To start, I will tell you that your result is definitely in the ballpark -- well done! Your procedure and calculations are very likely done correctly, although it's possible that you could change some things to make your measurements a bit more accurate. One thing you should be careful of when reporting results is significant figures, a concept that's related to error. For example, if you reported your result as 1883.5 nm (with no +/-), you would be implying that you're pretty sure the value you measured is closer to 1883.5 than 1883.4 or 1883.6 nm. On the other hand, you might decide to report it as 1900 nm, meaning that it's closer to 1900 than 1800 or 2000 nm. http://en.wikipedia.org/wiki/Significant_figures
It's even better to explicitly state the error, so you might end up with something like 1883.5 nm +/- 250 nm -- in which case you don't lose much by rounding to 1880 nm +/- 250 nm, because the last few sig figs don't mean much with that much uncertainly. 250 nm is just an example
-- you need to estimate the sources of error in your experiment, such as from reading the angle on the protractor, and then propagate them to your final result.http://en.wikipedia.org/wiki/Measurement_Uncertaintyhttp://en.wikipedia.org/wiki/Error_propagation
It's also worth googling both of those terms and checking the indices of your science books for different explanations. If the error-propagation math is too heavy, you can also do this: Say you measure the angle as 60 degrees +/- 2 degrees. Do the calculations using 58, 60, and 62 degrees. Your result is the one you got using 60 degrees, but you want to choose an error that sets the upper and lower bounds near the results for 58 and 62 degrees, respectively. It's a little more complicated if your result is the average of calculations from several different reflected beams, but you can at least do this for each calculation individually. If the errors are the same for each beam, the general rule of thumb is that the error in the average will be the error in one measurement divided by the square root of the number of measurements that went into the average -- but you should look this up, try to follow the reasoning behind it, and cite a good source (e.g. a statistical handbook) when you turn in your project.
At this point, you should have a decent idea of how much inaccuracy there is in your result due to error. Don't be discouraged by this -- there is error in every single scientific experiment, and it's impressive that you can measure such a tiny length scale with things around the house! Now, you want to look up the actual value. If it falls within your error bounds, then it means that your result is probably as good as it could be within the limitations of your measurement tools. If it's outside of your error bounds but still pretty close -- i.e. the same order of magnitude -- then it's likely that there's a source of error that you've neglected. If it's wildly different, then you should first check your calculations and then carefully repeat the experiment if necessary.
It's a good idea to estimate the error first, because then you won't be tempted to choose larger than reasonable error in your degree measurement, etc, just to make it work out. And if your 'blind' measurement of the track spacing is accurate within your error bounds, you really have something to be proud of. When you're ready, the track spacing can be found in this article:http://en.wikipedia.org/wiki/Compact_disc
(As a side note, the Greek letter mu, like a u with a funny tail going down in front, is used as a metric prefix meaning micro-, since m and M are taken for milli- and mega-. Micrometers are also called microns, perhaps because there are instruments called micrometers, though the stress is on the second syllable for the instrument when spoken aloud. Just a heads-up since it took me years to get this straight.)
Best wishes, and let us know how it turns out!