Express z = 1 + 2j in polar form then find z^3 then convert the answer to Cartesian
coordinates.
I am used to expressing complex numbers in the form a + bi, instead of your a + bj
if z = 1 + 2i
r = √(1^2+2^2) = √5
and if Ø is our angle, tan Ø = 2/1 , Ø is in I of your complex plane
Ø = appr 1.107 radians
then z = √5(cos 1.107 + i sin 1.107) or √5 cis 1.107
z^3 = √5^3 cis (3(1.107)) using De Moivre's Theorem
= √5^3 cis 3.3214..
or
= √5^3 (cos 3.3214.. + i sin 3.3214..)
= (-11 + (-2) i)
z^3 = -11 - 2i
....
we could have done this:
(1+2i)^3
= (1+2i)(1 + 4i + 4i^2)
= (1+2i)(-3 + 4i)
= -3+4i - 6i + 8i^2
= -11 - 2i
Can the equation be expressed in a polar form diagram?
To express the complex number, z = 1 + 2j, in polar form, we first need to calculate the magnitude (r) and the angle (θ).
Magnitude (r):
The magnitude (r) of a complex number is calculated using the equation:
r = sqrt(a^2 + b^2)
In this case, a = 1 and b = 2. Substituting the values into the equation:
r = sqrt((1^2) + (2^2))
r = sqrt(1 + 4)
r = sqrt(5)
Angle (θ):
The angle (θ) of a complex number is calculated using the equation:
θ = arctan(b/a)
In this case, a = 1 and b = 2. Substituting the values into the equation:
θ = arctan(2/1)
θ = arctan(2)
Now, let's calculate the values.
r = sqrt(5) ≈ 2.236
θ ≈ arctan(2) ≈ 1.107
Using the magnitude (r) and the angle (θ), we can express z in polar form as z = r * cis(θ).
z = 2.236 * cis(1.107)
Now, let's find z^3.
z^3 = (2.236 * cis(1.107))^3
To raise a complex number to a power, we can multiply its magnitude by the power and add the angles by the power. Thus, we have:
z^3 = 2.236^3 * cis(1.107 * 3)
Calculating the values:
z^3 ≈ 11.314 * cis(3.321)
To convert the answer back to Cartesian coordinates, we need to convert the polar form back to rectangular form.
The rectangular form of a complex number is given by:
z = r * cos(θ) + r * sin(θ) * i
Substituting the values:
z^3 ≈ 11.314 * cos(3.321) + 11.314 * sin(3.321) * i
Evaluating:
z^3 ≈ 11.314 * 0.998 + 11.314 * 0.061 * i
Simplifying:
z^3 ≈ 11.297 + 0.691 * i
Therefore, the answer in Cartesian coordinates is approximately z^3 ≈ 11.297 + 0.691 * i.