Complex numbers

Question 1

what’s the modulus-argument form of a complex number z?

Answer 1

z = r(cos θ + i sin θ)

where r is the modulus of z and θ is the principal argument

Question 2

how else can you express a complex number?

Answer 2

in the form z = r e^iθ

r is the modulus of z and a is the argument

Question 3

cos (-θ) =

Answer 3

cos θ

Question 4

sin (-θ) =

Answer 4

-sin(θ)

Question 5

sin (θ±ρ) =

Answer 5

sin θ cos ρ ± sin ρ cos θ

Question 6

cos (θ±ρ) =

Answer 6

cos θ cos ρ - /+ sin θ sin ρ

Question 7

sin^2 θ + cos^2 θ =

Answer 7

1

Question 8

multiplying z1= r1(cosθ +isinθ) and z2 = r2(cosρ+isinρ), z1 z2 =

Answer 8

z1z2= r1r2(cos(θ+ρ) + isin(θ+ρ))

Question 9

modulus and argument of z1z2?

Answer 9

modz1z1 = r1r2

argz1z2 = θ + ρ

Question 10

multiplying z1 = r1 e ^iθ and z2 = r2 e^iρ,

Answer 10

z1z2 = r1r2 e^i(θ+ρ)

Question 11

how can you express a comolex number?

Answer 11

exponential form

modulus argument form

standard form

Question 12

dividing complex numbers in modulus argument form

z1/z2 =

Answer 12

r1/r2 (cos(θ-ρ) +i sin (θ-ρ))

Question 13

z1/z2 has modulus and argument…

Answer 13

r1/r2

argz1 - argz2

Question 14

how do you prove de moivres’ theorem?

Answer 14

by induction

Question 15

z^n =

Answer 15

[r(cos θ +i sin θ)] ^n = r^n(cos nθ + i sin nθ)

Question 16

de moivres’ theorem in exponential form?

Answer 16

r^n e^inθ

Question 17

(x + y)^n =

Answer 17

x^n + nC1 x^n-1 y + nC2 x^n-2 y^2 + … + y^n

Question 18

nCk =

Answer 18

n!/k!(n-k)!

Question 19

z + 1/z =

Answer 19

2 cos θ

Question 20

z - 1/z =

Answer 20

2i sin θ

Question 21

z^n + 1/z^n =

Answer 21

2 cos nθ

Question 22

z^n - 1/z^n =

Answer 22

2i sin nθ

Question 23

cartesian equation of a circle center (a ,b) radius r?

Answer 23

(x-a)^2 + (y-b)^2 = r^2

Question 24

Example of a locus

Answer 24

A circle

Question 25

circle center (x1, y1) radius r?

Answer 25

| z - z1 | = r where z1 = x1 + iy1

Question 26

what does | z - z1 | = | z - z2 | represent?

Answer 26

a perpendicular bisector of the line segment joining points z1 to z2.

Question 27

angle in a semicircle =

Answer 27

90

Question 28

angles subtended at an arc in the same segment are

Answer 28

equal

Question 29

the angle subtended at the center of the circle is

Answer 29

twice the angle at the circumference

Question 30

what does arg (z -z1) = a represent?

Answer 30

a half-line form the fixed point z1 making an angle a with a line from the point z1 parallel to the real axis

Question 31

e^iθ =

Answer 31

cosθ + isinθ

Question 32

cos 2θ =

Answer 32

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

Question 33

sin2θ =

Answer 33

= 2sinθcosθ

Question 34

what does w = z + a + ib represent?

Answer 34

a translation with vector [ a, b] where a and b are real

Question 35

what does w = kz represent?

Answer 35

an enlargement scale factor k centre (0,0) where k > 0

Question 36

what does w = kz + a +ib represent?

Answer 36

enlargement scale factor k centre (0,0) followed by a translation with vector [ a, b ] where k > 0 and a, b are real

Question 37

three types of loci?

Answer 37

circle: | z - z1 | = r perpendicular bisector of line segment joining z1 to z2: | z - z1 | = | z - z2 | half line: arg (z - z1) = θ

Question 38

arg (z1 x z2) =

Answer 38

arg(z1) + arg(z2)

Question 39

arg (z1 ÷ z2) =

Answer 39

arg(z1) - arg (z2)

Question 40

|z1z2| =

Answer 40

|z1| |z2|

Question 41

|z1 ÷ z2| =

Answer 41

|z1| ÷ |z2|

Question 42

what does arg( z1/z1 + 4) = π/4 look like

Answer 42

arg(z1) - arg(z1 + 4) = π/4 looks like an arc of a circle

Question 43

area of a sector A =

Answer 43

1/2 r^2 θ θ in radians

Question 44

area of a segment A =

Answer 44

1/2 r^2 (θ - sinθ) θ in radians

Question 45

area of a circle =

Answer 45

π r^2