Imaginary...

Question 1

Prove sina+b = sinacosb …

Answer 1

Cos(a+b) + iSin(a+b) = e^(a+bi) = e^ai x e^bi = (cosa + isina)(cosb isinb) = isinacosb + isinbcosa + cosacosb - sinasinb

Question 2

de moivers theorem Z^n

Answer 2

Z^n = (r(cos0 +isin0))^n = r(cosn0 +isinn0)

Question 3

how to switch number into R(isin0+cos0)

Answer 3

4i = 4in magnitude and direction pi/2
thus r = 4
0 = pi/2 or arctan(IM/RE)

Question 4

answer to 3^(2+i) using moivers

Answer 4

e^(ln3)(2) x e^(ln3)i =
9 X (cos[ln3] +isin[ln3])

Question 5

e^io

Answer 5

cis 0

Question 6

find the other root if 1 and 2e^(i/3) are part of it

Answer 6

2e^(-i/3)

Question 7

steps to solve z^n = w

Answer 7

put w in modulus arg form
use de moivers for z^n = rcos…
compare modulus
compare argument n0 = arctan IM/RE

Question 8

prove the roots of unity sum to 0

Answer 8

let W = e^(2i[pi]/n) where 1+W1+W2+…W(n-1) are the roots of unity
thus Wn = 1
Wk = W^K thus roots = 1+ W + W^2 + W^3 +….
geometric sequence where r = W
1-W^n = 1-1 = 0 thus roots sum to 0

Question 9

how to add two transformations of imaginary vectors

Answer 9

compare magnitudes (multiple of) for magnification
compare argument (subtracting) for angle

Question 10

e^i0 +e^-i0

Answer 10

= 2cos 0