10.5 Polar form of complex numbers  (Page 4/8)

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Finding the n Th root of a complex number

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

$\begin{array}{l}{z}^{\frac{1}{3}}={8}^{\frac{1}{3}}\left[\mathrm{cos}\left(\frac{\frac{2\pi }{3}}{3}+\frac{2k\pi }{3}\right)+i\mathrm{sin}\left(\frac{\frac{2\pi }{3}}{3}+\frac{2k\pi }{3}\right)\right]\hfill \\ {z}^{\frac{1}{3}}=2\left[\mathrm{cos}\left(\frac{2\pi }{9}+\frac{2k\pi }{3}\right)+i\mathrm{sin}\left(\frac{2\pi }{9}+\frac{2k\pi }{3}\right)\right]\hfill \end{array}$

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

${z}^{\frac{1}{3}}=2\left(\mathrm{cos}\left(\frac{2\pi }{9}\right)+i\mathrm{sin}\left(\frac{2\pi }{9}\right)\right)$

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

$\begin{array}{l}\frac{\frac{2\pi }{3}}{3}+\frac{2\left(1\right)\pi }{3}=\frac{2\pi }{3}\left(\frac{1}{3}\right)+\frac{2\left(1\right)\pi }{3}\left(\frac{3}{3}\right)\hfill \\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=\frac{2\pi }{9}+\frac{6\pi }{9}\hfill \\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=\frac{8\pi }{9}\hfill \end{array}$

Find the four fourth roots of $\text{\hspace{0.17em}}16\left(\mathrm{cos}\left(120°\right)+i\mathrm{sin}\left(120°\right)\right).$

${z}_{0}=2\left(\mathrm{cos}\left(30°\right)+i\mathrm{sin}\left(30°\right)\right)$

${z}_{1}=2\left(\mathrm{cos}\left(120°\right)+i\mathrm{sin}\left(120°\right)\right)$

${z}_{2}=2\left(\mathrm{cos}\left(210°\right)+i\mathrm{sin}\left(210°\right)\right)$

${z}_{3}=2\left(\mathrm{cos}\left(300°\right)+i\mathrm{sin}\left(300°\right)\right)$

Access these online resources for additional instruction and practice with polar forms of complex numbers.

Key concepts

• Complex numbers in the form $\text{\hspace{0.17em}}a+bi\text{\hspace{0.17em}}$ are plotted in the complex plane similar to the way rectangular coordinates are plotted in the rectangular plane. Label the x- axis as the real axis and the y- axis as the imaginary axis. See [link] .
• The absolute value of a complex number is the same as its magnitude. It is the distance from the origin to the point: $\text{\hspace{0.17em}}|z|=\sqrt{{a}^{2}+{b}^{2}}.\text{\hspace{0.17em}}$ See [link] and [link] .
• To write complex numbers in polar form, we use the formulas $\text{\hspace{0.17em}}x=r\mathrm{cos}\text{\hspace{0.17em}}\theta ,y=r\mathrm{sin}\text{\hspace{0.17em}}\theta ,\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}r=\sqrt{{x}^{2}+{y}^{2}}.\text{\hspace{0.17em}}$ Then, $\text{\hspace{0.17em}}z=r\left(\mathrm{cos}\text{\hspace{0.17em}}\theta +i\mathrm{sin}\text{\hspace{0.17em}}\theta \right).\text{\hspace{0.17em}}$ See [link] and [link] .
• To convert from polar form to rectangular form, first evaluate the trigonometric functions. Then, multiply through by $\text{\hspace{0.17em}}r.\text{\hspace{0.17em}}$ See [link] and [link] .
• To find the product of two complex numbers, multiply the two moduli and add the two angles. Evaluate the trigonometric functions, and multiply using the distributive property. See [link] .
• To find the quotient of two complex numbers in polar form, find the quotient of the two moduli and the difference of the two angles. See [link] .
• To find the power of a complex number $\text{\hspace{0.17em}}{z}^{n},\text{\hspace{0.17em}}$ raise $\text{\hspace{0.17em}}r\text{\hspace{0.17em}}$ to the power $\text{\hspace{0.17em}}n,$ and multiply $\text{\hspace{0.17em}}\theta \text{\hspace{0.17em}}$ by $\text{\hspace{0.17em}}n.\text{\hspace{0.17em}}$ See [link] .
• Finding the roots of a complex number is the same as raising a complex number to a power, but using a rational exponent. See [link] .

