69. a grating with lines uniformly spaced at 775 nm is illuminated by a sodium discharge lamp.

for what range of wavelength will there only be three principal lines generated?

Only  two diffraction orders +1 and -1 are possible

Explanation:

The diffraction net is a system with many uniformly spaced lines, which is described by the expression

      d sin θ = m λ               m = 1, 2, 3,…

Where d is the spread between the lines, λ the wavelength and m is an integer that corresponds to the order of diffraction, all wavelengths

For this case the spacing is

          d = 775 nm

The sodium emission lines are 589.00 nm and 589.59 nm

With this data we can calculate how many diffraction orders are visible

         θ = sin⁻¹ (m λ / d)

m = 1

         λ1 = 589.00 nm

         θ1 = sin⁻¹ (1  589/775)

         θ1 = 49.46 °

         λ2 = 589.59 nm

        θ2 = sin⁻¹ ((1  589.59 / 775)

        θ2 = 49.53 °

m = 2

        λ1 = 589.00 nm

       θ1 = sin⁻¹ (2  589/775)

       θ1 = sin⁻¹ (1.5)

This order is not possible because the sin cannot be greater than 1

m = -1

       λ1 = 589 nm

       θ1 = sin⁻¹ (-1 589/775)

       θ1 = -49.46

       θ= sin⁻¹ ( -1 589.59/775)        

       λ2 = 589.59 nm

       θ2 = -49.53 °

m = -2

       λ1 = 589 nm

       θ1 = sin-1 (-2 589/775)

This order does not occur because the breast cannot be greater than 1

Only  two diffraction orders +1 and -1 are possible