The greatest and the least value of $a+b\sin \theta$ ($0^{\circ }\le \theta \le 90^{\circ }$) are $7$ and $3$ respectively. Find the values of $a$ and $b$.

For the given domain $0^{\circ }\le \theta \le 90^{\circ }$, the value of $\sin \theta$ varies continuously as:
$$0\le \sin \theta \le 1$$

Assuming $b>0$:

1. The least value occurs when $\sin \theta =0$:

$$\text{Least value}=a+b(0)=a=3$$

2. The greatest value occurs when $\sin \theta =1$:

$$\text{Greatest value}=a+b(1)=a+b=7$$

Substitute $a=3$ into the equation:

$$3+b=7⟹b=4$$

Thus, $a=3$ and $b=4$.