`Lap^{:-1:}{(s+4)/(s^2+9)}`

`=Lap^{:-1:}{s/(s^2+9)}` `+Lap^{:-1:}{4/(s^2+9)}`

`=Lap^{:-1:}{s/(s^2+9)}` `+4/3Lap^{:-1:}{3/(s^2+9)}`

`=cos 3t+4/3sin 3t`

For the sketch, recall that we can transform an expression involving 2 trigonometric terms

`a sin theta+b cos theta`

into

`R sin(theta+alpha)`

as follows:

For `g(t)=4/3sin 3t+cos 3t`, we have: `a=4/3,\ \ b=1, \ \ theta=3t`.

`R=sqrt(a^2+b^2)` `=sqrt((4//3)^2+1^2)` `=5/3`

`alpha=arctan{:1/(4//3):}` `=arctan{:3/4:}` `=0.6435`

So

`g(t)=4/3 sin 3t+cos 3t` `=5/3 sin(3t+0.6435)`

Here is the graph:

0.25π0.5π-0.25π0.511.5-0.5-1-1.5tg(t)Open image in a new page

Graph of `g(t)=5/3 sin(3t+0.6435)`.