complex numbers

Question 1

what is polar form

Answer 1

magnitude and direction, e.g. Re^iθ or n(cosx + isinx)

Question 2

what do you do when finding arguments

Answer 2

draw it and check you mug

Question 3

how to get roots of unity from z^n = 1

Answer 3

divide complex plane into equal chunks, starting at real axis e^i(2π/n)

Question 4

de moivres theorum

Answer 4

(r(cosθ = isinθ))^n = r^n(cos(nθ) + isin(nθ))

Question 5

how do you prove de moivres theorum

Answer 5

proof by induction, base case n = 0

Question 6

how to find cos(nθ) in terms of cosθ

Answer 6

de moivres theorum, binomially expand (c + is)^n

cos(nθ) = (c + is)^n

Question 7

how to find cos^n(θ) in terms of cosnθ

Answer 7

start with Z + Z^-1 = 2cosθ and raise to the power of n, cancel Zs and reapply formula Z^n + Z^-n = 2cos(nθ)

Question 8

2cosθ = ..

Answer 8

Z + Z^-1

Question 9

2isinθ = ..

Answer 9

Z - Z^-1

Question 10

2isin(nθ) = ..

Answer 10

Z^n - Z^-n

Question 11

2cos(nθ) = ..

Answer 11

Z^n + Z^-n

Question 12

split the difference

e^iθ - 1

Answer 12

e^(1/2)iθ ( e^(1/2)iθ - e^-(1/2)iθ)