percibaa8

2022-01-18

Find the quotient $\frac{{z}_{1}}{{z}_{2}}$ of the complex numbers
${z}_{1}=10\left({\mathrm{cos}10}^{\circ }+i{\mathrm{sin}10}^{\circ }\right)$
and ${z}_{2}=5\left({\mathrm{cos}5}^{\circ }+i{\mathrm{sin}5}^{\circ }\right)$

lovagwb

Expert

Step 1
$\frac{{z}_{1}}{{z}_{2}}=\frac{10\left({\mathrm{cos}10}^{\circ }+i{\mathrm{sin}10}^{\circ }\right)}{5\left({\mathrm{cos}5}^{\circ }+i{\mathrm{sin}5}^{\circ }\right)}$
$⇒\frac{{z}_{1}}{{z}_{2}}=\frac{2\left({\mathrm{cos}10}^{\circ }+i{\mathrm{sin}10}^{\circ }\right)}{\left({\mathrm{cos}5}^{\circ }+i{\mathrm{sin}5}^{\circ }\right)}$
$⇒\frac{{z}_{1}}{{z}_{2}}=2\left[\frac{\left({\mathrm{cos}10}^{\circ }+i{\mathrm{sin}10}^{\circ }\right)}{\left({\mathrm{cos}5}^{\circ }+i{\mathrm{sin}5}^{\circ }\right)}×\frac{\left({\mathrm{cos}5}^{\circ }-i{\mathrm{sin}5}^{\circ }\right)}{\left({\mathrm{cos}5}^{\circ }-i{\mathrm{sin}5}^{\circ }\right)}\right]$
$⇒\frac{{z}_{1}}{{z}_{2}}=2\left[\frac{{\mathrm{cos}10}^{\circ }{\mathrm{cos}5}^{\circ }-i{\mathrm{sin}5}^{\circ }{\mathrm{cos}10}^{\circ }+i{\mathrm{sin}10}^{\circ }{\mathrm{cos}5}^{\circ }-i{\mathrm{sin}10}^{\circ }i{\mathrm{sin}5}^{\circ }}{\left({\mathrm{cos}5}^{\circ }+i{\mathrm{sin}5}^{\circ }\right)\left({\mathrm{cos}5}^{\circ }-i{\mathrm{sin}5}^{\circ }\right)}\right]$
$⇒\frac{{z}_{1}}{{z}_{2}}=2\left[\frac{{\mathrm{cos}10}^{\circ }{\mathrm{cos}5}^{\circ }+i{\mathrm{sin}5}^{\circ }{\mathrm{cos}10}^{\circ }-i{\mathrm{sin}10}^{\circ }{\mathrm{cos}5}^{\circ }+i{\mathrm{sin}10}^{\circ }i{\mathrm{sin}5}^{\circ }}{\left({\mathrm{cos}5}^{\circ }+i{\mathrm{sin}5}^{\circ }\right)\left({\mathrm{cos}5}^{\circ }-i{\mathrm{sin}5}^{\circ }\right)}\right]$

Lynne Trussell

Expert

Step 1
The quotient $\frac{{z}_{1}}{{z}_{2}}$ of the complex number, and leave the answer in Polar form.
Given: The complex numbers are ${z}_{1}=10\left({\mathrm{cos}10}^{\circ }+i{\mathrm{sin}10}^{\circ }\right)$ and ${z}_{2}=5\left({\mathrm{cos}5}^{\circ }+i{\mathrm{sin}5}^{\circ }\right)$
Concept used:
$\left(\mathrm{cos}\theta +i\mathrm{sin}\theta \right)={e}^{i\theta }$
Step 2
Calculation:
The quotient $\frac{{z}_{1}}{{z}_{2}}$ can be obtained as,
$\frac{{z}_{1}}{{z}_{2}}=\frac{10\left({\mathrm{cos}10}^{\circ }+i{\mathrm{sin}10}^{\circ }\right)}{5\left({\mathrm{cos}5}^{\circ }+i{\mathrm{sin}5}^{\circ }\right)}$
$=\frac{2{e}^{10i}}{{e}^{{5}^{\circ }i}}$
$=2{e}^{\left(10i-5i\right)}$
$=2\left({\mathrm{cos}5}^{\circ }+i{\mathrm{sin}5}^{\circ }\right)$
Thus, the quotient $\frac{{z}_{1}}{{z}_{2}}$ of the complex numbers is $\frac{{z}_{1}}{{z}_{2}}=2\left({\mathrm{cos}5}^{\circ }+i{\mathrm{sin}5}^{\circ }\right)$

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