By @GregLocock

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

Early study of triangles can be traced to the 2nd millennium BC, in Egyptian mathematics (Rhind Mathematical Papyrus) and Babylonian mathematics.
Systematic study of trigonometric functions began in Hellenistic mathematics, reaching India as part of Hellenistic astronomy. In Indian astronomy, the study of trigonometric functions flourished in the Gupta period, especially due to Aryabhata (sixth century CE), who discovered the sine function. During the Middle Ages, the study of trigonometry con Th...