Argument and modulus

Question

Question 1

 

Question 2

 

Question 3

 

 

Answer 1

Q1)

c) i - 1

- 1 + i

Compare it to x + iy

x = - 1 , y = 1

Since x < 0, use the formula for the argument  θ = tan^-1(y/x) + π

θ = tan^-1(1/-1) + π

θ = tan^-1(-1) + π

θ = -45 + π

θ = - π/4 + π

θ = 3π/4

Argument θ = 3π/4.

e) - 2 - 2 i

Compare it to x + iy

x = - 2 , y = - 2

Since x < 0, use the formula for the argument  θ = tan^-1(y/x) + π

 θ = tan^-1(-2/-2) + π

θ = tan^-1(1) + π

θ = 45 + π

θ =  π/4 + π

θ = 5π/4

Argument θ = 5π/4.

Answer 2

Q2)

a) 1 + 2i

Compare it to x + iy

x = 1 , y = 2

Since x > 0, use the formula for the argument  θ = tan^-1(y/x)

θ = tan^-1(2/1)

θ = tan^-1(2)

θ =  63.4349^o

θ = 63.4349^o

Argument is 1. 1071 radians.

b) - 4 + 2i

Compare it to x + iy

x = - 4 , y = 2

Since x < 0, use the formula for the argument

 θ = tan^-1(y/x) + 180^o

θ = tan^-1(2/-4) + 180^o

θ = tan^-1(-2) + 180^o

θ =  63.4349^o+ 180^o

θ = 243.4349^o

Argument is 4.2487 radians.

 

Answer 3

Q3)

a)

• w = - 1 - 3i

Compare it to x + iy

x = - 1 , y = - 3

Since x < 0, use the formula for the argument

 θ = tan^-1(y/x) + π

 θ = tan^-1(-3/-1) + π

θ = tan^-1(3) + π

tan(θ) =tan (π + tan^-1(3))

            =tan (tan^-1(3))            (Since tan( π +θ) =tan θ)

            = 3

tan(θ) =  3

• z = 2 +√3 i

Compare it to x + iy

x = 2 , y = √3

Since x > 0, use the formula for the argument

 ∅ = tan^-1(y/x)

 ∅ = tan^-1(√3/2)

tan(∅) = √3/2.

b) w = - 1 - 3i

 | w | = √ [(-1)²  + (-3)²]

= √( 1 + 9)

| w | = √10

z = 2 +√3 i

| z | = √ [(2)²  + (√3)²]

= √ [4 + 3]

| z | = √7.

Answer 4

Q1)

a) 4

Rewrite as 4 + 0(i)

Compare it to complex number z = x + i y

x = 4, y = 0

Since x > 0, use the formula for the argument  θ = tan^-1(y/x)

θ = tan^-1(0/4)

θ = tan^-1(0)

θ = 0

Argument is 0.

b) 3i

 it can be written as 0 + 3i

Compare it to complex number  x + i y

x = 0, y = 3

In this case x = 0, y > 0

Since x = 0, y > 0 use the formula b = tan^-1(y/x)

θ = tan^-1 (3/0)

θ = tan^-1 (∞)

θ =  π/2

The argument is π/2.

d) - 12 i

 it can be written as 0 - 12i

Compare it to complex number x + i y

x = 0, y = -12

In this case x = 0, y < 0

Since x = 0, y < 0 use the formula b = tan^-1(y/x)

θ = tan^-1 (-12/0)

θ = tan^-1 (-∞)

θ = - π/2

The argument is - π/2.

Comments

Thank you, i am still unsure of how you are getting pi/2 for Q1) b) and d) would you just use a calculator because it isnt showing up for me