Magnetic Field of A Current Loop

[jameson2]

Isn't exactly a homework question, but it seems like it should go here.
This is a page from my notes that I'm having trouble understanding. Basically, it's calculating the magnetic field of a dipole by considering a current loop and using the Biot Savart law. http://img257.imageshack.us/i/magneticp.jpg/
I can see that B_A will just be 4 times the contribution from one side. I don't see why it's sin though. I take that to mean that you're taking the horizontal effect of the magnetic field. But wouldn't this cancel out taking the 4 sides' contribution? I'm also wondering how the vertical effect of the magnetic field at A is taken into account.
I'm also wondering what the epsilon the B_B calculation is referring to.
Finally I don't quite understand how to get from the B_A and B_B answers to the last two lines of equations.

Thanks

 

[NateD]

for your help.The sin in B_A takes into account the angle between the current loop and the point A. Since the dipole is essentially two current loops, the magnetic field at point A can be assumed to be the sum of the two components. The vertical component is cancelled out by taking the 4 sides' contribution. The epsilon in the B_B calculation refers to the angle between the current loop and the point B. The last two lines of equations are calculated by using the properties of vector addition. Here, the B_A and B_B components are added together to get a single vector representing the total magnetic field of the dipole.