Find z1z2 and z1/z2 for the pair of complex numbers using trigonometric form.
z1=4+9i, z2=-5-5i
I got z1z2 = 25-65i
But cannot find the z1/z2.
z1/z2 = (4+9i)/(-5-5i)
Multiply top and bottom by the denominator's conjugate:
(4+9i)(-5+5i) / (-5-5i)(-5+5i)
(-65-25i) / (5^2+5^2)
-65/50 - 25/50 i
-13/10 - 1/2 i
To find z1z2, you just need to multiply the two complex numbers together.
Given:
z1 = 4 + 9i
z2 = -5 - 5i
To multiply complex numbers, we can use the trigonometric form which is expressed as:
z = r * (cos(θ) + i * sin(θ))
In trigonometric form, we can express z1 as:
z1 = r1 * (cos(θ1) + i * sin(θ1))
Similarly, we can express z2 as:
z2 = r2 * (cos(θ2) + i * sin(θ2))
To find z1z2, we can multiply their magnitudes and add their angles.
First, let's calculate the magnitudes of z1 and z2:
The magnitude is given by:
r = sqrt(a^2 + b^2), where a and b are the real and imaginary parts of the complex number respectively.
For z1:
a = 4
b = 9
|r1| = sqrt(4^2 + 9^2) = sqrt(16 + 81) = sqrt(97)
For z2:
a = -5
b = -5
|r2| = sqrt((-5)^2 + (-5)^2) = sqrt(25 + 25) = sqrt(50)
Next, let's calculate the angles of z1 and z2:
The angle is given by:
θ = arctan(b/a), where a and b are the real and imaginary parts of the complex number respectively.
For z1:
a = 4
b = 9
θ1 = arctan(9/4)
For z2:
a = -5
b = -5
θ2 = arctan(-5/-5) = arctan(1) = π/4 (Since the argument is the same for both numbers)
So now we have:
z1 = sqrt(97) * (cos(θ1) + i * sin(θ1))
z2 = sqrt(50) * (cos(θ2) + i * sin(θ2))
Now, multiply z1 and z2 together:
z1z2 = sqrt(97 * 50) * (cos(θ1 + θ2) + i * sin(θ1 + θ2))
After simplification, we get:
z1z2 = sqrt(4850) * (cos(θ1 + π/4) + i * sin(θ1 + π/4))
Note that we need to combine the angles θ1 and π/4 in this case.
To calculate z1/z2, we can divide z1 by z2:
z1/z2 = (sqrt(97) * (cos(θ1) + i * sin(θ1))) / (sqrt(50) * (cos(θ2) + i * sin(θ2)))
To divide complex numbers, we multiply the numerator and denominator by the complex conjugate of the denominator.
The complex conjugate of z2 is:
z2* = sqrt(50) * (cos(θ2) - i * sin(θ2))
So we have:
z1/z2 = (sqrt(97) * (cos(θ1) + i * sin(θ1))) * (sqrt(50) * (cos(θ2) - i * sin(θ2))) / (sqrt(50) * (cos(θ2) + i * sin(θ2))) * (sqrt(50) * (cos(θ2) - i * sin(θ2)))
After simplification, we get:
z1/z2 = (sqrt(4850) * (cos(θ1 - θ2) + i * sin(θ1 - θ2))) / 50
So the final result is:
z1z2 = sqrt(4850) * (cos(θ1 + π/4) + i * sin(θ1 + π/4))
z1/z2 = (sqrt(4850) * (cos(θ1 - θ2) + i * sin(θ1 - θ2))) / 50
You can calculate the values for z1z2 and z1/z2 using a calculator by substituting the values for θ1 and θ2 that we calculated earlier.