Answer
$17^{\circ}$ and $64^{\circ}$
Work Step by Step
For destructive interference to occur, the path difference is a half-wavelength, three half-wavelengths, five half-wavelengths, etc. This is stated in equation 24–2b, where m = 0, 1, 2, 3...
$$d sin \theta = (m+\frac{1}{2})\lambda $$
Solve for the possible angles.
$$sin \theta = \frac{(m+\frac{1}{2})\lambda }{d}$$
$$\theta = sin^{-1}(\frac{(m+\frac{1}{2})\lambda }{d})$$
Find the first few angles, relative to the straight-through direction, at which destructive interference occurs. Let m = 0, then 1, then 2, etc.
$$\theta = sin^{-1}(\frac{(0+\frac{1}{2})(4.5cm) }{7.5cm})=17^{\circ}$$
$$\theta = sin^{-1}(\frac{(1+\frac{1}{2})(4.5cm) }{7.5cm})=64^{\circ}$$
There are no solutions for m=3, or higher, because the maximum of the sine function is 1.