Hyperbolic Sine and Cosine
The hyperbolic trig functions are even easier than the regular ones because they can be decomposed into two real-valued functions. Thus, I will only provide the general result and all other particular results can be derived when needed.
Hyperbolic Sine
Given function $f(t)$ with Laplace Transform $F(s)$ and constant $b \in \mathbb{R}$
Recall the identity \(\sinh(t) = \frac{e^{t} - e^{-t}}{2}\).
\[\begin{align} \mathcal{L}\{ f(t) \sin(bt) \} &= \mathcal{L} \left \{ f(t) \cdot \frac{1}{2} (e^{bt} - e^{-bt}) \right \} \\[10pt] &= \frac{1}{2} \left ( \mathcal{L}\{ f(t) e^{bt} \} - \mathcal{L}\{ f(t) e^{-bt} \} \right ) \\[10pt] &= \frac{F(s-b) - F(s+b)}{2} \end{align}\]Hyperbolic Cosine
Given function $f(t)$ with Laplace Transform $F(s)$ and constant $b \in \mathbb{R}$
Recall the identity \(\cosh(t) = \frac{e^{t} + e^{-t}}{2}\).
\[\begin{align} \mathcal{L}\{ f(t) \sin(bt) \} &= \mathcal{L} \left \{ f(t) \cdot \frac{1}{2} (e^{bt} + e^{-bt}) \right \} \\[10pt] &= \frac{1}{2} \left ( \mathcal{L}\{ f(t) e^{bt} \} + \mathcal{L}\{ f(t) e^{-bt} \} \right ) \\[10pt] &= \frac{F(s-b) + F(s+b)}{2} \end{align}\]