Polar Form of Complex Numbers

“God made the integers; all else is the work of man.” This rather famous quote by nineteenth-century German mathematician Leopold Kronecker sets the stage for this section on the polar form of a complex number. Complex numbers were invented by people and represent over a thousand years of continuous investigation and struggle by mathematicians such as Pythagoras, Descartes, De Moivre, Euler, Gauss, and others. Complex numbers answered questions that for centuries had puzzled the greatest minds in science.

We first encountered complex numbers in Complex Numbers. In this section, we will focus on the mechanics of working with complex numbers: translation of complex numbers from polar form to rectangular form and vice versa, interpretation of complex numbers in the scheme of applications, and application of De Moivre’s Theorem.

Plotting Complex Numbers in the Complex Plane

Plotting a complex number $a + b i$ is similar to plotting a real number, except that the horizontal axis represents the real part of the number, $a,$ and the vertical axis represents the imaginary part of the number, $b i .$

Example 1

Plotting a Complex Number in the Complex Plane

Plot the complex number $2 - 3 i$ in the complex plane.

Solution

From the origin, move two units in the positive horizontal direction and three units in the negative vertical direction. See Figure 1.

Plot of 2-3i in the complex plane (2 along the real axis, -3 along the imaginary axis). — Figure 1

Finding the Absolute Value of a Complex Number

The first step toward working with a complex number in polar form is to find the absolute value. The absolute value of a complex number is the same as its magnitude, or $| z | .$ It measures the distance from the origin to a point in the plane. For example, the graph of $z = 2 + 4 i,$ in Figure 2, shows $| z | .$

Plot of 2 + 4i in the complex plane and its magnitude, |z| = rad 2 squared + 4 squared = rad 4 + 16 = rad 20 = 2 rad 5. — Figure 2

Example 2

Finding the Absolute Value of a Complex Number with a Radical

Find the absolute value of $z = \sqrt{5} - i .$

Solution

Using the formula, we have

\begin{array}{l} | z | = \sqrt{x^{2} + y^{2}} \\ | z | = \sqrt{{(\sqrt{5})}^{2} + {(- 1)}^{2}} \\ | z | = \sqrt{5 + 1} \\ | z | = \sqrt{6} \end{array}

See Figure 3.

Plot of z=(rad5 - i) in the complex plane and its magnitude rad6. — Figure 3

Example 3

Finding the Absolute Value of a Complex Number

Given $z = 3 - 4 i,$ find $| z | .$

Solution

Using the formula, we have

\begin{array}{l} | z | = \sqrt{x^{2} + y^{2}} \\ | z | = \sqrt{{(3)}^{2} + {(- 4)}^{2}} \\ | z | = \sqrt{9 + 16} \\ \begin{array}{l} | z | = \sqrt{25} \\ | z | = 5 \end{array} \end{array}

The absolute value $z$ is 5. See Figure 4.

Plot of (3-4i) in the complex plane and its magnitude |z| =5. — Figure 4

Writing Complex Numbers in Polar Form

The polar form of a complex number expresses a number in terms of an angle $θ$ and its distance from the origin $r .$ Given a complex number in rectangular form expressed as $z = x + y i,$ we use the same conversion formulas as we do to write the number in trigonometric form:

\begin{array}{l} x = r \cos θ \\ y = r \sin θ \\ r = \sqrt{x^{2} + y^{2}} \end{array}

We review these relationships in Figure 5.

Triangle plotted in the complex plane (x axis is real, y axis is imaginary). Base is along the x/real axis, height is some y/imaginary value in Q 1, and hypotenuse r extends from origin to that point (x+yi) in Q 1. The angle at the origin is theta. There is an arc going through (x+yi). — Figure 5

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

\begin{array}{l} z = x + y i \\ z = r \cos θ + (r \sin θ) i \\ z = r (\cos θ + i \sin θ) \end{array}

Example 4

Expressing a Complex Number Using Polar Coordinates

Express the complex number $4 i$ using polar coordinates.

Solution

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

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

Next, we look at $x .$ If $x = r \cos θ,$ and $x = 0,$ then $θ = \frac{π}{2} .$ In polar coordinates, the complex number $z = 0 + 4 i$ can be written as $z = 4 (\cos (\frac{π}{2}) + i \sin (\frac{π}{2}))$ or $4 cis (\frac{π}{2}) .$ See Figure 6.

