For z1 = 4cis (7pi/6) and z2 = 3cis(pi/3), find z1 * z2 in rectangular form.
short cut
4cis (7pi/6)*3cis(pi/3)
= 12 cis(7π/6+π/3) , by De Moivre's theorem
= 12cis(3π/2)
but you wanted it in rectangular form
= 12(cos 3π/2 + i sin 3π/2)
= 12(0 - 1 i)
= -12i
long way (if you don't know the above law):
4cis (7pi/6)
= 4(cos 7π/6 + i sin 7π/6)
= 4(-√3/2 - 1/2 i)
= (-2√3 - 2i)
3cis(pi/3)
= 3(1/2 + √3/2 i)
= (3/2 + 3√3/2 i)
4cis (7pi/6)*3cis(pi/3)
= (-2√3 - 2i)(3/2 + 3√3/2 i)
= -3√3 - 9i - 3i - 6√3 i^2
= -12 i
To find the product of two complex numbers in rectangular form, we can multiply their magnitudes and add their arguments.
Given:
z1 = 4cis(7pi/6)
z2 = 3cis(pi/3)
First, let's convert the complex numbers from polar form to rectangular form:
For z1:
Magnitude (r1) = 4
Argument (θ1) = 7pi/6
Using Euler's formula, we can convert this to rectangular form:
z1 = r1 * cos(θ1) + r1 * sin(θ1) * i
z1 = 4 * cos(7pi/6) + 4 * sin(7pi/6) * i
z1 = 4 * (-√3/2) + 4 * (-1/2) * i
z1 = -2√3 - 2i
For z2:
Magnitude (r2) = 3
Argument (θ2) = pi/3
Using Euler's formula, we can convert this to rectangular form:
z2 = r2 * cos(θ2) + r2 * sin(θ2) * i
z2 = 3 * cos(pi/3) + 3 * sin(pi/3) * i
z2 = 3 * (1/2) + 3 * (√3/2) * i
z2 = 3/2 + 3√3/2 i
Now, let's find the product of z1 and z2 in rectangular form:
z1 * z2 = (-2√3 - 2i) * (3/2 + 3√3/2 i)
To multiply complex numbers in rectangular form, we multiply the real parts and the imaginary parts separately:
z1 * z2 = -2√3 * 3/2 + (-2i) * 3/2 + (-2√3) * 3√3/2 i + (-2i) * 3√3/2 i
Simplifying, we get:
z1 * z2 = -3√3 - 3i√3 - 9 + 3√3 i
z1 * z2 = -12 - 6√3 i
Therefore, the product of z1 and z2 in rectangular form is -12 - 6√3 i.
To find the multiplication of complex numbers in rectangular form, we need to convert the given numbers from polar form to rectangular form, perform the multiplication, and then convert the result back to polar form if necessary.
Given:
z₁ = 4cis(7π/6)
z₂ = 3cis(π/3)
Step 1: Convert z₁ to rectangular form.
To convert from polar form to rectangular form, we use the following formulas:
x = r * cos(θ)
y = r * sin(θ)
For z₁:
r = 4 (magnitude or modulus)
θ = 7π/6 (angle or argument)
x₁ = 4 * cos(7π/6)
y₁ = 4 * sin(7π/6)
To calculate these values, we need to know the cosine and sine values of 7π/6. We can use the reference angle of π/6:
cos(π/6) = √3/2
sin(π/6) = 1/2
Since 7π/6 is in the third quadrant, the cosine will be negative and the sine will be negative.
cos(7π/6) = -cos(π/6) = -√3/2
sin(7π/6) = -sin(π/6) = -1/2
x₁ = 4 * (-√3/2) = -2√3
y₁ = 4 * (-1/2) = -2
Therefore, z₁ in rectangular form is:
z₁ = -2√3 - 2i
Step 2: Convert z₂ to rectangular form.
For z₂:
r = 3
θ = π/3
x₂ = 3 * cos(π/3)
y₂ = 3 * sin(π/3)
Using the known values:
cos(π/3) = 1/2
sin(π/3) = √3/2
x₂ = 3 * (1/2) = 3/2
y₂ = 3 * (√3/2) = (3√3)/2
Therefore, z₂ in rectangular form is:
z₂ = (3/2) + ((3√3)/2)i
Step 3: Multiply the numbers in rectangular form.
To multiply complex numbers, we can use the FOIL method just like multiplying binomials.
z = z₁ * z₂
z = (-2√3 - 2i) * ((3/2) + ((3√3)/2)i)
Using the distributive property, we can simplify this expression:
z = (-2√3 * (3/2)) + (-2√3 * ((3√3)/2)i) + (-2i * (3/2)) + (-2i * ((3√3)/2)i)
Simplifying further:
z = -3√3 - 9/2i - 3i + (9√3)/2i²
Since i² = -1, we can simplify:
z = -3√3 - 9/2i - 3i - (9√3)/2
Combining like terms:
z = -3√3 - (9√3)/2 - (9/2)i - 3i
Finally, we have:
z = -3√3 - (9√3)/2 - (9/2 + 3)i
So the multiplication of z₁ and z₂ in rectangular form is:
z = -3√3 - (9√3)/2 - (9/2 + 3)i