Vers- is just from versus "opposite/against". But versine isn't opposite of sine, if anything it's opposing cosine. I too am curious this how this naming convention arose.
It is opposite the sine, it is at a right angle. Look at the circle diagram: the "sinus rectus" goes vertically and the "sinus versus" goes horizontally.
It always bugged me that secant is 1/cosine, cosecant is 1/sine, and cotangent is 1/tangent. Like what does co- mean?
Trigonometry never really clicked for me. I can remember the formulas and such, but I never really understood what they meant. It was an exercise in remembering but never knowing.
The prefix 'co-' stands for complementary. Two complementary angles add up to 90°. The cosine of a angle is therefore the sine of its complementary angle, etc.
For the secant and tangent, it originates from the geometry of the unit circle. The first image in the article explains it well.
