB is the line FG, now I was thinking that it would be a tangent that goes through the tangent point B, but it isn't. Can someone please explain to me why not? I read their explanation but I just do not get it. If we look at the angle AOB, then the tangent line goes through the tangent point A and not B, then why would the tangent line of the FOB go through the tangent point F and not the tangent point B. Also, if the angle FOB has a tangent line that goes through tangent point F then what angle has a tangent line that goes through tangent point B?

Kept looking at it for over an hour, after turning my head left and right, up and down, I saw the symmetry. I see what they meant by symmetry now.

Answer 1

First of all, the word tangent in mathematics in English has two different meanings: the tangent line to a curve (or similar but more general concepts in more advance math) and the trigonometric tangent function. That's a bit unfortunate, unlike in other languages, including my native language, where these are two different words. It's not a random coincidence — after all, the values of the tangent function can be found on a tangent line, which explains the origin of the trigonometric use of the word. But it still can be confusing.

And I can see that your problem here is precisely that you're confusing the two meanings of the word. To make matters worse, the explanation on that webpage uses some poor word choices too. Where it says

the tangent of the angle $FOB$ is the line $FG$

it should really be

the tangent of the angle $FOB$ is the (length of the) segment $FG$.

In this sentence, the word tangent refers to the trigonometric tangent function, not to a tangent line. And it's very confusing (and strictly speaking, incorrect) that they then say "line $FG$" when they don't mean that at all. Note that $\angle FOB$ and $\angle FOG$ are exactly the same angle simply because $B$ and $G$ lie on the same ray going from the point $O$. So the quoted sentence simply says that

$$\tan(\angle FOB)=\tan(\angle FOG)=|FG|.$$

And it's true because the circle shown on the picture is a unit circle, so using the right triangle definitions of trigonometric functions:

$$\tan(\angle FOB)=\tan(\angle FOG)=\frac{\text{opposite}}{\text{adjacent}}=\frac{|FG|}{|OF|}=\frac{|FG|}{1}=|FG|.$$