Using mathematically precise language, explain in detail how you would multiply the complex number z1=r1(cos theta1+ i sin theta 1 with the complex number .

Answer 1

To multiply two complex numbers z1 and z2, we can use the properties of complex numbers in polar form. Let z1 = r1(cosθ1 + i sinθ1) and z2 = r2(cosθ2 + i sinθ2).

The product of z1 and z2 can be calculated as follows:

z1 * z2 = r1(cosθ1 + i sinθ1) * r2(cosθ2 + i sinθ2)
= r1 * r2 * (cosθ1 + i sinθ1)(cosθ2 + i sinθ2)

Using the trigonometric identity for the product of two angles, we can expand the product of the trigonometric functions in the form: cos(A + B) = cosAcosB - sinAsinB and sin(A + B) = sinAcosB + cosAsinB. Applying these identities to the above expression, we get:

z1 * z2 = r1 * r2 * [(cosθ1cosθ2 - sinθ1sinθ2) + i(sinθ1cosθ2 + cosθ1sinθ2)]
= r1 * r2 * [cos(θ1 + θ2) + i sin(θ1 + θ2)]

Therefore, the product of z1 and z2 is given by z1 * z2 = r1r2(cos(θ1 + θ2) + i sin(θ1 + θ2). This is the result of multiplying two complex numbers in polar form.