Betsy Rhone

2022-01-16

Find the quotient
$\frac{{z}_{1}}{{z}_{2}}$
of the complez numbers
${z}_{1}50\left({\mathrm{cos}80}^{\circ }+i{\mathrm{sin}80}^{\circ }\right)$
and
${z}_{2}=4\left({\mathrm{cos}25}^{\circ }+i{\mathrm{sin}25}^{\circ }$
Leave answers in polar form.

hysgubwyri3

Expert

Step 1
So rewriting the complex numbers,
${z}_{1}=50\left({\mathrm{cos}80}^{\circ }+i{\mathrm{sin}80}^{\circ }\right)$
$=50\left({e}^{i{80}^{\circ }}\right)$
and
${z}_{2}=4\left({\mathrm{cos}25}^{\circ }+i{\mathrm{sin}25}^{\circ }\right)$
$=4\left({e}^{i{25}^{\circ }}\right)$
Step 2
Finding quotient $\frac{{z}_{1}}{{z}_{2}}$ using above results, we get
$\frac{{z}_{1}}{{z}_{2}}=\frac{50\left({e}^{i{80}^{\circ }}\right)}{4\left({e}^{i{25}^{\circ }}\right)}$
$=\frac{25}{2}{e}^{i{80}^{\circ }-i{25}^{\circ }}$
$=\frac{25}{2}{e}^{i{55}^{\circ }}$
$=\frac{25}{2}\left({\mathrm{cos}55}^{\circ }+i{\mathrm{sin}55}^{\circ }\right)$
Hence, value of quotient $\frac{{z}_{1}}{{z}_{2}}$ is $\frac{25}{2}\left({\mathrm{cos}55}^{\circ }+i{\mathrm{sin}55}^{\circ }\right)$

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