Light of wavelength 600 nm illuminates a diffraction grating. the second-order maximum is at angle 39.5 ∘. part a how many lines per millimeter does this grating have?

Diffraction equation applies in this case:

d*Sin x = m*wavelength, where d = spacing of lines, x = angle = 39.5°, m = order of maximum = 2

Substituting;
d* Sin 39.5 = 2*600*10^-9
d = (2*600*10^-9)/Sin 39.5 = 1.88656*10^-6 m

In 1 mm (or 0.001 m), the number of lines is given as;
Number of lines = 0.001/d = 0.001/(1.88656*10^-6) = 530.065 ≈ 530 lines