We started learning about critical points on a graph in my calculus class, and it got me wondering about tangents that touch multiple critical points on this graph. For example, the equation $y = sinx$ has an infinite number of critical points with a period of $\pi$, and infinite local maxima with a period of $2\pi$. Any tangent at a point that is a multiple of $2\pi$ will be tangent to every other point that is a multiple of $2\pi$.
Is there some specific term for tangents with this property? Is there a special use for them?
Up until now I don't think I've heard any special "names" for lines of this property.
We can use this property to make observations but I've never heard a name given to the actual line. have a look at Rolle's Theorem it might provide some additional insight into tangents and functions.