Answer
(a) $y(t)=a\cdot sin(\omega x)$
(b) $y(t)=a\cdot cos(\omega x)$
Work Step by Step
General forms $y(t)=a\cdot sin(\omega(x-b))$ and $y(t)=a\cdot cos(\omega(x-b))$
(a) Since $y(0)=0$ choose since with $b=0$, we have $y(t)=a\cdot sin(\omega x)$
(b) Since $y(0)=a$ choose cosine with $b=0$, we have $y(t)=a\cdot cos(\omega x)$