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Evaluate the cube roots of $\text{\hspace{0.17em}}z=8\left(\mathrm{cos}\left(\frac{2\pi}{3}\right)+i\mathrm{sin}\left(\frac{2\pi}{3}\right)\right).$
We have
There will be three roots: $\text{\hspace{0.17em}}k=0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}1,\text{\hspace{0.17em}}\text{\hspace{0.17em}}2.\text{\hspace{0.17em}}$ When $\text{\hspace{0.17em}}k=0,\text{\hspace{0.17em}}$ we have
When $\text{\hspace{0.17em}}k=1,\text{\hspace{0.17em}}$ we have
When $\text{\hspace{0.17em}}k=2,\text{\hspace{0.17em}}$ we have
Remember to find the common denominator to simplify fractions in situations like this one. For $\text{\hspace{0.17em}}k=1,\text{\hspace{0.17em}}$ the angle simplification is
Find the four fourth roots of $\text{\hspace{0.17em}}16(\mathrm{cos}(\mathrm{120\xb0})+i\mathrm{sin}(\mathrm{120\xb0})).$
${z}_{0}=2(\mathrm{cos}(\mathrm{30\xb0})+i\mathrm{sin}(\mathrm{30\xb0}))$
${z}_{1}=2(\mathrm{cos}(\mathrm{120\xb0})+i\mathrm{sin}(\mathrm{120\xb0}))$
${z}_{2}=2(\mathrm{cos}(\mathrm{210\xb0})+i\mathrm{sin}(\mathrm{210\xb0}))$
${z}_{3}=2(\mathrm{cos}(\mathrm{300\xb0})+i\mathrm{sin}(\mathrm{300\xb0}))$
Access these online resources for additional instruction and practice with polar forms of complex numbers.
A complex number is $\text{\hspace{0.17em}}a+bi.\text{\hspace{0.17em}}$ Explain each part.
a is the real part, b is the imaginary part, and $\text{\hspace{0.17em}}i=\sqrt{-1}$
What does the absolute value of a complex number represent?
How is a complex number converted to polar form?
Polar form converts the real and imaginary part of the complex number in polar form using $\text{\hspace{0.17em}}x=r\mathrm{cos}\theta \text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}y=r\mathrm{sin}\theta .$
How do we find the product of two complex numbers?
What is De Moivre’s Theorem and what is it used for?
${z}^{n}={r}^{n}\left(\mathrm{cos}\left(n\theta \right)+i\mathrm{sin}\left(n\theta \right)\right)\text{\hspace{0.17em}}$ It is used to simplify polar form when a number has been raised to a power.
For the following exercises, find the absolute value of the given complex number.
$5+\text{}3i$
For the following exercises, write the complex number in polar form.
$8-4i$
$4\sqrt{5}\mathrm{cis}\left(\mathrm{333.4\xb0}\right)$
$-\frac{1}{2}-\frac{1}{2}\text{}i$
For the following exercises, convert the complex number from polar to rectangular form.
$z=7\mathrm{cis}\left(\frac{\pi}{6}\right)$
$\frac{7\sqrt{3}}{2}+i\frac{7}{2}$
$z=2\mathrm{cis}\left(\frac{\pi}{3}\right)$
$z=4\mathrm{cis}\left(\frac{7\pi}{6}\right)$
$-2\sqrt{3}-2i$
$z=7\mathrm{cis}\left(\mathrm{25\xb0}\right)$
$z=3\mathrm{cis}\left(\mathrm{240\xb0}\right)$
$-1.5-i\frac{3\sqrt{3}}{2}$
$z=\sqrt{2}\mathrm{cis}\left(\mathrm{100\xb0}\right)$
For the following exercises, find $\text{\hspace{0.17em}}{z}_{1}{z}_{2}\text{\hspace{0.17em}}$ in polar form.
