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

Danelle Albright

Danelle Albright

Answered question

2021-12-13

Find the quotient z1z2 of the complex numbers
z1=50(cos4π3+isin4π3) and
z2=5(cosπ3+isinπ3)
Leave the answer in polar form.

Answer & Explanation

Shawn Kim

Shawn Kim

Beginner2021-12-14Added 25 answers

Step 1
If the complex numbers z1 and z2S have the polar forms
z1=r1(cosθ1+isinθ1) and z2=r2(cosθ2+isinθ2)
Then the product and the quotient of two complex numbers is
z1z2=r1r2(cos(θ1θ2)+isin(θ1θ2)), z20
The given complex numbers are
z1=50(cos4π3+isin4π3) and z2=5(cosπ3+isinπ3)
Substitute: r1=50, r2=5, θ1=4π3, θ2=π3 in
z1z2=r1r2(cos(θ1θ2)+isin(θ1θ2))
z1z2=505(cos(4π3π3)+isin(4π3π3))
=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

Serita Dewitt

Beginner2021-12-15Added 41 answers

Step 1
Apply complex arithmetric rule:
a+bic+di=(cdi)(a+bi)(cdi)(c+di)=(ac+bd)+(bcad)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

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