Question

A diffraction grating has 5540lines/cm . When a beam of monochromatic light goes through it, the second pair of bright spots occurs at ±26.0cm from the central spot on a screen that is 42.5cm past the grating.

To see if the 'small angle approximation' is sufficient, Calculate tan θ using the quantities given. If sin θ is very different from tan θ then the approximation is not going to be useful. (In the context of these questions, very different means different if we calculate three significant figures.)
question 1- What is the wavelength of this light?
question 2- What angle is between a line from the grating to the central spot, and a line from the grating to the third bright spot (m = 3) on the screen?

Answer 1

N=5540 lines/cm=554000 lines/m
d=1/N=1.8•10⁻⁶ m
1)
m=2
tanφ₂= x₂/L=26/42.5 =0.61
φ₂=tan⁻¹0.61= 31.5⁰
sinφ₂=sin 31.5⁰=0.52
dsinφ =mλ
dsinφ₂=2λ
λ= dsinφ₂/m=1.8•10⁻⁶ •0.52/2=4.5•10⁻⁷ m
2)
m=3
dsinφ₃=3λ
sinφ₃=3 λ/d=3 λ/1.8•10⁻⁶=
=0.75,
φ₃=sin⁻¹0.75=48.8⁰