How do people do Trigonometry in their head so fast?
If you ask somebody tan(pi/4) and if they say 1 pretty quickly, that means they have memorized the unit circle! The unit circle is a really helpful tool that allows you to compute a variety of trigonometric functions without a calculator.
What should I study for the AP Calculus test?
You should be pretty safe if you know limits, derivatives, and integrals pretty well. And memorizing formulas really helped me as well.
I am having a hard time computing imaginary numbers with high powers. How do I do this more efficiently?
All you have to do is subtract the closest multiple of 4 from the power that is no greater than the power. For example: i^50. Looking at the power, the closest multiple of 4 less than 50 is 48. 50 - 48 = 2. Now we know i^2 = -1. So i^50 = -1. "How does this work?" You may ask. Note that in general, i^2 = -1 and i^4 = i^2 * i^2. This is helpful since we get i^4 = -1 * -1 meaning i^4 = 1. This allows us to deal with larger powers on imaginary numbers!