# Test: Complex Numbers III - Ambitious

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Question 1:   Write the modified polar form of $z=3\left(\mathrm{cos}240°+i\mathrm{sin}240°\right)$ using the shorthand notation
$z=3\overline{)120°}$
$z=3\overline{)240°}$
$z=3\overline{)240°}$
$z=3\overline{)120°}$
Question 2:   Write the proper polar form $z=2\left(\mathrm{cos}25°-i\mathrm{sin}25°\right)$ using the shorthand notation
$-3\overline{)25°}$
$3\overline{)25°}$
$2\overline{)335°}$
$3\overline{)-25°}$
Question 3:   Multiply the following: ${z}_{1}=x\left(\mathrm{cos}\theta +i\mathrm{sin}\theta \right)$and ${z}_{2}=y\left(\mathrm{cos}\phi +i\mathrm{sin}\phi \right)$
${z}_{1}{z}_{2}=xy\left(\mathrm{cos}\left(\theta +\phi \right)+i\mathrm{sin}\left(\theta +\phi \right)\right)$
${z}_{1}{z}_{2}=\left(x+y\right)\left(\mathrm{cos}\left(\theta +\phi \right)+i\mathrm{sin}\left(\theta +\phi \right)\right)$
${z}_{1}{z}_{2}=xy\left(\mathrm{cos}\left(\theta \phi \right)+i\mathrm{sin}\left(\theta \phi \right)\right)$
${z}_{1}{z}_{2}=xy\left(\mathrm{cos}\left(\theta +\phi \right)-i\mathrm{sin}\left(\theta +\phi \right)\right)$
Question 4:   Find ${z}_{1}{z}_{2}$, where ${z}_{1}=3\left(\mathrm{cos}15°+i\mathrm{sin}15°\right)$ and ${z}_{2}=5\left(\mathrm{cos}63°+i\mathrm{sin}63°\right)$
${z}_{1}{z}_{2}=8\left(\mathrm{cos}78°+i\mathrm{sin}78°\right)$
${z}_{1}{z}_{2}=15\left(\mathrm{cos}78°-i\mathrm{sin}78°\right)$
${z}_{1}{z}_{2}=15\left(\mathrm{cos}78°+i\mathrm{sin}78°\right)$
${z}_{1}{z}_{2}=8\left(\mathrm{cos}78°-i\mathrm{sin}78°\right)$
Question 5:   Find ${z}_{1}{z}_{2}$, where ${z}_{1}=4\left(\mathrm{cos}31°+i\mathrm{sin}31°\right)$ and ${z}_{2}=7\left(\mathrm{cos}\left(-12°\right)+i\mathrm{sin}\left(-12°\right)\right)$
${z}_{1}{z}_{2}=28\left(\mathrm{cos}19°+i\mathrm{sin}19°\right)$
${z}_{1}{z}_{2}=28\left(\mathrm{cos}43°-i\mathrm{sin}43°\right)$
${z}_{1}{z}_{2}=11\left(\mathrm{cos}19°-i\mathrm{sin}19°\right)$
${z}_{1}{z}_{2}=28\left(i\mathrm{sin}43°-\mathrm{cos}43°\right)$
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