For z1 = 4cis(7pi/6)and z2 = 3ci(pi/3), find z1 · z2 in rectangular form.
A.12 − 12i
B.-12 − 12i
C.12 + 12i
D.12i
E.-12i
just multiply moduli and add arguments:
(4)(3)cis(7pi/6 + pi/3)
12 cis 3pi/2
(E)
To find the product of complex numbers in rectangular form, you need to multiply their real and imaginary parts separately.
Given: z1 = 4cis(7π/6) and z2 = 3cis(π/3)
First, let's convert both complex numbers from polar form to rectangular form:
For z1:
- The magnitude of z1, denoted as r1, is 4.
- The argument of z1, denoted as θ1, is 7π/6.
To convert from polar to rectangular form, we can use the following formulas:
x = r * cos(θ)
y = r * sin(θ)
For z1, we have:
x1 = 4 * cos(7π/6)
y1 = 4 * sin(7π/6)
Now, let's calculate x1 and y1:
x1 = 4 * cos(7π/6) ≈ 4 * (-0.866) ≈ -3.464
y1 = 4 * sin(7π/6) ≈ 4 * 0.5 ≈ 2
Therefore, z1 in rectangular form is approximately -3.464 + 2i.
Now, let's convert z2 to rectangular form:
For z2:
- The magnitude of z2, denoted as r2, is 3.
- The argument of z2, denoted as θ2, is π/3.
Using the formulas for conversion:
x2 = 3 * cos(π/3)
y2 = 3 * sin(π/3)
Calculating x2 and y2:
x2 = 3 * cos(π/3) ≈ 3 * 0.5 ≈ 1.5
y2 = 3 * sin(π/3) ≈ 3 * sqrt(3)/2 ≈ 2.598
Therefore, z2 in rectangular form is approximately 1.5 + 2.598i.
To find z1 · z2, we can multiply the real and imaginary parts of both z1 and z2.
Using the given values:
z1 · z2 = (-3.464 + 2i) * (1.5 + 2.598i)
Now, let's perform the multiplication:
z1 · z2 = -3.464 * 1.5 + (-3.464) * 2.598i + 2i * 1.5 + 2i * 2.598i
Simplifying the result:
z1 · z2 = -5.196 - 8.986i + 3i + 5.196i^2
= -5.196 - 8.986i + 3i - 5.196 (-1)
= -5.196 - 8.986i + 3i + 5.196
= -5.196 + 5.196 - 8.986i + 3i
= 0 - 5.79i
Therefore, z1 · z2 in rectangular form is approximately 0 - 5.79i.
Since option E is the correct answer (-12i), the correct answer is not among the choices provided.