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 find their magnitude (r) and argument (θ). The magnitude (r) can be calculated using the formula √(a^2 + b^2), where 'a' is the real part and 'b' is the imaginary part. The argument (θ) can be calculated using the formula tan^(-1)(b/a), but we need to consider the quadrant to give the correct angle.
Let's calculate the polar form of each complex number:
A. To find the polar form of z1 (3 - 3√3i), we first calculate the magnitude:
r = √(3^2 + (-3√3)^2) = √(9 + 27) = √36 = 6
Next, we need to find the argument by considering the quadrant:
The real part (a) is 3, and the imaginary part (b) is -3√3.
Since both a and b are positive, we can use the formula θ = tan^(-1)(b/a):
θ = tan^(-1)(-3√3/3) = -π/3.
Therefore, z1 in polar form is 6∠(-π/3).
B. To find the polar form of z2 (-11 + i), we calculate the magnitude:
r = √((-11)^2 + 1^2) = √(121 + 1) = √122.
Next, find the argument considering the quadrant:
The real part (a) is -11, and the imaginary part (b) is 1.
Since a is negative and b is positive, we need to add π to the result of θ = tan^(-1)(b/a):
θ = tan^(-1)(1/-11) + π = -0.089 + π.
Therefore, z2 in polar form is √122∠(-0.089 + π).
C. To find z1z2, we can use the polar form property that says: r1∠θ1 * r2∠θ2 = (r1 * r2)∠(θ1 + θ2).
In this case, z1 * z2 = (6 * √122)∠((-π/3) + (-0.089 + π)).
Simplifying, z1z2 = (6 * √122)∠(-4π/3 - 0.089).
D. To find z1/z2, we can use the polar form property: (r1 / r2)∠(θ1 - θ2).
In this case, z1 / z2 = (6 / √122)∠((-π/3) - (-0.089 + π)).
Simplifying, z1/z2 = (6 / √122)∠(-4π/3 + 0.089).
E. To find (z1)^5, we can use the De Moivre's theorem, which states: (r∠θ)^n = r^n∠(nθ).
For z1^5, we have (6∠(-π/3))^5 = 6^5∠(5 * (-π/3)).
Simplifying, (z1)^5 = 7776∠(-5π/3).
F. To find (z2)^(1/4), we take the fourth root of the magnitude and divide the argument by 4.
For z2^(1/4), we have √122^(1/4)∠((-0.089 + π)/4).
Simplifying, (z2)^(1/4) = √√122∠((-0.089 + π)/4).
Note: Please double-check all calculations to ensure accuracy.