Diffraction Grating, calculating rulings/mm from spectrum

[Oijl]

Homework Statement

Assume that the limits of the visible spectrum are arbitrarily chosen as 430 and 680 nm. Calculate the number of rulings per millimeter of a grating that will spread the first-order spectrum through an angle of 20.0 degrees.

Homework Equations

(y is lambda)
dsinø = my (maxima)

The Attempt at a Solution

So, the first-order spectrum of white light has a line (maxima) the lowest wavelength and another line at the highest wavelength - right?

And for the first-order spectrum would have a m of 1, right?

If this is true, then,
dsin(ø1) = 430
dsin(ø2) = 680

And, ø2 - ø1 = 20 degrees

And
ø1 = arcsin(430/d)
ø2 = arcsin(680/d)

But, I want to find out d. How can I extract d from these equations?

Or, alternatively, do I not understand what's going on?

 

[Redbelly98]

You have the right approach, but we can't solve for d explicitly; a numerical approach is needed here. What value of d results in ø1 and ø2 that are 20 degrees apart?
If you can set this up in a spreadsheet, it won't take very long to find d that gives the required 20 degree spread.