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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.

Wikipedia explains this clearly, https://en.wikipedia.org/wiki/Versine#Versine_and_coversine

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.

If you like visual explanations, you might enjoy this from 3blue1brown: https://www.youtube.com/watch?v=yBw67Fb31Cs