A few special relativity questions before my exam

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Ok so my exam is here tomorrow and I don't feel like I have the grasp on SR I would like to have, most of the books I ordered recently got here so I haven't had time to read them yet. So if I could get any help with some questions I would really appreciate it.

1.) When using the general Doppler shift formula [tex]\nu'=\frac{v_0}{\gamma(1-\beta Cos\theta)}[/tex] I am a little confused on how its derived by resolving the problem into radial and transverse motion. I would think just the radial would give the above result as the only difference is that the Cos that is taken as the velocity is only a component of the full velocity source velocity. I don't know what to do with the time dilated only transverse part though as it doesn't seem to be needed. Hopefully that makes sense.

2.) When thinking of which to use, full Lorenz transformations or just time dilation/length contraction formulas what is the best way to decide which to use? It is giving me a bit of trouble even though it seems to be pretty straight forward.

Thanks for any help, hopefully I can have everything straight out by tomorow.
 
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