Laplace Transform with unit step function

Question

Here is the link to the question and the answer: https://gyazo.com/d98918d0e0eaabe88606c0314aff0aca

Here is the link to what I have done: https://gyazo.com/20ddc3822c0c7bca888cd3334dd78281

What happens to 3PI? Where does it go?

Answer 1

You can simplify that:$$\sin(\pi t+3\pi)=-\sin(\pi t)$$ since $$\sin(a\pm b)=\sin a\cos b\mp \sin b\cos a$$ Alternatively, notice that the period of $\sin(\pi t)$ is $\dfrac{2\pi}{\pi}=2$, which means values of sine repeat for $t+2k$ where $k\in \mathbb{Z}$, while the sign alternates for $t+k$.