10.5 Polar form of complex numbers  (Page 2/8)

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Given $\text{\hspace{0.17em}}z=1-7i,\text{\hspace{0.17em}}$ find $\text{\hspace{0.17em}}|z|.$

$|z|=\sqrt{50}=5\sqrt{2}$

Writing complex numbers in polar form

The polar form of a complex number    expresses a number in terms of an angle $\text{\hspace{0.17em}}\theta \text{\hspace{0.17em}}$ and its distance from the origin $\text{\hspace{0.17em}}r.\text{\hspace{0.17em}}$ Given a complex number in rectangular form expressed as $\text{\hspace{0.17em}}z=x+yi,\text{\hspace{0.17em}}$ we use the same conversion formulas as we do to write the number in trigonometric form:

$\begin{array}{l}\text{\hspace{0.17em}}x=r\mathrm{cos}\text{\hspace{0.17em}}\theta \hfill \\ \text{\hspace{0.17em}}y=r\mathrm{sin}\text{\hspace{0.17em}}\theta \hfill \\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}r=\sqrt{{x}^{2}+{y}^{2}}\hfill \end{array}$

We review these relationships in [link] .

We use the term modulus to represent the absolute value of a complex number, or the distance from the origin to the point $\text{\hspace{0.17em}}\left(x,y\right).\text{\hspace{0.17em}}$ The modulus, then, is the same as $\text{\hspace{0.17em}}r,\text{\hspace{0.17em}}$ the radius in polar form. We use $\text{\hspace{0.17em}}\theta \text{\hspace{0.17em}}$ to indicate the angle of direction (just as with polar coordinates). Substituting, we have

$\begin{array}{l}z=x+yi\hfill \\ z=r\mathrm{cos}\text{\hspace{0.17em}}\theta +\left(r\mathrm{sin}\text{\hspace{0.17em}}\theta \right)i\hfill \\ z=r\left(\mathrm{cos}\text{\hspace{0.17em}}\theta +i\mathrm{sin}\text{\hspace{0.17em}}\theta \right)\hfill \end{array}$

Polar form of a complex number

Writing a complex number in polar form involves the following conversion formulas:

$\begin{array}{l}\hfill \\ x=r\mathrm{cos}\text{\hspace{0.17em}}\theta \hfill \\ y=r\mathrm{sin}\text{\hspace{0.17em}}\theta \hfill \\ r=\sqrt{{x}^{2}+{y}^{2}}\hfill \end{array}$

Making a direct substitution, we have

$\begin{array}{l}z=x+yi\hfill \\ z=\left(r\mathrm{cos}\text{\hspace{0.17em}}\theta \right)+i\left(r\mathrm{sin}\text{\hspace{0.17em}}\theta \right)\hfill \\ z=r\left(\mathrm{cos}\text{\hspace{0.17em}}\theta +i\mathrm{sin}\text{\hspace{0.17em}}\theta \right)\hfill \end{array}$

where $\text{\hspace{0.17em}}r\text{\hspace{0.17em}}$ is the modulus    and $\theta$ is the argument    . We often use the abbreviation $\text{\hspace{0.17em}}r\text{cis}\text{\hspace{0.17em}}\theta \text{\hspace{0.17em}}$ to represent $\text{\hspace{0.17em}}r\left(\mathrm{cos}\text{\hspace{0.17em}}\theta +i\mathrm{sin}\text{\hspace{0.17em}}\theta \right).$

Expressing a complex number using polar coordinates

Express the complex number $\text{\hspace{0.17em}}4i\text{\hspace{0.17em}}$ using polar coordinates.

On the complex plane, the number $\text{\hspace{0.17em}}z=4i\text{\hspace{0.17em}}$ is the same as $\text{\hspace{0.17em}}z=0+4i.\text{\hspace{0.17em}}$ Writing it in polar form, we have to calculate $\text{\hspace{0.17em}}r\text{\hspace{0.17em}}$ first.

$\begin{array}{l}r=\sqrt{{x}^{2}+{y}^{2}}\hfill \\ r=\sqrt{{0}^{2}+{4}^{2}}\hfill \\ r=\sqrt{16}\hfill \\ r=4\hfill \end{array}$

Next, we look at $\text{\hspace{0.17em}}x.\text{\hspace{0.17em}}$ If $\text{\hspace{0.17em}}x=r\mathrm{cos}\text{\hspace{0.17em}}\theta ,\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}x=0,\text{\hspace{0.17em}}$ then $\text{\hspace{0.17em}}\theta =\frac{\pi }{2}.\text{\hspace{0.17em}}$ In polar coordinates, the complex number $\text{\hspace{0.17em}}z=0+4i\text{\hspace{0.17em}}$ can be written as $\text{\hspace{0.17em}}z=4\left(\mathrm{cos}\left(\frac{\pi }{2}\right)+i\mathrm{sin}\left(\frac{\pi }{2}\right)\right)\text{\hspace{0.17em}}$ or $\text{\hspace{0.17em}}4\text{cis}\left(\text{\hspace{0.17em}}\frac{\pi }{2}\right).\text{\hspace{0.17em}}$ See [link] .

