"tangent" (the trig ratio) vs "tangent" (the geometric concept) Do the terms "tangent" (the trigonometric ratio), which is defined as "opposite-side / adjacent-side", and "tangent" (of the curve), which is a line that touches a curve at a point, have the SAME meaning? If yes, how? Also: Is this same for "secant" and "cosecant"?
 A: No, they are not the same.
The tangent function $\tan$ is a function defined on numbers: you put in one number (namely an angle) and obtain another number (namely the quotient between opposite and adjacent leg in a right triangle with the given angle). The property of a line to be tangent to a given curve is something different. As is the operation of finding the tangent to a curve at a given point on the curve. The input here is a curve and one point on it, the output is a line. Since a line is not a number (at least in general), the two things are different.
That doesn't mean the two are not related. The post What reasoning is behind the names of the trigonometric functions “sine”, “secant” and “tangent”? pointed out by Michael Seifert contains a picture which demonstrates how various trigonometric functions can be seen as lengths related to a right triangle of unit length hypothenuse. There the value of the trigonometric tangent function is indeed the length of a certain segment along the tangent of the unit circle around one corner. But being related is not being the same.
