To compute the Laplace transform of the Heaviside step function, the function $f(t)$ in front of it has to be shifted to $f(t-c)$ for a Heaviside step function given by $u_c(t)$. In the case of $\cos (t)u_c(t)$ this can be done with the knowledge that $$\mathscr{L}\{\cos(at+b)\}=\frac{s\cos(b)-a\sin(b)}{s^2+a^2}$$however I am interested if there is another way to get functions of $t-c$ for cosine for the cases where this won't work. I have tried a few methods with the addition formula but I always end up with $\sin(t)$ which obviously has the same problem (perhaps I'm not being imaginative enough).
Essentially this is a question on how to shift $\cos(t)$ into functions, $f(t-c)$.
Thanks in advance.