Find the quotient z1z2 of the complex numbers z1=50(cos⁡4π3+isin⁡4π3) and z2=5(cos⁡π3+isin⁡π3) Leave the answer in polar form.

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Find the quotient z1z2 of the complex numbers  z1=50(cos⁡4π3+isin⁡4π3) and  z2=5(cos⁡π3+isin⁡π3)  Leave the answer in polar form.

Shawn Kim · Accepted Answer

Step 1  If the complex numbers z1 and z2S have the polar forms  z1=r1(cos⁡θ1+isin⁡θ1) and z2=r2(cos⁡θ2+isin⁡θ2)  Then the product and the quotient of two complex numbers is  z1z2=r1r2(cos⁡(θ1−θ2)+isin⁡(θ1−θ2)), z2≠0  The given complex numbers are  z1=50(cos⁡4π3+isin⁡4π3) and z2=5(cos⁡π3+isin⁡π3)  Substitute: r1=50, r2=5, θ1=4π3, θ2=π3 in  z1z2=r1r2(cos⁡(θ1−θ2)+isin⁡(θ1−θ2))  z1z2=505(cos⁡(4π3−π3)+isin⁡(4π3−π3))  =10(cos⁡(3π3)+isin⁡(3π3))  =10(cos⁡(π)+isin⁡(π))  The quotient of the complex number is z1z2=10(cos⁡(π)+isin⁡(π))

Serita Dewitt · Accepted Answer

Step 1  Apply complex arithmetric rule:  a+bic+di=(c−di)(a+bi)(c−di)(c+di)=(ac+bd)+(bc−ad)ic2+d2  So,  50(cos⁡(4π3)+isin⁡(4π3))5(cos⁡(π3)+isin⁡(π3))  a=−25, b=−253, c=52, d=532  =(−25×52+(−253)532)+(−25352−(−25)532)i(52)2+(532)2  Refine  =−25025  −25025=−10  =−10

Find the quotient \frac{z_{1}}{z_{2}} of the complex numbers z_{1}=50(\cos\frac{4\pi}{3}+i \sin\frac{4\pi}{3}) and z_{2}=5(\cos

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