Plot of z=4i in the complex plane, also shows that the in polar coordinate it would be (4,pi/2). — Figure 6

Example 5

Finding the Polar Form of a Complex Number

Find the polar form of $- 4 + 4 i .$

Solution

First, find the value of $r .$

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

Find the angle $θ$ using the formula:

\begin{array}{l} \cos θ = \frac{x}{r} \\ \cos θ = \frac{- 4}{4 \sqrt{2}} \\ \cos θ = - \frac{1}{\sqrt{2}} \\ θ = \cos^{- 1} (- \frac{1}{\sqrt{2}}) = \frac{3 π}{4} \end{array}

Thus, the solution is $4 \sqrt{2} cis (\frac{3 π}{4}) .$

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 $z = r (\cos θ + i \sin θ),$ first evaluate the trigonometric functions $\cos θ$ and $\sin θ .$ Then, multiply through by $r .$

Example 6

Converting from Polar to Rectangular Form

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

z = 12 (\cos (\frac{π}{6}) + i \sin (\frac{π}{6}))

Solution

We begin by evaluating the trigonometric expressions.

\cos (\frac{π}{6}) = \frac{\sqrt{3}}{2} and \sin (\frac{π}{6}) = \frac{1}{2}

After substitution, the complex number is

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

We apply the distributive property:

\begin{array}{l} z = 12 (\frac{\sqrt{3}}{2} + \frac{1}{2} i) \\ = (12) \frac{\sqrt{3}}{2} + (12) \frac{1}{2} i \\ = 6 \sqrt{3} + 6 i \end{array}

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

Example 7

Finding the Rectangular Form of a Complex Number

Find the rectangular form of the complex number given $r = 13$ and $\tan θ = \frac{5}{12} .$ Assume the number is in the first quadrant.

Solution

If $\tan θ = \frac{5}{12},$ and $\tan θ = \frac{y}{x},$ we first confirm $r = \sqrt{x^{2} + y^{2}} = \sqrt{12^{2} + 5^{2}} = 13 .$ We then find $\cos θ = \frac{x}{r}$ and $\sin θ = \frac{y}{r} .$

\begin{array}{l} z = 13 (\cos θ + i \sin θ) \\ = 13 (\frac{12}{13} + \frac{5}{13} i) \\ = 12 + 5 i \end{array}

The rectangular form of the given number in complex form is $12 + 5 i .$

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.

Products of Complex Numbers in Polar Form

If $z_{1} = r_{1} (\cos θ_{1} + i \sin θ_{1})$ and $z_{2} = r_{2} (\cos θ_{2} + i \sin θ_{2}),$ then the product of these numbers is given as:

\begin{array}{l} z_{1} z_{2} = r_{1} r_{2} [\cos (θ_{1} + θ_{2}) + i \sin (θ_{1} + θ_{2})] \\ z_{1} z_{2} = r_{1} r_{2} cis (θ_{1} + θ_{2}) \end{array}

Notice that the product calls for multiplying the moduli and adding the angles.

Example 8

Finding the Product of Two Complex Numbers in Polar Form

Find the product of $z_{1} z_{2},$ given $z_{1} = 4 (\cos (80°) + i \sin (80°))$ and $z_{2} = 2 (\cos (145°) + i \sin (145°)) .$

Solution

Follow the formula

\begin{array}{l} z_{1} z_{2} = 4 \cdot 2 [\cos (80° + 145°) + i \sin (80° + 145°)] \\ z_{1} z_{2} = 8 [\cos (225°) + i \sin (225°)] \\ z_{1} z_{2} = 8 [\cos (\frac{5 π}{4}) + i \sin (\frac{5 π}{4})] \\ z_{1} z_{2} = 8 [- \frac{\sqrt{2}}{2} + i (- \frac{\sqrt{2}}{2})] \\ z_{1} z_{2} = - 4 \sqrt{2} - 4 i \sqrt{2} \end{array}

Finding Quotients of Complex Numbers in Polar Form

The quotient of two complex numbers in polar form is the quotient of the two moduli and the difference of the two arguments.

Quotients of Complex Numbers in Polar Form

If $z_{1} = r_{1} (\cos θ_{1} + i \sin θ_{1})$ and $z_{2} = r_{2} (\cos θ_{2} + i \sin θ_{2}),$ then the quotient of these numbers is

\begin{array}{l} \frac{z_{1}}{z_{2}} = \frac{r_{1}}{r_{2}} [\cos (θ_{1} - θ_{2}) + i \sin (θ_{1} - θ_{2})], z_{2} \neq 0 \\ \frac{z_{1}}{z_{2}} = \frac{r_{1}}{r_{2}} cis (θ_{1} - θ_{2}), z_{2} \neq 0 \end{array}

Notice that the moduli are divided, and the angles are subtracted.

