Is there any relation between the tangent to a curve and the trigonometric $~\tan(x)~$ function?

Question

This question has pestered me in the past, on whether the trigonometric identity $~\tan(x)~$ has anything to do with the tangent line to a curve or whether the naming is purely a coincidence and unrelated.

It has recently been brought back to my mind when I was reading about tangent spaces.

Answer 1

You've already got the right answer in a comment but I thought a picture might help:

The tangent of the red angle is the length of the red line, which is tangent to the unit circle.

Answer 2

Tangent

In differential calculus the slope of secant (s) becomes the slope of a special line when intersection points are made to be coincident.. forming a tangent line (t) shown when the point is a double point.

Angle of slope $\alpha_1$ becomes $\alpha_2$. The line and curve have same slope but different curvature.

The tangent straight line has slope equalling $ \dfrac{dy}{dx}= \tan \alpha_2$

From tangential contact's unique appearance probably the name of trig function tangent has been derived or adopted.

Answer 3

The answer explaining it in terms of slopes has a good approach, but a more graphical way of answering your question is using the unit circle. I thought it would be easy to find a video explaining this, but it looks like most videos just define the tangent as the division of sine and cosine, namely $$ \tan \theta= \frac{\sin \theta}{\cos \theta} $$ , including the videos that focus on the unit circle.

However, I found the video titled "Is a tangent to a circle related to the trig function tan?", by someone named Todd Federman, and I think it is a good visual explanation.