There are four complex fourth roots to the number 4−43√i. 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)?
Details and assumptions
i is the imaginary unit satisfying i2=−1.
If the smallest angle is θ1, then the other angles are
θ1+90,θ1+180,θ1+270, so
θ1+θ2+θ3+θ4 = 4θ1 + 540
So, what is θ1?
Well, √i = (1+i)/√2, so
4−43√i = 4 - (43+43i)/√2 = 40.27 cis 229.027
So, the 4th root with smallest angle is
2.52 cis 57.26
So, θ1+θ2+θ3+θ4 = 4(57.26) + 540 = 769
Z_(1 ) and Z_(2 )are given by Z_1=5*j(-〖60〗^0) Z_2=4*j〖45〗^0
To find the value of θ1 + θ2 + θ3 + θ4, we need to find the polar form of the complex number 4 - 43√i.
First, let's find the modulus (r) and argument (θ) of this complex number using the absolute value and inverse tangent functions:
The modulus (r) of a complex number a + bi can be found using the formula: |z| = √(a^2 + b^2).
In this case, a = 4 and b = -43√i. Since the imaginary part is -43√i, we can write it as -43 * √(-1) = -43i.
The absolute value or modulus of 4 - 43i is:
|4 - 43√i| = √((4)^2 + (-43)^2) = √(16 + 1849) = √1865.
The argument (θ) of a complex number a + bi can be found using the inverse tangent function: θ = atan(b/a).
In this case, a = 4 and b = -43. Therefore, the argument of 4 - 43i is: θ = atan((-43)/4).
Now that we have the polar representation of the complex number 4 - 43√i, let's find its four complex fourth roots.
To find the complex fourth roots, we can express the number in exponential form:
4 - 43√i = √1865 * (cos(θ) + isin(θ))
The complex fourth roots can be obtained by taking the fourth root of the modulus and dividing the argument by 4, and we will have:
z1 = (r1)^(1/4) * [cos((θ1)/4) + isin((θ1)/4)]
z2 = (r2)^(1/4) * [cos((θ2)/4) + isin((θ2)/4)]
z3 = (r3)^(1/4) * [cos((θ3)/4) + isin((θ3)/4)]
z4 = (r4)^(1/4) * [cos((θ4)/4) + isin((θ4)/4)]
Now, to find the value of θ1 + θ2 + θ3 + θ4, let's add the arguments of these complex fourth roots:
(θ1 + θ2 + θ3 + θ4) = (θ/4) + (θ/4) + (θ/4) + (θ/4) = 4(θ/4) = θ
Therefore, the value of θ1 + θ2 + θ3 + θ4 is equal to θ.