tangent - Latin, tangere, to touch
secant - Latin secare - to cut
[That’s what Chambers 20th C says, anyway]
Euler pretty much nailed modern trig terminology in 1748. And you don’t argue with Euler.
Here’s a great diagram, showing why they are named that way:
And, here’s wiki’s take on history
He was likely the father of Topology… the study of mathematical surfaces.
I love that diagram of a circle of unit radius. It makes everything so clear and actually gives us a rough idea of approximately what size the various trig functions should be.