Let z1= 3-3sqrt 3i and z2= -11+i. write all answers in polar form with 0< theta< 2pi
A. what is z1 in polar form?
B. what is z2 in polar form?
C. what is z1z2
D. what is z1/z2
E. what is (z1)^5
F. what is (z2) ^1/4
To write complex numbers in polar form, we need to convert them to their magnitude and argument form. Here's how to do it for each part of the question:
A. To convert z1 to polar form, we need to find its magnitude (r) and argument (theta).
z1 = 3 - 3√3i
First, calculate the magnitude (r):
|z1| = sqrt((Re(z1))^2 + (Im(z1))^2)
= sqrt((3)^2 + (-3√3)^2)
= sqrt(9 + 27)
= sqrt(36)
Therefore, |z1| = 6.
Second, calculate the argument (theta):
theta = arctan(Im(z1) / Re(z1))
= arctan((-3√3) / 3)
Note: Since the real part is positive, we need to add 180 degrees (π radians) to the argument to get it within the range of 0 to 2π.
theta = arctan(-√3) + π
= -π/3 + π
= 2π/3
Therefore, theta = 2π/3.
Therefore, z1 in polar form is 6(cos(2π/3) + isin(2π/3)).
B. To convert z2 to polar form, we follow the same steps as in part A:
z2 = -11 + i
First, calculate the magnitude (r):
|z2| = sqrt((-11)^2 + (1)^2)
= sqrt(121 + 1)
= sqrt(122)
Therefore, |z2| = √122.
Second, calculate the argument (theta):
theta = arctan(Im(z2) / Re(z2))
= arctan(1 / (-11))
= arctan(-1/11)
Note: Since the real part is negative, we need to add 180 degrees (π radians) to the argument to get it within the range of 0 to 2π.
theta = arctan(-1/11) + π
= -0.087266 + π
= 3.05433
Therefore, theta ≈ 3.05433.
Therefore, z2 in polar form is (√122) * (cos(3.05433) + isin(3.05433)).
C. To multiply z1 and z2 (z1z2), we can simply multiply their magnitudes and add their arguments:
z1z2 = |z1| * |z2| * (cos(theta1 + theta2) + isin(theta1 + theta2))
= 6 * √122 * (cos(2π/3 + 3.05433) + isin(2π/3 + 3.05433))
D. To divide z1 by z2 (z1/z2), we can divide their magnitudes and subtract their arguments:
z1/z2 = (|z1| / |z2|) * (cos(theta1 - theta2) + isin(theta1 - theta2))
= (6 / √122) * (cos(2π/3 - 3.05433) + isin(2π/3 - 3.05433))
E. To raise z1 to the power of 5 (z1^5), we can raise its magnitude to the power of 5 and multiply its argument by 5:
z1^5 = (|z1|^5) * (cos(5 * theta1) + isin(5 * theta1))
= 6^5 * (cos(5 * (2π/3)) + isin(5 * (2π/3)))
F. To find the square root of z2 ((z2)^(1/4)), we can take the square root of its magnitude and divide its argument by 4:
(z2)^(1/4) = (√|z2|) * (cos(theta/4) + isin(theta/4))
= (√√122) * (cos(3.05433/4) + isin(3.05433/4))
Therefore, the polar forms for each part are:
A. z1 in polar form: 6(cos(2π/3) + isin(2π/3))
B. z2 in polar form: (√122) * (cos(3.05433) + isin(3.05433))
C. z1z2 in polar form: 6 * √122 * (cos(2π/3 + 3.05433) + isin(2π/3 + 3.05433))
D. z1/z2 in polar form: (6 / √122) * (cos(2π/3 - 3.05433) + isin(2π/3 - 3.05433))
E. (z1)^5 in polar form: 6^5 * (cos(5 * (2π/3)) + isin(5 * (2π/3)))
F. (z2)^(1/4) in polar form: (√√122) * (cos(3.05433/4) + isin(3.05433/4))