  Let $\mathfrak{g}$ be a complex simple Lie algebra and
$U_q\hat{\mathfrak{g}}$ the corresponding quantum affine algebra. We prove that
every irreducible finite-dimensional $U_q\hat{\mathfrak{g}}$-module gives rise
to a family of trigonometric K-matrices, i.e., rational solutions of
Cherednik's generalized reflection equation. The result depends upon the choice
of a quantum affine symmetric pair $U_q\mathfrak{k}\subset
U_q\hat{\mathfrak{g}}$. It hinges on the construction of universal K-matrices
for arbitrary quantum symmetric pairs, and relies on proving that every
irreducible $U_q\hat{\mathfrak{g}}$-module is generically irreducible under
restriction to $U_q{\mathfrak{k}}$. In the case of small modules and
Kirillov-Reshetikhin modules, we obtain new solutions of the standard and the
transposed reflection equations.
