Complex Numbers

Question 1

De moivres theorum

Answer 1

(cosθ+isin θ)^n = cos nθ + isin nθ

Question 2

(r(cosθ+isin θ))^n =

Answer 2

r^n(cos nθ + isin nθ)

Question 3

Nth root of unity

Answer 3

where a^n = 1

Question 4

finding the Nth root of unity

Answer 4

1. apply de moivres to the complex number
2. knowing 1 has modulus 1 and argument 0 equate 0 with nθ to get the n angles for θ
3. put this angles back into the original complex number

Question 5

what is always a root of unity?

Answer 5

1

Question 6

Where do the complex roots of unity all lie?

Answer 6

equally spaced on the unit circle

Question 7

finding the general Nth roots

Answer 7

1. find 1 root (i.e. find the argument of one number and sub back in)
2. split a circle into n+1 parts and work out the arguments of the other roots
3. sub these arguments back into the complex number

OR
argument = argz +2πk/n sub these back into general de moivres form

Question 8

uses of de moivre

Answer 8

• give multiple angle formula in terms of powers

- to express powers of sine and cosine in terms of multiple angles

Question 9

sin nθ =

Answer 9

z^n - z^-n / 2i

Question 10

cos nθ =

Answer 10

z^n + z^-n / 2

Question 11

how do you come up with the sin nθ and cos nθ expressions

Answer 11

by finding a general z^n and z^-n and adding/ subracting the expression

Question 12

e^iθ =

Answer 12

cosθ + isinθ

Question 13

double angle formula cos

Answer 13

= cos^2θ - sin^2θ
= 2cos^2θ -1
= 1 - 2sin^θ

Question 14

double angle formula sin

Answer 14

= 2cosθsinθ

Question 15

using complex numbers to sum real series

Answer 15

1. introduce a complex term to make it a known series
2. manipulate to it becomes a de moivres expression
3. equation real or imaginary parts to give the original value of the summation

Question 16

express multiple angle formula in terms of powers

Answer 16

1. use de moivres to show what it should be
2. expand the expression using binomial expansion
3. equate real and imaginary terms

Question 17

useful method

Answer 17

equating the real and imaginary parts

Question 18

dividing complex numbers

Answer 18

• set equal to x + yi then equate real/ imaginary parts
  OR
• multiply by the complex conjugate

Question 19

y axis of argand diagram

Answer 19

imaginary

Question 20

x axis of an argand diagram

Answer 20

real

Question 21

|wz| =

Answer 21

|w||z|

Question 22

|w/z|

Answer 22

|w|/|z|

Question 23

arg(zw)

Answer 23

arg(z)+arg(w)

Question 24

arg(z/w)

Answer 24

arg(z)-arg(w)

Question 25

|z-z1|=r

Answer 25

``` |z-(a+bi)|=r circle centre (a,b) radius r ```

Question 26

|z-z1| less than r

Answer 26

interior of the circle

Question 27

|z-z1|>r

Answer 27

exterior of the circle

Question 28

arg(z-z1) = θ

Answer 28

arg(z-(a+bi))= θ | consist of a half line from point a +bi in the direction θ

Question 29

arg(z-z1) < θ

Answer 29

all the point below the half line and the horizontal line to the point z1

Question 30

arg(z-z1) > θ

Answer 30

all the point above the half line and the horizontal line to the point z1

Question 31

|z-z1|=|z-z2|

Answer 31

perpendicular bisector of the line joining z1 and z2

Question 32

|z-z1| < |z-z2| | less than

Answer 32

represents the region closer to z1 from the bisector

Question 33

|z-z1|>|z-z2|

Answer 33

represemts the region closer to z2 from the bisector

Question 34

>=

Answer 34

line is included

Question 35

>

Answer 35

line not included

Question 36

how to show a line is not included in the region

Answer 36

dot it

Question 37

expanding sinnθ or cosnθ

Answer 37

- use de moivres and take the imaginary part or real part depending on cos/sin i.e. im[] - expand the power n from de moivres using binomial expansion - after multiplying out the i's keep only the real/imaginary expressions - use identities to get in terms of just sin or cos

Question 38

how to express powers of sin or cos in terms of multiple angles

Answer 38

1. express sin/ cos in terms of z^n (the sin/ cos will have a 2 infront don't forget the power this number) 2. expand the z^n terms via bionomial expansion 3. factorise out z^ns with their corresponding minus powers, replace these expressions with sin/cos 4. divide by the 2^n

Question 39

2cos nθ=

Answer 39

Z^n + Z^-n

Question 40

2isin nθ =

Answer 40

Z^n - Z^-n

Question 41

when equating real and imaginary parts and a real number on the bottom of the fraction

Answer 41

the bottom of the fraction appears for both the real and imaginary number