${z}_{1}=2\sqrt{3}\mathrm{cis}\left(\mathrm{116\xb0}\right);\text{\hspace{0.17em}}\text{}{z}_{2}=2\mathrm{cis}\left(\mathrm{82\xb0}\right)$
$4\sqrt{3}\mathrm{cis}\left(\mathrm{198\xb0}\right)$
${z}_{1}=\sqrt{2}\mathrm{cis}\left(\mathrm{205\xb0}\right);\text{}{z}_{2}=2\sqrt{2}\mathrm{cis}\left(\mathrm{118\xb0}\right)$
${z}_{1}=3\mathrm{cis}\left(\mathrm{120\xb0}\right);\text{}{z}_{2}=\frac{1}{4}\mathrm{cis}\left(\mathrm{60\xb0}\right)$
$\frac{3}{4}\mathrm{cis}\left(\mathrm{180\xb0}\right)$
${z}_{1}=3\mathrm{cis}\left(\frac{\pi}{4}\right);\text{}{z}_{2}=5\mathrm{cis}\left(\frac{\pi}{6}\right)$
${z}_{1}=\sqrt{5}\mathrm{cis}\left(\frac{5\pi}{8}\right);\text{}{z}_{2}=\sqrt{15}\mathrm{cis}\left(\frac{\pi}{12}\right)$
$5\sqrt{3}\mathrm{cis}\left(\frac{17\pi}{24}\right)$
${z}_{1}=4\mathrm{cis}\left(\frac{\pi}{2}\right);\text{}{z}_{2}=2\mathrm{cis}\left(\frac{\pi}{4}\right)$
For the following exercises, find $\text{\hspace{0.17em}}\frac{{z}_{1}}{{z}_{2}}\text{\hspace{0.17em}}$ in polar form.
${z}_{1}=21\mathrm{cis}\left(\mathrm{135\xb0}\right);\text{}{z}_{2}=3\mathrm{cis}\left(\mathrm{65\xb0}\right)$
$7\mathrm{cis}\left(\mathrm{70\xb0}\right)$
${z}_{1}=\sqrt{2}\mathrm{cis}\left(\mathrm{90\xb0}\right);\text{}{z}_{2}=2\mathrm{cis}\left(\mathrm{60\xb0}\right)$
${z}_{1}=15\mathrm{cis}\left(\mathrm{120\xb0}\right);\text{}{z}_{2}=3\mathrm{cis}\left(\mathrm{40\xb0}\right)$
$5\mathrm{cis}\left(\mathrm{80\xb0}\right)$
${z}_{1}=6\mathrm{cis}\left(\frac{\pi}{3}\right);\text{}{z}_{2}=2\mathrm{cis}\left(\frac{\pi}{4}\right)$
${z}_{1}=5\sqrt{2}\mathrm{cis}\left(\pi \right);\text{}{z}_{2}=\sqrt{2}\mathrm{cis}\left(\frac{2\pi}{3}\right)$
$5\mathrm{cis}\left(\frac{\pi}{3}\right)$
${z}_{1}=2\mathrm{cis}\left(\frac{3\pi}{5}\right);\text{}{z}_{2}=3\mathrm{cis}\left(\frac{\pi}{4}\right)$
For the following exercises, find the powers of each complex number in polar form.
Find $\text{\hspace{0.17em}}{z}^{3}\text{\hspace{0.17em}}$ when $\text{\hspace{0.17em}}z=5\mathrm{cis}\left(\mathrm{45\xb0}\right).$
$125\mathrm{cis}\left(\mathrm{135\xb0}\right)$
Find $\text{\hspace{0.17em}}{z}^{4}\text{\hspace{0.17em}}$ when $\text{\hspace{0.17em}}z=2\mathrm{cis}\left(\mathrm{70\xb0}\right).$
Find $\text{\hspace{0.17em}}{z}^{2}\text{\hspace{0.17em}}$ when $\text{\hspace{0.17em}}z=3\mathrm{cis}\left(\mathrm{120\xb0}\right).$
$9\mathrm{cis}\left(\mathrm{240\xb0}\right)$
Find $\text{\hspace{0.17em}}{z}^{2}\text{\hspace{0.17em}}$ when $\text{\hspace{0.17em}}z=4\mathrm{cis}\left(\frac{\pi}{4}\right).$
Find $\text{\hspace{0.17em}}{z}^{4}\text{\hspace{0.17em}}$ when $\text{\hspace{0.17em}}z=\mathrm{cis}\left(\frac{3\pi}{16}\right).$
$\mathrm{cis}\left(\frac{3\pi}{4}\right)$
Find $\text{\hspace{0.17em}}{z}^{3}\text{\hspace{0.17em}}$ when $\text{\hspace{0.17em}}z=3\mathrm{cis}\left(\frac{5\pi}{3}\right).$
For the following exercises, evaluate each root.
