What math is used to find a tangent to a point on a curve?

You may find something to help you in High School Geometry -> Circles -> Properties of tangents

I would say differential calculus.

Here is the “real” problem:

If I have a right triangle and wish to place the intersection of the legs of that triangle, at a “tangent” to a point on a curve - how to make sure* the intersection of the triangle and that point is perpendicular to one of he legs of that triangle?

*proven mathematically

What is the accuracy of the angle of the intersection?