Why is it called the cotangent bundle? We all know that the cotangent of an angle is the tangent of the complement of that angle. What is the etymology of a cotangent bundle? In the sense of mechanics, the coordinates of the tangent bundle are positions and velocities, while the coordinates of the cotangent bundle are positions and conjugate momenta.
What is the meaning of the "co-" prefix in this case?
 A: The relationship between the cotangent and the tangent in trigonometry is something of a red herring here when it comes to cotangent and tangent vectors; the co- there (and in cosine) is an abbreviation of "complement"  (etymonline). However, elsewhere the co-/com- means "together" or "counterpart."  For instance, complex numbers are the "braiding together" of real and imaginary numbers (aside: the calque "symplectic" was created by Weyl because "complex" had too many uses already).
The "co-" for vectors is related to dualization, the process of taking a vector space and considering, as a vector space, the set of all linear functionals on that vector space.  Cotangent vectors (conjugate momenta) are $1$-forms which take tangent vectors as arguments, producing real numbers.  Dualization gives a pairing $\langle \alpha, v\rangle = \alpha(v)$ which can be used to associate a finite basis of vectors with a finite basis of covectors, leading to the association between them.
Co- tends to appear when there is a metaphor of dualization.  Dualization is a contravariant functor in that linear maps $f:V\to W$ when dualized reverse as in $f^*:W^*\to V^*$, so whenever the "arrows are reversed" in the corresponding diagram in category theory, you can be sure you will find a co- describing it: coproduct, colimit, cohomology, etc.