Express $\text{\hspace{0.17em}}z=3i\text{\hspace{0.17em}}$ as $\text{\hspace{0.17em}}r\text{\hspace{0.17em}}\text{cis}\text{\hspace{0.17em}}\theta \text{\hspace{0.17em}}$ in polar form.

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

Finding the polar form of a complex number

Find the polar form of $\text{\hspace{0.17em}}-4+4i.$

First, find the value of $\text{\hspace{0.17em}}r.$

$\begin{array}{l}r=\sqrt{{x}^{2}+{y}^{2}}\hfill \\ r=\sqrt{{\left(-4\right)}^{2}+\left({4}^{2}\right)}\hfill \\ r=\sqrt{32}\hfill \\ r=4\sqrt{2}\hfill \end{array}$

Find the angle $\text{\hspace{0.17em}}\theta \text{\hspace{0.17em}}$ using the formula:

$\begin{array}{l}\mathrm{cos}\text{\hspace{0.17em}}\theta =\frac{x}{r}\hfill \\ \mathrm{cos}\text{\hspace{0.17em}}\theta =\frac{-4}{4\sqrt{2}}\hfill \\ \mathrm{cos}\text{\hspace{0.17em}}\theta =-\frac{1}{\sqrt{2}}\hfill \\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\theta ={\mathrm{cos}}^{-1}\left(-\frac{1}{\sqrt{2}}\right)=\frac{3\pi }{4}\hfill \end{array}$

Thus, the solution is $\text{\hspace{0.17em}}4\sqrt{2}\text{cis}\left(\frac{3\pi }{4}\right).$

Write $\text{\hspace{0.17em}}z=\sqrt{3}+i\text{\hspace{0.17em}}$ in polar form.

$z=2\left(\mathrm{cos}\left(\frac{\pi }{6}\right)+i\mathrm{sin}\left(\frac{\pi }{6}\right)\right)$

Converting a complex number from polar to rectangular form

Converting a complex number from polar form to rectangular form is a matter of evaluating what is given and using the distributive property. In other words, given $\text{\hspace{0.17em}}z=r\left(\mathrm{cos}\text{\hspace{0.17em}}\theta +i\mathrm{sin}\text{\hspace{0.17em}}\theta \right),\text{\hspace{0.17em}}$ first evaluate the trigonometric functions $\text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}\theta \text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\mathrm{sin}\text{\hspace{0.17em}}\theta .\text{\hspace{0.17em}}$ Then, multiply through by $\text{\hspace{0.17em}}r.$

Converting from polar to rectangular form

Convert the polar form of the given complex number to rectangular form:

$z=12\left(\mathrm{cos}\left(\frac{\pi }{6}\right)+i\mathrm{sin}\left(\frac{\pi }{6}\right)\right)$

We begin by evaluating the trigonometric expressions.

$\mathrm{cos}\left(\frac{\pi }{6}\right)=\frac{\sqrt{3}}{2}\text{\hspace{0.17em}}\text{and}\text{\hspace{0.17em}}\mathrm{sin}\left(\frac{\pi }{6}\right)=\frac{1}{2}\text{\hspace{0.17em}}$

After substitution, the complex number is

$z=12\left(\frac{\sqrt{3}}{2}+\frac{1}{2}i\right)$

We apply the distributive property:

The rectangular form of the given point in complex form is $\text{\hspace{0.17em}}6\sqrt{3}+6i.$

Finding the rectangular form of a complex number

Find the rectangular form of the complex number given $\text{\hspace{0.17em}}r=13\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\mathrm{tan}\text{\hspace{0.17em}}\theta =\frac{5}{12}.$

If $\text{\hspace{0.17em}}\mathrm{tan}\text{\hspace{0.17em}}\theta =\frac{5}{12},\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\mathrm{tan}\text{\hspace{0.17em}}\theta =\frac{y}{x},\text{\hspace{0.17em}}$ we first determine $\text{\hspace{0.17em}}r=\sqrt{{x}^{2}+{y}^{2}}=\sqrt{{12}^{2}+{5}^{2}}=13\text{.}$ We then find $\text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}\theta =\frac{x}{r}\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\mathrm{sin}\text{\hspace{0.17em}}\theta =\frac{y}{r}.$

$\begin{array}{l}z=13\left(\mathrm{cos}\text{\hspace{0.17em}}\theta +i\mathrm{sin}\text{\hspace{0.17em}}\theta \right)\hfill \\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=13\left(\frac{12}{13}+\frac{5}{13}i\right)\hfill \\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=12+5i\hfill \end{array}$

The rectangular form of the given number in complex form is $\text{\hspace{0.17em}}12+5i.$

Convert the complex number to rectangular form:

$z=4\left(\mathrm{cos}\frac{11\pi }{6}+i\mathrm{sin}\frac{11\pi }{6}\right)$

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

Finding products of complex numbers in polar form

Now that we can convert complex numbers to polar form we will learn how to perform operations on complex numbers in polar form. For the rest of this section, we will work with formulas developed by French mathematician Abraham de Moivre (1667-1754). These formulas have made working with products, quotients, powers, and roots of complex numbers much simpler than they appear. The rules are based on multiplying the moduli and adding the arguments.

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