Using mathematically precise language, explain in detail how you would multiply the complex number z1=r1(cos theta1 + I sin theta1) with the complex number z2=r2(cos theta 2 + I sin theta 2)

Answer 1

To multiply these two complex numbers, we first expand them using Euler's formula:

z1 = r1(cos θ1 + i sin θ1) = r1e^(iθ1)
z2 = r2(cos θ2 + i sin θ2) = r2e^(iθ2)

Now, multiply the two complex numbers together:

z1 * z2 = r1e^(iθ1) * r2e^(iθ2)
= r1 * r2 * e^(i(θ1 + θ2))

Using Euler's formula again, we can rewrite this product as:

z1 * z2 = r1 * r2 * (cos(θ1 + θ2) + i sin(θ1 + θ2))

Therefore, the product of the two complex numbers z1 and z2 is equal to r1 * r2 * (cos(θ1 + θ2) + i sin(θ1 + θ2).