Q:

Calculating wavelengths from angles in a diffraction grating?

I am having some problems calculating wavelengths from some given information about a grating spectrum.
A diffraction grating with a spacing of 3μm is used in a spectrometer to investigate the emission spectrum of a mercury vapour discharge lamp.
The spectrum is examined over the range of angles from 30° to 50°, and maxima of intensity are observed at the angles and with the colours shown in the table. No other maxima are observed in this range of angles.
Violet: 32.7°, 42.4°  
Blue:   35.5°, 46.6°  
Green:  33.1°, 46.7°  
Yellow: 35.2°, 35.4°

To calculate wavelengths, I know that: $n\lambda = dsin \theta$.
So for the first angle: $n\lambda = 3 \times 10^{-6} \times sin 32.7$
However, I'm not sure where to go from there as I don't know what order of maxima they are as 0° to 30° is not examined. I have tried guessing numbers to make sure the wavelengths are in the range 400nm to 700nm. For example, with values of n=4 (35.5°) and n=5 (46.6°) for the blue light I get wavelength values that both round to 436nm, but this is apparently incorrect.
Also I don't understand how the two yellow maxima are so close together because this would normally mean there is a tiny wavelength, much smaller than that of visible light.
Any pointers would be much appreciated.

A:

For each wavelength you have been given (exactly) two values $\theta$, so
$$\begin{align}n\lambda &= d\sin\theta_1\\
(n+1)\lambda &= d\sin\theta_2\end{align}$$
subtracting these two equations, we get
$$\lambda = d\left(\sin\theta_2-\sin\theta_1\right)$$
You should be able to figure it out from there...