Example 9

Finding the Quotient of Two Complex Numbers

Find the quotient of $z_{1} = 2 (\cos (213°) + i \sin (213°))$ and $z_{2} = 4 (\cos (33°) + i \sin (33°)) .$

Solution

Using the formula, we have

\begin{array}{l} \frac{z_{1}}{z_{2}} = \frac{2}{4} [\cos (213° - 33°) + i \sin (213° - 33°)] \\ \frac{z_{1}}{z_{2}} = \frac{1}{2} [\cos (180°) + i \sin (180°)] \\ \frac{z_{1}}{z_{2}} = \frac{1}{2} [- 1 + 0 i] \\ \frac{z_{1}}{z_{2}} = - \frac{1}{2} + 0 i \\ \frac{z_{1}}{z_{2}} = - \frac{1}{2} \end{array}

Finding Powers of Complex Numbers in Polar Form

Finding powers of complex numbers is greatly simplified using De Moivre’s Theorem. It states that, for a positive integer $n, z^{n}$ is found by raising the modulus to the $n th$ power and multiplying the argument by $n .$ It is the standard method used in modern mathematics.

Example 10

Evaluating an Expression Using De Moivre’s Theorem

Evaluate the expression ${(1 + i)}^{5}$ using De Moivre’s Theorem.

Solution

Since De Moivre’s Theorem applies to complex numbers written in polar form, we must first write $(1 + i)$ in polar form. Let us find $r .$

\begin{array}{l} r = \sqrt{x^{2} + y^{2}} \\ r = \sqrt{{(1)}^{2} + {(1)}^{2}} \\ r = \sqrt{2} \end{array}

Then we find $θ .$ Using the formula $\tan θ = \frac{y}{x}$ gives

\begin{array}{l} \tan θ = \frac{1}{1} \\ \tan θ = 1 \\ θ = \frac{π}{4} \end{array}

Use De Moivre’s Theorem to evaluate the expression.

\begin{array}{l} {(a + b i)}^{n} = r^{n} [\cos (n θ) + i \sin (n θ)] \\ {(1 + i)}^{5} = {(\sqrt{2})}^{5} [\cos (5 \cdot \frac{π}{4}) + i \sin (5 \cdot \frac{π}{4})] \\ {(1 + i)}^{5} = 4 \sqrt{2} [\cos (\frac{5 π}{4}) + i \sin (\frac{5 π}{4})] \\ {(1 + i)}^{5} = 4 \sqrt{2} [- \frac{\sqrt{2}}{2} + i (- \frac{\sqrt{2}}{2})] \\ {(1 + i)}^{5} = - 4 - 4 i \end{array}

Finding Roots of Complex Numbers in Polar Form

To find the nth root of a complex number in polar form, we use the $n th$ Root Theorem or De Moivre’s Theorem and raise the complex number to a power with a rational exponent. There are several ways to represent a formula for finding $n th$ roots of complex numbers in polar form.

Example 11

Finding the nth Root of a Complex Number

Evaluate the cube roots of $z = 8 (\cos (\frac{2 π}{3}) + i \sin (\frac{2 π}{3})) .$

Solution

We have

\begin{array}{l} z^{\frac{1}{3}} = 8^{\frac{1}{3}} [\cos (\frac{\frac{2 π}{3}}{3} + \frac{2 k π}{3}) + i \sin (\frac{\frac{2 π}{3}}{3} + \frac{2 k π}{3})] \\ z^{\frac{1}{3}} = 2 [\cos (\frac{2 π}{9} + \frac{2 k π}{3}) + i \sin (\frac{2 π}{9} + \frac{2 k π}{3})] \end{array}

There will be three roots: $k = 0, 1, 2.$ When $k = 0,$ we have

z^{\frac{1}{3}} = 2 (\cos (\frac{2 π}{9}) + i \sin (\frac{2 π}{9}))

When $k = 1,$ we have

\begin{array}{l} z^{\frac{1}{3}} = 2 [\cos (\frac{2 π}{9} + \frac{6 π}{9}) + i \sin (\frac{2 π}{9} + \frac{6 π}{9})] \begin{matrix}  \end{matrix} Add \frac{2 (1) π}{3} to each angle. \\ z^{\frac{1}{3}} = 2 (\cos (\frac{8 π}{9}) + i \sin (\frac{8 π}{9})) \end{array}

When $k = 2,$ we have

\begin{array}{l} z^{\frac{1}{3}} = 2 [\cos (\frac{2 π}{9} + \frac{12 π}{9}) + i \sin (\frac{2 π}{9} + \frac{12 π}{9})] \begin{matrix}  \end{matrix} & Add \frac{2 (2) π}{3} to each angle. \\ z^{\frac{1}{3}} = 2 (\cos (\frac{14 π}{9}) + i \sin (\frac{14 π}{9})) \end{array}

