Find the quotient z1z2 of the complex numbers z1=20(cos⁡75∘+isin⁡75∘)z2=4(cos⁡25∘+isin⁡25∘)Leave answers in polar form.

Step 1For&nbsp;z=z+iyPolar form&nbsp;⇒z=reiθTherefore for&nbsp;z1,r=(20cos⁡75∘)2+(20sin⁡75∘)2=202(cos275∘+sin275∘)=202⇒r=20θ=tan−1(20sin⁡75∘20cos⁡75∘)=75∘∴z1=20ei75∘Similary,&nbsp;z2=4ei25∘z1z2=20ei75∘4ei25∘=5ei(75∘−25∘)⇒z1z2=5ei50∘&nbsp;(Polar form)orz1z2=5(cos⁡50∘+isin⁡50∘)

Expert Answer to "Find the quotient z1z2 of the complex numbers..."

Gretchen Barker

Answered question

2022-04-06

Find the quotient

\frac{z_{1}}{z_{2}}

of the complex numbers

z_{1} = 20 ({\cos 75}^{\circ} + i {\sin 75}^{\circ})

z_{2} = 4 ({\cos 25}^{\circ} + i {\sin 25}^{\circ})

Leave answers in polar form.

Answer & Explanation

tralhavahr9c

Beginner2022-04-07Added 16 answers

Step 1
For $z = z + i y$
Polar form $\Rightarrow z = r e^{i θ}$
Therefore for $z_{1}$ ,
$r = \sqrt{{(20 {\cos 75}^{\circ})}^{2} + {(20 {\sin 75}^{\circ})}^{2}}$
$= \sqrt{20^{2} (\cos^{2} 75^{\circ} + \sin^{2} 75^{\circ})}$
$= \sqrt{20^{2}}$
$\Rightarrow r = 20$
$θ = \tan^{- 1} (\frac{20 {\sin 75}^{\circ}}{20 {\cos 75}^{\circ}}) = 75^{\circ}$
$∴ z_{1} = 20 e^{i 75^{\circ}}$
Similary, $z_{2} = 4 e^{i 25^{\circ}}$
$\frac{z_{1}}{z_{2}} = \frac{20 e^{i 75^{\circ}}}{4 e^{i 25^{\circ}}}$
$= 5 e^{i (75^{\circ} - 25^{\circ})}$
$\Rightarrow \frac{z_{1}}{z_{2}} = 5 e^{i 50^{\circ}}$ (Polar form)
or
$\frac{z_{1}}{z_{2}} = 5 ({\cos 50}^{\circ} + i {\sin 50}^{\circ})$

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