Y2, C1 - Complex Numbers

Question 1

If z = x + iy, what is the argument

Answer 1

θ = arg(z) = tan^-1 (y / x)

Question 2

If z = x + iy, what is the modulus

Answer 2

r = l z l = root(x^2 + y^2)

Question 3

What is the range for the principal argument

Answer 3

-pi < theta < pi

Question 4

What is the modulus argument form of a complex number

Answer 4

z = r(cos(θ) + isin(θ)

Question 5

What is the cartesian form of a complex number

Answer 5

z = x + iy

Question 6

What is the exponential form of a complex number

Answer 6

z = re^iθ
Where r is the modulus
θ is the argument

Question 7

Using Euler’s formula how can e^ix be written

Answer 7

e^ix = cosx + i sinx

Question 8

What is Euler’s identity

Answer 8

e^ipi + 1 = 0

Question 9

What constant is cosθ + isinθ equal to (exponential form)

Answer 9

e^iθ

Question 10

How do you multiply complex numbers

Answer 10

Multiply the moduli and add the arguments

Question 11

How do you divide complex numbers

Answer 11

Divide the moduli and subtract the arguments

Question 12

How can cosθ - isinθ be written

Answer 12

cos(-θ) + isin(-θ)

Question 13

What is De Moivre’s theorem

Answer 13

z^n = r^n (cos(nθ) + isin(nθ))
z^n = r^n * e^inθ

Question 14

Steps for expressing cos(3θ) in terms of powers of cos(θ)

Answer 14

1) Create a De Moivre statement that includes cos(3θ) on the RHS
2) Binomial expansion
3) Compare real / imaginary parts

Question 15

What is (cosθ + isinθ)^6 equal to

Answer 15

cos6θ +isin6θ

Question 16

What is the De Moivre statement when expressing cos(3θ) in terms of powers of cos(θ)

Answer 16

(cosθ + isinθ)^3 = cos3θ + isin3θ

Question 17

If z = cosθ + isinθ, what is z + 1/z

Answer 17

2cosθ

Question 18

If z = cosθ + isinθ, what is z - 1/z

Answer 18

2isinθ

Question 19

If z = cosθ + isinθ, what is z^n + 1/z^n

Answer 19

2cos(nθ)

Question 20

If z = cosθ + isinθ. what is z^n - 1/z^n

Answer 20

2isin(nθ)

Question 21

How would you express cos^5(θ) in the form acos5θ + bcos3θ + ccosθ

Answer 21

1) Raise RHS to required power (2cosθ)^5)
2) Raise LHS to the same power
(z + 1/z)^5)
3) Binomial expansion
4) Use the identities once again
5) Remember to isolate by dividing by any coefficient on LHS

Question 22

What is 2cosnθ equal to
(exponential form)

Answer 22

e^niθ + e^-niθ

Question 23

What is 2sinnθ equal to
(exponential form)

Answer 23

e^niθ - e^-niθ

Question 24

How would you express 3 / (e^2iθ - 1) in hyperbolic form

Answer 24

Multiply by e to the power of half the negated power (e^-iθ)
Then sub in 2isinθ as the denominator

Question 25

How would you express 3 / (e^4iθ + 2) in hyperbolic form

Answer 25

Multiply by the same expression but with the power of e negated (e^-4iθ + 2)

Question 26

When would you multiply by e to the power of half the negated power to get an imaginary number into hyperbolic form

Answer 26

When there is a 1+, 1-, or -1 with the coefficient of e^inθ as 1

Question 27

How to solve z^3 = 1

Answer 27

z^3 = e^0i
z = (e^0i)^1/3, +2pi/3 = z = (e^i2pi)^1/3
z = 1, z = e^i2pi/3, z = e^i4pi/3

Question 28

How would solutions to z^n = 1 look on an Argand diagram

Answer 28

n-sided polygon
centre at origin
1 always a root

Question 29

What is the difference in argument for each root of a root of unity (z^n = 1)

Answer 29

2pi / n

Question 30

What is omega (w) in roots of unity

Answer 30

The first root (z1 = 1)
e^i2pi/n = w

Question 31

What do the roots of unity sum to

Answer 31

0

Question 32

What does multiplying by omega (w) do

Answer 32

Rotates the complex number by 2pi / n radians
Moves it along to the next vertex of the root

Question 33

How do you find the vertexes of a polygon which doesn’t have its centre at the origin

Answer 33

Move the centre to the origin
Move the known vertex to the corresponding point
Work out the other vertexes
Transform them back to the correct place

Question 34

How many times do you have to multiply by omega (w) to reach z1 (1)

Answer 34

n times
w^n = 1
If z^6 = 1
w^6 = 1