A Diffraction Grating

First off, I want to say hi. Im taking a summer physics course, I have really bad math anxiety and there isnt alot of help sources here so I will prob be posting a lot. Here is my first question.

It says a diffraction grating is illuminated simultaneously with red light of wavelength 660 nm and light of an unknown wavelength. The 5th order maximum of the unknown wavelength exactly overlaps the 3rd order maximum of the red light. What is the unknown wavelength.

So the only things I have to work with are λ=660nm λ=? and the unknown is at 5th order max and the red is at 3rd order max.

The equation d sinθ_m=mλ I thought that d and sin θ would be the same due to the fact they overlap (and be in phase) so they cancel out and i can use mλ=mλ which got me 3(660nm)=5(λ) which made λ equal to 396nm is that correct or did i totally mess that up. I couldnt think of anything else to do all the other formulas had variables which I had absolutely no info for so this was the best I could come up with..

Well done.

If they are in phase they will add.

BB means since they overlap, θ_5 = θ_3 and d = d, so
d sin θ_5 = d sin θ_3 and so
5(λ) = d sin θ_5 = d sin θ_3 = 3(660 nm)
5(λ) = 3(660 nm)
λ = (3/5)(660 nm) = 396 nm

Okay. Thank you so much for your help!

Thanks for you information i newly join and your post help me.