Evaluate $L\{F(t)\}$ if $$ F(t) = \begin{cases} \sin(t-\pi/3), &t> \pi/3 \\ 0, & t <\pi/3. \end{cases}$$
I'm stuck on this question on Laplace Transforms.
I've broken the limits from $0$ to $\pi/3$ and from $\pi/3$ to infinity. The term with limit $0$ to $\pi/3$ zeroes out. I'm on a stalemate with the second term with limit $\pi/3$ to infinity.
Your function can be re-written using the unit step function $$u(t)= \begin{cases} 1,&t\ge 0\\ 0,&t<0 \end{cases} $$ as $$F(t)=\sin(t-\pi/3)\,u(t-\pi/3).$$ Then can you finish with the time-shifting property: $$L[f(t-a)\,u(t-a)]=e^{-as}\,L[f(t)]$$ and the LT of the $\sin$ function: $$L[\sin(t)]=\frac{1}{s^2+1}?$$