Today, I am tutoring a student in complex numbers, so I am going to use my journal to type up some notes before the study session:
*Radian = degrees * (pi/180)
*Degree = Radian * (180/pi)
*Euler's Formula: e^(ix) = cis(x) & e^(nix) = cos(nx) + isin(nx)
0 = tan^-1(y/x) if x > 0 & 0 = tan^-1(y/x) + pi or 0 = tan^-1(y/x) + 180 degrees for x < 0
**If z= 1 + j, what is z^10?
r = sqrt(x^2 + y^2) = sqrt(1^2 + 1 ^2) = √2
arctan(1/1) = tan^-1(1) = pi/4, so z = √2(cos(pi/4) + jsin(pi/4))
z^10 = √2^10 * cis (10 * pi/4) = 32cis(5pi / 2) = 32cis(2pi + (pi/2)) = 32cis(pi/2) = 32 (
cos(pi/2) 0 + j sin(pi/2) 1) = 32j
*A complex number z = r(cos(θ) + ısin(θ) has exactly nnth roots given by the equation [cos() + ısin()], where n is a positive integer, and k = 0, 1, 2,..., n - 2, n - 1.
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