There are four complex fourth roots to the number 4−4√3i. These can be expressed in polar form as
z1=r1(cosθ1+isinθ1)
z2=r2(cosθ2+isinθ2)
z3=r3(cosθ3+isinθ3)
z4=r4(cosθ4+isinθ4),
where ri is a real number and 0∘≤θi<360∘. What is the value of θ1+θ2+θ3+θ4 (in degrees)?
wrong man
To find the value of θ1 + θ2 + θ3 + θ4, we first need to express the given number 4 - 4√3i in polar form.
The magnitude r of the complex number is given by:
r = sqrt(a^2 + b^2)
where a = 4 and b = -4√3.
r = sqrt((4)^2 + (-4√3)^2)
r = sqrt(16 + 4*3)
r = sqrt(16 + 12)
r = sqrt(28)
r = 2√7
To find the angle θ, we can use the tangent function:
tan(θ) = b/a
tan(θ) = (-4√3)/4
tan(θ) = -√3/1
θ = -60°
Since we want the angle in the range 0° ≤ θ < 360°, we can add 360° to the angle until it falls within this range:
θ1 = -60°
θ2 = -60° + 360° = 300°
θ3 = -60° + 2*360° = 660° - 360° = 300°
θ4 = -60° + 3*360° = 1020° - 2*360° = 300°
Now, we can calculate the value of θ1 + θ2 + θ3 + θ4:
θ1 + θ2 + θ3 + θ4 = -60° + 300° + 300° + 300°
θ1 + θ2 + θ3 + θ4 = 840°
Therefore, the value of θ1 + θ2 + θ3 + θ4 is 840°.
To find the value of θ1+θ2+θ3+θ4, we need to determine the values of θ1, θ2, θ3, and θ4.
First, let's express the given number, 4 - 4√3i, in polar form. We can do this by converting it to its magnitude (r) and argument (θ). The magnitude of the complex number is given by:
r = sqrt(a^2 + b^2) = sqrt(4^2 + (-4√3)^2) = 4√4 = 8
The argument (θ) can be found using the formula:
θ = arctan(b/a) = arctan(-4√3/4) = -60° (note that the negative sign is because the complex number falls in the third quadrant)
Now that we have the polar form of the given number: 4 - 4√3i = 8(cos(-60°) + isin(-60°)), we can express it as:
z = r(cosθ + isinθ) = 8(cos(-60°) + isin(-60°))
Now, we need to find the fourth roots of z. Fourth roots are obtained by raising z to the power of 1/4.
z1/4 = [8(cos(-60°) + isin(-60°))]^(1/4)
To find the four roots, we need to find the four solutions to the equation (θ/4)+360k, where k is an integer. In this case, we have four roots, so k ranges from 0 to 3.
For the first root (k=0):
θ1/4 = (θ/4) + 360k = (-60°/4) + 360(0) = -15°
For the second root (k=1):
θ2/4 = (θ/4) + 360k = (-60°/4) + 360(1) = 345°
For the third root (k=2):
θ3/4 = (θ/4) + 360k = (-60°/4) + 360(2) = 705°
For the fourth root (k=3):
θ4/4 = (θ/4) + 360k = (-60°/4) + 360(3) = 1065°
Now, to find θ1 + θ2 + θ3 + θ4, we sum all the values:
θ1 + θ2 + θ3 + θ4 = -15° + 345° + 705° + 1065° = 2100°
Therefore, θ1 + θ2 + θ3 + θ4 is equal to 2100 degrees.
let z = 4-4√3 i
then by De Moivre's theorem
tanØ = -4√3/4 = -√3
so that Ø = 120° or Ø = 300°
z = 8(cos 120° + i sin120°) or z = 8(cos 300° + i sin300°)
case1:
then z^(1/4) = 8^(1/4) (cos 30° + i sin30°)
but for tanØ = -4√3/4 , recall that the period of tanØ is 180°
so adding 180° to our angle yields another solution
making z(1/4) = 8^(1/4) (cos 210° + i sin 210°)
so far we have Ø1=30° , Ø2 = 210°
cose2:
z^(1/4) = 8(1/2)( cos 300/4 + i sin 300/4) = 8(1/4) (cos 75 + i sin 75)
and with a period of 180° again,
z^(1/4) could also be 8^(1/4)(cos 255° + i sin 255°)
giving us Ø3 = 75 and Ø4 = 255
Ø1+Ø2+Ø3+Ø4 = 30+210+75+255 = 570°