Do functions exist, for which the the cosine transform does not exist, but the sine transform exists?

Question

Is there a real valued function

$f: (0, \infty) \rightarrow \mathbb{R}$

s.th.

$\int_{0}^\infty \cos(kx) f(x) dx$

does not exist as a reasonable function/distribution, but

$\int_{0}^\infty \sin(kx) f(x) dx$

exists (or vice versa)?