The product complex number is z1z2 = r1r2[cos (θ1+θ2) + isin (θ1+θ2)]
[tex]Z_1\times Z_2= r_1(cos \theta_1 + isin \theta_1) \times r_2(cos \theta_2 + isin \theta_2)[/tex]
r-radius/radii
m-modulus/moduli
theta-argument represent angle theta
[tex]= r_1r_2(cos \theta_1 + i sin \theta_1) (cos \theta_2 + isin \theta_2)[/tex]
[tex]= r_1r_2(cos \theta_1 cos \theta_2 + i cos \theta_1 sin \theta_2 + i sin \theta_1 cos \theta_2 + i^2sin\theta_1sin\theta_2[/tex]
What is the value of i square in complex number?
The value of i square is
[tex]i^2=-1[/tex]
Therefore,we get,
[tex]= r_1r_2(cos \theta_1 cos \theta_2 + i cos \theta_1 sin \theta_2 + i sin \theta_1 cos \theta_2 + (-1)sin \theta_1 sin \theta_2)[/tex]
[tex]= r_1r_2(cos \theta_1 cos \theta_2 + i cos \theta_1 sin \theta_2 + i sin \theta_1 cos \theta_2 - sin \theta_1 sin \theta_2)[/tex]
Since we know that,
[tex]cos(\theta_1+\theta_2)=cos\theta_1cos\theta_2-sin\theta_1sin\theta_2[/tex]
[tex]sin(\theta_1+\theta_2)=sin\theta_1cos\theta_2+cos\theta_1sin\theta_2[/tex]
So using above value we get,
[tex]= r_1r_2(cos \theta_1 cos \theta_2 - sin \theta_1 sin \theta_2) + i (cos \theta_1 sin \theta_2 + sin \theta_1 cos \theta_2)[/tex]
[tex]=r_1r_2cos(\theta_1+\theta_2)+isin(\theta_1+\theta_2)[/tex]
Therfore the product of the complex number is z1z2 = r1r2[cos (θ1+θ2) + isin (θ1+θ2)].
To learn more about the product of complex number visit:
https://brainly.com/question/1462345
