# 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
$z=3\overline{)120°}$
$z=3\overline{)120°}$
$z=3\overline{)240°}$
$z=3\overline{)240°}$
Question 2:   Write the proper polar form $z=2\left(\mathrm{cos}25°-i\mathrm{sin}25°\right)$ using the shorthand notation
$3\overline{)25°}$
$2\overline{)335°}$
$3\overline{)-25°}$
$-3\overline{)25°}$
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 ${z}_{2}=5\left(\mathrm{cos}63°+i\mathrm{sin}63°\right)$
${z}_{1}{z}_{2}=8\left(\mathrm{cos}78°+i\mathrm{sin}78°\right)$
${z}_{1}{z}_{2}=15\left(\mathrm{cos}78°-i\mathrm{sin}78°\right)$
${z}_{1}{z}_{2}=8\left(\mathrm{cos}78°-i\mathrm{sin}78°\right)$
${z}_{1}{z}_{2}=15\left(\mathrm{cos}78°+i\mathrm{sin}78°\right)$
Question 5:   Find ${z}_{1}{z}_{2}$, where ${z}_{1}=4\left(\mathrm{cos}31°+i\mathrm{sin}31°\right)$ and ${z}_{2}=7\left(\mathrm{cos}\left(-12°\right)+i\mathrm{sin}\left(-12°\right)\right)$
${z}_{1}{z}_{2}=11\left(\mathrm{cos}19°-i\mathrm{sin}19°\right)$
${z}_{1}{z}_{2}=28\left(i\mathrm{sin}43°-\mathrm{cos}43°\right)$
${z}_{1}{z}_{2}=28\left(\mathrm{cos}43°-i\mathrm{sin}43°\right)$
${z}_{1}{z}_{2}=28\left(\mathrm{cos}19°+i\mathrm{sin}19°\right)$
Question 6:   Find $\frac{{z}_{1}}{{z}_{2}}$, where ${z}_{1}=x\left(\mathrm{cos}\theta +i\mathrm{sin}\theta \right)$ and ${z}_{2}=y\left(\mathrm{cos}\phi +i\mathrm{sin}\phi \right)$
$\frac{{z}_{1}}{{z}_{2}}=\frac{x}{y}\left(\mathrm{cos}\left(\theta -\phi \right)+i\mathrm{sin}\left(\theta -\phi \right)\right)$
$\frac{{z}_{1}}{{z}_{2}}=\frac{x}{y}\left(\mathrm{cos}\frac{\theta }{\phi }+i\mathrm{sin}\frac{\theta }{\phi }\right)$
$\frac{{z}_{1}}{{z}_{2}}=\frac{x}{y}\left(\mathrm{cos}\left(\theta -\phi \right)-i\mathrm{sin}\left(\theta -\phi \right)\right)$
$\frac{{z}_{1}}{{z}_{2}}=xy\left(\mathrm{cos}\left(\theta -\phi \right)+i\mathrm{sin}\left(\theta -\phi \right)\right)$
Question 7:   Find $\frac{{z}_{1}}{{z}_{2}}$, where ${z}_{1}=12\left(\mathrm{cos}87°+i\mathrm{sin}87°\right)$ and ${z}_{2}=4\left(\mathrm{cos}29°+i\mathrm{sin}29°\right)$
$\frac{{z}_{1}}{{z}_{2}}=3\left(\mathrm{cos}56°+i\mathrm{sin}56°\right)$
$\frac{{z}_{1}}{{z}_{2}}=8\left(\mathrm{cos}56°+i\mathrm{sin}56°\right)$
$\frac{{z}_{1}}{{z}_{2}}=3\left(\mathrm{cos}3°+i\mathrm{sin}3°\right)$
$\frac{{z}_{1}}{{z}_{2}}=8\left(\mathrm{cos}56°-i\mathrm{sin}56°\right)$
Question 8:   Find ${z}_{1}{z}_{2}{z}_{3}$, where ${z}_{1}=2\left(\mathrm{cos}15°+i\mathrm{sin}15°\right)$, ${z}_{2}=3\left(\mathrm{cos}16°+i\mathrm{sin}16°\right)$ and ${z}_{3}=4\left(\mathrm{cos}17°+i\mathrm{sin}17°\right)$
${z}_{1}{z}_{2}{z}_{3}=24\left(\mathrm{cos}120°-i\mathrm{sin}120°\right)$
${z}_{1}{z}_{2}{z}_{3}=24\left(\mathrm{cos}48°+i\mathrm{sin}48°\right)$
${z}_{1}{z}_{2}{z}_{3}=9\left(\mathrm{cos}48°+i\mathrm{sin}48°\right)$
${z}_{1}{z}_{2}{z}_{3}=24\left(\mathrm{cos}120°+i\mathrm{sin}120°\right)$
Question 9:   Find ${z}^{n}$, where $z=r\left(\mathrm{cos}\theta +i\mathrm{sin}\theta \right)$
${z}^{n}={r}^{n}\left(\mathrm{cos}n\theta +i\mathrm{sin}n\theta \right)$
${z}^{n}={r}^{n}\left({\mathrm{cos}}^{n}\theta -i{\mathrm{sin}}^{n}\theta \right)$
${z}^{n}={r}^{n}\left({\mathrm{cos}}^{n}n\theta +i{\mathrm{sin}}^{n}n\theta \right)$
${z}^{n}=nr\left(n\mathrm{cos}\theta +in\mathrm{sin}\theta \right)$
Question 10:   Find ${z}^{3}$, where $z=3\left(\mathrm{cos}25°+i\mathrm{sin}25°\right)$
$27\left(3\mathrm{cos}25°+i3\mathrm{sin}25°\right)$
$9\left(\mathrm{cos}75°+i\mathrm{sin}75°\right)$
$27\left(\mathrm{cos}75°+i\mathrm{sin}75°\right)$
$27\left({\mathrm{cos}}^{3}25°+i{\mathrm{sin}}^{3}25°\right)$