Why is tan(x) denote a line perpendicular to the tangent, not the tangent itself?

Question

I asked a question before, and was directed here instead. :^( Hopefully, someone reading on this thread might still be able to help answer my question.

I am trying to understand how the word "tangent" was decided in trigonometric nomenclature, why did mathematicians choose the name of the exact opposite of the tangent line? In geometry, the definition of Tangent is "a straight line or plane that touches a curve or curved surface at a point, but if extended does not cross it at that point".Geometric definition of Tangent

BTW - In Doug M's response, he has 2 pics of unit circles. (It is nice to have a visual to work with, by the way) On the 1st one, he shows a red line and says the vertical line that intersects with the line radiating from the unit circle is "has a measure of tanx". I get what he is trying to get across with this statement. That is the definition of x/y in a unit circle (1/1=1), but not in every quadrant. Tanx of 135° is negative, there is a sign associated with that measure. TanX is the slope of the line that contains the radius perpendicular to the circle. The way I read Doug's statement seems to only be a true (between 0-90° and at exactly 180°). Of course, I will clearly accept corrections to any wayward logic.

Answer 1

"a straight line or plane that touches a curve or curved surface at a point, but if extended does not cross it at that point"

That is wrong. The tangent line to the graph of $y = \sin x$ at $x=0$ crosses the graph at the point at which it is tangent to the graph.

The blue line in the picture is tangent to the circle.

The angle $\theta$ corresponds to a line passing through the center and crossing the circle at two points. The point where that line intersection the blue tangent line corresponds to a number, which is the tangent of the angle. That is true in either the left or the right half of the circle and either above or below the horizontal axis. At the top and bottom of the circle, you get a line not intersecting the blue tangent line. That can be considered to intersect it at $\infty,$ and that is neither $+\infty$ nor $-\infty,$ but is the $\infty$ that is approached by going in either direction along the line.

Answer 2

It is true now that the tangent is now taught as the slope of a radius.

But that is not the historical understanding of things.

“Slope” is a concept from coordinate (aka Cartesian) geometry.

Trigonometric functions were studied long before coordinate geometry. All six functions were used by the 10th century, long before Descarte.

In Euclidean geometry, the key measures are angles and lengths. And several people here have already indicated a length of $\tan x$ that:

1. Is a length, of a line segment,
2. Which is a tangent to the circle,
3. And is directly related to the angle in question

Names in general have murky origins. Sometimes names make no sense at all. For an example from this answe: Why do we call coordinate geometry “Cartesian” when his name was Descarte.

But here, we have a perfectly fine reason for the name. It is the length of a tangent line segment.

Yes, the tangent function, as defined now, is signed, but we don’t know if the name came before or after that notion of a signed function, and the sign of a tangent can still be explained by giving a direction to the circle (counterclockwise, usually) and talk about the counterclockwise angle and “directional distance” of the circle tangent in the counterclockwise direction.

While that makes mathematical sense, that is probably ahistorical, too.