Let z= √(3 - i)
i) Plot z on an Argand diagram.
ii) Let w = az where a > 0, a E R. Express w in polar form
iii) Express w^8 in the form ka^n(x + i√y) where k,x,y E Z
let u = 3-i
|u| = √(9+1) = √10
tanθ = -1/3
θ = 5.96143.. since θ in in quad IV in the Argand plane
u = √10(cos 5.96143.. + i sin 5.96143)
then z = √u = √√10(cos 2.980717... + i sin 2.980717...) by De Moivre
= -1.7553 + .28485 i <----- the primary square root of 3-i
by adding π to 2.980717.. you can find a 2nd square root
ii) w = az just extends the maginitude of z, it does not change θ
iii) use De Moivre
Please help me with the solutions to this question
To answer these questions, we'll need to understand some concepts related to complex numbers and their graphical representation. Let's break it down step by step:
i) Plotting z on an Argand diagram:
Step 1: Identify the real and imaginary parts of z.
In this case, z = √(3 - i), so the real part is 3 and the imaginary part is -1.
Step 2: Plotting on an Argand diagram.
An Argand diagram is a graphical representation of complex numbers where the real part is represented on the x-axis and the imaginary part is represented on the y-axis.
To plot z = √(3 - i), find the point (3, -1) on the Argand diagram.
ii) Expressing w in polar form:
Step 1: Write z in polar form.
Polar form of a complex number z = r(cosθ + isinθ), where r is the magnitude of z and θ is the angle between the positive x-axis and the line connecting the origin and z.
To find r and θ:
r = |z| = √(3^2 + (-1)^2) = √10
θ = tan^(-1)(-1/3)
So, z = √10 (cosθ + isinθ).
Step 2: Express w in polar form.
w = az, where a > 0 and a ∈ R.
Let's assume a = 2. (You can use any other positive real number for a.)
So, w = 2z = 2√10 (cosθ + isinθ).
iii) Expressing w^8 in the form ka^n(x + i√y):
Step 1: Simplify w^8.
w^8 = (2√10 (cosθ + isinθ))^8
= (2√10)^8 (cos(8θ) + isin(8θ))
= 2560 (cos(8θ) + isin(8θ))
Step 2: Write w^8 in the desired form.
Comparing with ka^n(x + i√y), we have:
k = 2560,
n = 1,
x = cos(8θ),
y = sin(8θ).
Therefore, w^8 can be expressed in the form ka^n(x + i√y) as 2560(cos(8θ) + isin(8θ)).
Note: The values of θ depend on the chosen value of a in step ii). If you choose a different value, the angles and overall answer may vary.