For mechanical systems, the phase space usually consists of all possible values of position and momentum variables (i.e. the cotangent space of configuration space).
The cosine, cotangent, and cosecant are so named because they are respectively the sine, tangent, and secant of the complementary angle abbreviated to co-.
The tangent space and the cotangent space at a point are both real vector spaces of the same dimension and therefore isomorphic to each other via many possible isomorphisms.
Derived algebraic geometry can be thought of as an extension of this, and provides a natural setting for cotangent complexes and such in deformation theory.
