+564 Find the four fourth roots of -3 + 4i and express the roots in polar coordinates.
+33659 If we have a general point (x, y) in Cartesian coordinates, the polar form is (R, Θ) where R = √(x2 + y2) and
Θ = tan-1(y/x).
And if we are in the complex plane, the point z = x + iy is z = ReiΘ in polar coordinates.
Your point is z = -3 + 4i so, in polar form it is z = ReiΘ where R = 5 and Θ = tan-1(4/-3) (note that this is an angle in the 2nd quadrant, because the x-value is negative while the y-value is positive). Because you can add any multiple of 2pi (360°) to Θ without changing the value of eiΘ, we can also write z = rei(Θ+2pi*k) where k is an integer.
To find the fourth root of a complex number in polar form we simply take the fourth root of r and divide the angle by 4. That is: z1/4 = r1/4ei(θ+2pi*k)/4. So z1/4 = 51/4ei(tan-1(4/-3)+2pi*k).
We can let k = 1, 2, 3 and 4 to get the four roots.
We can write this as z1/4 = reiθ, where r = 51/4 and θ = tan-1(4/-3) + 2pi*k.
The result of doing this is summarised in the image below:
+33659 If we have a general point (x, y) in Cartesian coordinates, the polar form is (R, Θ) where R = √(x2 + y2) and
Θ = tan-1(y/x).
And if we are in the complex plane, the point z = x + iy is z = ReiΘ in polar coordinates.
Your point is z = -3 + 4i so, in polar form it is z = ReiΘ where R = 5 and Θ = tan-1(4/-3) (note that this is an angle in the 2nd quadrant, because the x-value is negative while the y-value is positive). Because you can add any multiple of 2pi (360°) to Θ without changing the value of eiΘ, we can also write z = rei(Θ+2pi*k) where k is an integer.
To find the fourth root of a complex number in polar form we simply take the fourth root of r and divide the angle by 4. That is: z1/4 = r1/4ei(θ+2pi*k)/4. So z1/4 = 51/4ei(tan-1(4/-3)+2pi*k).
We can let k = 1, 2, 3 and 4 to get the four roots.
We can write this as z1/4 = reiθ, where r = 51/4 and θ = tan-1(4/-3) + 2pi*k.
The result of doing this is summarised in the image below:
