When we give a proof that the tangent is the sine to cosine ratio of an oriented angle,
$$\bbox[5px,border:2px solid #C0A000]{\tan \alpha=\frac{\sin\alpha}{\cos \alpha}}$$
with $\cos \alpha \neq 0$, we take the tangent $t$ in $A(1,0)\equiv S$ to the circle of center in $O(0,0)$ ad radius $r=1$. See the image
The name tangent has been given because we consider the tangent to the circle of radius $1$ at point $A\equiv S$ or for another reason?
Best Answer
This is a good reason and it is consistent also with the other equivalent definition for the tangent
(credit)
Refer also to The Etymology of Trig Functions.