Question

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 theta2 + i sin theta2).

Answer 1

$r_1$ and $r_2$ are, respectively, the moduli, while $\theta_1$ and $\theta_2$ are, respectively, the arguments of $z_1$ and $z_2$.

The product $z_1z_2$ can be found by multiplying the moduli and adding the arguments to get a number of the form

$z_1z_2=r_1r_2(\cos(\theta_1+\theta_2)+i\sin(\theta_1+\theta_2))$

Answer 2

Answer:

Step-by-step explanation:

Given: One complex number is $z_{1}=r_{1}(cos{\theta}_{1}+isin{\theta}_{1})$ and the another complex number is $z_{2}=r_{2}(cos{\theta}_{2}+isin{\theta}_{2})$

To find: The product of the two complex numbers.

Solution:

One complex number is $z_{1}=r_{1}(cos{\theta}_{1}+isin{\theta}_{1})$ and the another complex number is $z_{2}=r_{2}(cos{\theta}_{2}+isin{\theta}_{2})$, the product of two complex numbers is given as:

$z_{1}\times}z_{2}=(r_{1}(cos{\theta}_{1}+isin{\theta}_{1}){\times}r_{2}(cos{\theta}_{2}+isin{\theta}_{2}))$

$z_{1}z_{2}=r_{1}r_{2}(cos{\theta}_{1}+isin{\theta}_{1})(cos{\theta}_{2}+isin{\theta}_{2})$

$z_{1}z_{2}=r_{1}r_{2}(cos({\theta}_{1}+{\theta}_{2})+isin(({\theta}_{1}+{\theta}_{2}))$

which is the required product of the two given complex numbers.