Verbal

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.

Algebraic

For the following exercises, find the absolute value of the given complex number.

$5+\text{​}3i$

$-7+\text{​}i$

$5\sqrt{2}$

$-3-3i$

$\sqrt{2}-6i$

$\sqrt{38}$

$2i$

$2.2-3.1i$

$\sqrt{14.45}$

For the following exercises, write the complex number in polar form.

$2+2i$

$8-4i$

$4\sqrt{5}\mathrm{cis}\left(333.4°\right)$

$-\frac{1}{2}-\frac{1}{2}\text{​}i$

$\sqrt{3}+i$

$2\mathrm{cis}\left(\frac{\pi }{6}\right)$

$3i$

For the following exercises, convert the complex number from polar to rectangular form.

$z=7\mathrm{cis}\left(\frac{\pi }{6}\right)$

$\frac{7\sqrt{3}}{2}+i\frac{7}{2}$

$z=2\mathrm{cis}\left(\frac{\pi }{3}\right)$

$z=4\mathrm{cis}\left(\frac{7\pi }{6}\right)$

$-2\sqrt{3}-2i$

$z=7\mathrm{cis}\left(25°\right)$

$z=3\mathrm{cis}\left(240°\right)$

$-1.5-i\frac{3\sqrt{3}}{2}$

$z=\sqrt{2}\mathrm{cis}\left(100°\right)$

For the following exercises, find $\text{\hspace{0.17em}}{z}_{1}{z}_{2}\text{\hspace{0.17em}}$ in polar form.

$4\sqrt{3}\mathrm{cis}\left(198°\right)$

$\frac{3}{4}\mathrm{cis}\left(180°\right)$

$5\sqrt{3}\mathrm{cis}\left(\frac{17\pi }{24}\right)$

For the following exercises, find $\text{\hspace{0.17em}}\frac{{z}_{1}}{{z}_{2}}\text{\hspace{0.17em}}$ in polar form.

$7\mathrm{cis}\left(70°\right)$

$5\mathrm{cis}\left(80°\right)$

$5\mathrm{cis}\left(\frac{\pi }{3}\right)$

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 $\text{\hspace{0.17em}}z=5\mathrm{cis}\left(45°\right).$

$125\mathrm{cis}\left(135°\right)$

Find $\text{\hspace{0.17em}}{z}^{4}\text{\hspace{0.17em}}$ when $\text{\hspace{0.17em}}z=2\mathrm{cis}\left(70°\right).$

Find $\text{\hspace{0.17em}}{z}^{2}\text{\hspace{0.17em}}$ when $\text{\hspace{0.17em}}z=3\mathrm{cis}\left(120°\right).$

$9\mathrm{cis}\left(240°\right)$

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 $\text{\hspace{0.17em}}z=27\mathrm{cis}\left(240°\right).$

$\text{\hspace{0.17em}}3\mathrm{cis}\left(80°\right),3\mathrm{cis}\left(200°\right),3\mathrm{cis}\left(320°\right)\text{\hspace{0.17em}}$

Evaluate the square root of $\text{\hspace{0.17em}}z\text{\hspace{0.17em}}$ when $\text{\hspace{0.17em}}z=16\mathrm{cis}\left(100°\right).$

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 $\text{\hspace{0.17em}}z=8\mathrm{cis}\left(\frac{7\pi }{4}\right).$

$2\sqrt{2}\mathrm{cis}\left(\frac{7\pi }{8}\right),2\sqrt{2}\mathrm{cis}\left(\frac{15\pi }{8}\right)$

Graphical

For the following exercises, plot the complex number in the complex plane.

$2+4i$

$-3-3i$

$5-4i$

$-1-5i$

$3+2i$

$2i$

$-4$

$6-2i$

$-2+i$

$1-4i$

Technology

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 $\text{\hspace{0.17em}}4\mathrm{cis}\left(120°\right)\text{\hspace{0.17em}}$ 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 $\text{\hspace{0.17em}}2\mathrm{cis}\left(45°\right)\text{\hspace{0.17em}}$ to rectangular form.

Use the polar to rectangular feature on the graphing calculator to change $\text{\hspace{0.17em}}5\mathrm{cis}\left(210°\right)\text{\hspace{0.17em}}$ to rectangular form.

$\text{\hspace{0.17em}}-4.33-2.50i\text{\hspace{0.17em}}$

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