I have one trigonometric function which is simplified as
f[t_,RV_,H_]:=0.000308148 H Cos[ 0.0523599 - 0.0862193 RV - 0.172439 RV Sin[4.71239 - 6.28319 t]] + 0.032 Sin[2 \[Pi] (-0.1 + t)] + H (-0.129421 - 0.0153853 Sin[ 0.0523599 - 0.0862193 RV - 0.172439 RV Sin[4.71239 - 6.28319 t]]) This function acts as input to one simple ODE which I want to solve with Laplace transform where RV and H are just two arbitrary parameters.
My question is how to find the Laplace transform of this function. One idea I am trying to do is using the following rules to transform the identities.
Rules = {Cos[ p1_*Sin[p2_*u_]] -> Sum[((-1)^k (p1 Sin[p2 u])^(2 k))/(2 k)!, {k, 0, Num}], Sin[ p1_*Sin[p2_*u_]] -> Sum[((-1)^k (p1 Sin[p2 u])^(2 k + 1))/(2 k + 1)!, {k, 0, Num}], Sin[ p1_*Cos[p2_*u_]] -> Sum[((-1)^k (p1 Cos[p2 u])^(2 k + 1))/(2 k + 1)!, {k, 0, Num}], Cos[ p1_*Cos[p2_*u_]] -> Sum[((-1)^k (p1 Cos[p2 u])^(2 k))/(2 k)!, {k, 0, Num}]} Then apply the Laplace transform. Is there any other way, since I have several of these functions and I do not know exactly how to apply these rules repeatedly to functions like sin(sin(sin…))) and write them such that I can take the Laplace transform.
