find zw
z=8(cospi/2+isinpi/2)
w=4(cospi/4+isinpi/4)
To find zw, we can use the following formula for multiplying complex numbers:
(z1 * z2) = (r1 * r2) * (cos(θ1 + θ2) + i * sin(θ1 + θ2))
where r1 and r2 are the magnitudes (or moduli) of the complex numbers, and θ1 and θ2 are the arguments (or angles).
Given z = 8(cos(pi/2) + i * sin(pi/2)) and w = 4(cos(pi/4) + i * sin(pi/4)), let's calculate zw.
First, let's write z and w in their polar form:
z = 8cis(pi/2)
w = 4cis(pi/4)
Now, let's multiply the moduli and add the angles:
r1 * r2 = 8 * 4 = 32
θ1 + θ2 = (pi/2) + (pi/4) = 3pi/4
So, we have:
zw = 32 * cis(3pi/4)
Now, let's convert it back to rectangular form:
zw = 32 * [cos(3pi/4) + i * sin(3pi/4)]
We can simplify cos(3pi/4) and sin(3pi/4) as follows:
cos(3pi/4) = -sqrt(2)/2
sin(3pi/4) = sqrt(2)/2
Now, substitute the values:
zw = 32 * [-sqrt(2)/2 + i * sqrt(2)/2]
Calculating the expression:
zw = -16sqrt(2) + 16i * sqrt(2)
Therefore, the product of z and w is -16sqrt(2) + 16i * sqrt(2).