Evaluate the cube root of $\text{\hspace{0.17em}}z\text{\hspace{0.17em}}$ when $\text{\hspace{0.17em}}z=27\mathrm{cis}\left(\mathrm{240\xb0}\right).$
$\text{\hspace{0.17em}}3\mathrm{cis}\left(\mathrm{80\xb0}\right),3\mathrm{cis}\left(\mathrm{200\xb0}\right),3\mathrm{cis}\left(\mathrm{320\xb0}\right)\text{\hspace{0.17em}}$
Evaluate the square root of $\text{\hspace{0.17em}}z\text{\hspace{0.17em}}$ when $\text{\hspace{0.17em}}z=16\mathrm{cis}\left(\mathrm{100\xb0}\right).$
Evaluate the cube root of $\text{\hspace{0.17em}}z\text{\hspace{0.17em}}$ when $\text{\hspace{0.17em}}z=32\mathrm{cis}\left(\frac{2\pi}{3}\right).$
$\text{\hspace{0.17em}}2\sqrt[3]{4}\mathrm{cis}\left(\frac{2\pi}{9}\right),2\sqrt[3]{4}\mathrm{cis}\left(\frac{8\pi}{9}\right),2\sqrt[3]{4}\mathrm{cis}\left(\frac{14\pi}{9}\right)$
Evaluate the square root of $\text{\hspace{0.17em}}z\text{\hspace{0.17em}}$ when $\text{\hspace{0.17em}}z=32\text{cis}\left(\pi \right).$
Evaluate the cube root of $\text{\hspace{0.17em}}z\text{\hspace{0.17em}}$ when $\text{\hspace{0.17em}}z=8\mathrm{cis}\left(\frac{7\pi}{4}\right).$
$2\sqrt{2}\mathrm{cis}\left(\frac{7\pi}{8}\right),2\sqrt{2}\mathrm{cis}\left(\frac{15\pi}{8}\right)$
For the following exercises, plot the complex number in the complex plane.
For the following exercises, find all answers rounded to the nearest hundredth.
Use the rectangular to polar feature on the graphing calculator to change $\text{\hspace{0.17em}}5+5i\text{\hspace{0.17em}}$ to polar form.
Use the rectangular to polar feature on the graphing calculator to change $\text{\hspace{0.17em}}3-2i\text{\hspace{0.17em}}$ to polar form.
$\text{\hspace{0.17em}}3.61{e}^{-0.59i}\text{\hspace{0.17em}}$
Use the rectangular to polar feature on the graphing calculator to change $-3-8i\text{\hspace{0.17em}}$ to polar form.
Use the polar to rectangular feature on the graphing calculator to change $\text{\hspace{0.17em}}4\mathrm{cis}\left(\mathrm{120\xb0}\right)\text{\hspace{0.17em}}$ to rectangular form.
$\text{\hspace{0.17em}}-2+3.46i\text{\hspace{0.17em}}$
Use the polar to rectangular feature on the graphing calculator to change $\text{\hspace{0.17em}}2\mathrm{cis}\left(\mathrm{45\xb0}\right)\text{\hspace{0.17em}}$ to rectangular form.
Use the polar to rectangular feature on the graphing calculator to change $\text{\hspace{0.17em}}5\mathrm{cis}\left(\mathrm{210\xb0}\right)\text{\hspace{0.17em}}$ to rectangular form.
$\text{\hspace{0.17em}}-4.33-2.50i\text{\hspace{0.17em}}$
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