j548831
Answered

Welcome to Westonci.ca, your ultimate destination for finding answers to a wide range of questions from experts. Join our platform to connect with experts ready to provide precise answers to your questions in various areas. Discover in-depth answers to your questions from a wide network of professionals on our user-friendly Q&A platform.

20 POINTS
Using mathematically precise language, explain in detail how you would multiply the complex number
z1=r1 (cos theta1+isin theta 1)
with the complex number
z2=r2 (cos theta 2+isin theta 2)

r-radius/radii
m-modulus/moduli
theta-argument represent angle theta

Sagot :

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

Thanks for using our platform. We're always here to provide accurate and up-to-date answers to all your queries. Thank you for your visit. We're dedicated to helping you find the information you need, whenever you need it. We're here to help at Westonci.ca. Keep visiting for the best answers to your questions.