Complex

Question 1

Converting a+bi to e^ix

Answer 1

a + bi = sqrt(a^2 + b^2)*e^[iarg(b/a) + 2nπ]

Question 2

cos(arctan(x)) =

Answer 2

1/sqrt(x^2 + 1)

Question 3

sin(arctan(x)) =

Answer 3

x/sqrt(x^2 + 1)

Question 4

How do you work out the locus of |z-3| = |z-6i| ?

Answer 4

x + iy - 3 | = | x + iy - 6i |
| (x-3) + iy | = | x + i(y-6) |
(x-3)^2 + y^2 = x^2 + (y-6)^2

Question 5

Roots of z^n = 1

Answer 5

e^(2πki/n)

Question 6

Given two complex roots a and b, what is the quadratic?

Answer 6

z^2 - (a+b)z + ab = 0

Question 7

What is the locus of arg(z - (a+bi) ) = θ?

Answer 7

A half-line through (a,b) that makes an angle θ through the x-axis

Question 8

What is the locus of | z - (a+bi) | = r?

Answer 8

A circle with centre (a,b) and radius r

Question 9

1/z =

Answer 9

cos(θ) - isin(θ)

Question 10

z^n + z^-n =

Answer 10

2cos(nθ)

Question 11

z^n - z^-n =

Answer 11

2isin(nθ)

Question 12

De Moivre’s theorem

Answer 12

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

Question 13

Find an expression for cos(x)^5 using De Moivre’s

Answer 13

z^n + z^-n = 2cos(nx)

z + 1/z = 2cos(x)

32cos^5(x) = (z + z^-1)^5 = z^5 + 5z^4z^-1 + 10z^3z^-2 + 10z^2z^-3 + 5zz^-4 + z^-5

32cos^5(x) = z^5 + 5z^3 + 10z + 10^-z + 5z^-3 + z^-5

32cos^5(x) = (z^5 + z^-5) + 5(z^3 + z^-3) + 10(z + z^-1)

32cos^5(x) = 2cos(5x) + 10cos(3x) + 20cos(x)

cos^5(x) = cos(5x)/16 + 5cos(3x)/16 + 5cos(x)/8

Question 14

Find an expression for cos(5x) using De Moivre’s

Answer 14

cos(5x) + isin(5x) = (c + is)^5

cos(5x) + isin(5x) = c^5 + 5c^4is + 10c^3i^2s^2 + 10c^2i^3s^3 + 5ci^4s^4 + i^5s^5

cos(5x) + isin(5x) = c^5 + 5c^4is - 10c^3s^2 - 10c^2is^3 + 5cs^4 + is^5

cos(5x) + isin(5x) = c^5 - 10c^3s^2 + 5cs^4 + i(s^5 - 10c^2s^3 + 5sc^4)

cos(5x) = c^5 - 10c^3s^2 + 5cs^4 = cos(x)^5 - 10cos(x)^3sin(x)^2 + 5cos(x)sin(x)^4

Question 15

If C + iS = 2/(2 - e^ix), find C and S

Answer 15

Flip the sign of the exponent on the bottom and multiply top and bottom by that

2/(2 - e^ix) * (2 - e^-ix)/(2 - e^-ix)

(4 - 2e^-ix)/(5 - 2e^ix - 2e^-ix)

Simplify the top and use z + 1/z and z - 1/z to simplify the bottom (c = cosx, s = sinx)

(4 - 2c + 2is)/(5 - 4c)

Therefore C = (4 - 2cosx)/(5 - 4cosx)
S = 2sinx/(5 - 4cosx)

Question 16

How do you get all the vertices of an n-gon centered on the origin of the complex plane?

Answer 16

Remember that for an n-gon, w = e^(2πi/n). Multiply the vertex you’re given by w, n times to get all the vertices