The magnetic field is proportional to the current in the coil and the number of turns of the coil. More turns means a greater magnetic field. It would probably be better to make all the layers of the coil from one continuous length of wire.
Wikipedia article for Solenoid
Numerous high-end laptop builders/resellers are building systems based on barebones chassis produced by a Taiwanese manufacturer called Clevo. I have also recently been researching on buying a higher-end laptop (not AW). B. Elliot mentioned Sager, and they are one of the builders using the Clevo...
You can do away with the whole unit circle business. Use Taylor series expansions for sin(A) and cos(A). It's an infinite series polynomial expansion, so you can choose how many terms you want to calculate and get accurate output to an arbitrary level of precision.
See the following...