Answer
a) Decreasing $W$ would increase $\theta$
b) $20^{\circ}$
Work Step by Step
(a) We know that $sin\theta=\frac{m\lambda}{W}$. This equation shows that the angle of diffraction and the width of the slit are inversely proportional. Thus, if the width of the slit is decreased, then the angle to the first dark fringe above the central bright fringe will also increase.
(b) We know that
$sin\theta=\frac{m\lambda}{W}$
We plug in the known values to obtain:
$sin\theta=\frac{511\times 10^{-9}m}{1.50\times 10^{-6}m}$
$sin\theta=0.3406$
$\implies \theta=sin^{-1}(0.3406)=20^{\circ}$
