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.