Remember to find the common denominator to simplify fractions in situations like this one. For $k = 1,$ the angle simplification is

\begin{array}{l} \frac{\frac{2 π}{3}}{3} + \frac{2 (1) π}{3} = \frac{2 π}{3} (\frac{1}{3}) + \frac{2 (1) π}{3} (\frac{3}{3}) \\ = \frac{2 π}{9} + \frac{6 π}{9} \\ = \frac{8 π}{9} \end{array}

Key Concepts

Complex numbers in the form $a + b i$ 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 Example 1.
The absolute value of a complex number is the same as its magnitude. It is the distance from the origin to the point: $| z | = \sqrt{a^{2} + b^{2}} .$ See Example 2 and Example 3.
To write complex numbers in polar form, we use the formulas $x = r \cos θ, y = r \sin θ,$ and $r = \sqrt{x^{2} + y^{2}} .$ Then, $z = r (\cos θ + i \sin θ) .$ See Example 4 and Example 5.
To convert from polar form to rectangular form, first evaluate the trigonometric functions. Then, multiply through by $r .$ See Example 6 and Example 7.
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 Example 8.
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 Example 9.
To find the power of a complex number $z^{n},$ raise $r$ to the power $n,$ and multiply $θ$ by $n .$ See Example 10.
Finding the roots of a complex number is the same as raising a complex number to a power, but using a rational exponent. See Example 11.

Section Exercises

Verbal

Exercise 1

A complex number is $a + b i .$ Explain each part.

Solution

a is the real part, b is the imaginary part, and $i = \sqrt{- 1}$

Exercise 2

What does the absolute value of a complex number represent?

Exercise 3

How is a complex number converted to polar form?

Solution

Polar form converts the real and imaginary part of the complex number in polar form using $x = r \cos θ$ and $y = r \sin θ .$

Exercise 4

How do we find the product of two complex numbers?

Exercise 5

What is De Moivre’s Theorem and what is it used for?

Solution

$z^{n} = r^{n} (\cos (n θ) + i \sin (n θ))$ 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.

Exercise 6

$5 + 3 i$

Exercise 7

$- 7 + i$

Solution

$5 \sqrt{2}$

Exercise 8

$- 3 - 3 i$

Exercise 9

$\sqrt{2} - 6 i$

Solution

$\sqrt{38}$

Exercise 10

$2 i$

Exercise 11

$2.2 - 3.1 i$

Solution

$\sqrt{14.45}$

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

Exercise 12

$2 + 2 i$

Exercise 13

$8 - 4 i$

Solution

$4 \sqrt{5} cis (333.4°)$

Exercise 14

$- \frac{1}{2} - \frac{1}{2} i$

Exercise 15

$\sqrt{3} + i$

Solution

$2 cis (\frac{π}{6})$

Exercise 16

$3 i$

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

Exercise 17

$z = 7 cis (\frac{π}{6})$

Solution

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

Exercise 18

$z = 2 cis (\frac{π}{3})$

Exercise 19

$z = 4 cis (\frac{7 π}{6})$

Solution

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

Exercise 20

$z = 7 cis (25°)$

Exercise 21

$z = 3 cis (240°)$

Solution

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

Exercise 22

$z = \sqrt{2} cis (100°)$

For the following exercises, find $z_{1} z_{2}$ in polar form.

Exercise 23

$z_{1} = 2 \sqrt{3} cis (116°); z_{2} = 2 cis (82°)$

Solution

$4 \sqrt{3} cis (198°)$

Exercise 24

$z_{1} = \sqrt{2} cis (205°); z_{2} = 2 \sqrt{2} cis (118°)$

Exercise 25

$z_{1} = 3 cis (120°); z_{2} = \frac{1}{4} cis (60°)$

Solution

$\frac{3}{4} cis (180°)$

Exercise 26

$z_{1} = 3 cis (\frac{π}{4}); z_{2} = 5 cis (\frac{π}{6})$

Exercise 27

$z_{1} = \sqrt{5} cis (\frac{5 π}{8}); z_{2} = \sqrt{15} cis (\frac{π}{12})$

Solution

$5 \sqrt{3} cis (\frac{17 π}{24})$

Exercise 28

$z_{1} = 4 cis (\frac{π}{2}); z_{2} = 2 cis (\frac{π}{4})$

For the following exercises, find $\frac{z_{1}}{z_{2}}$ in polar form.

Exercise 29

$z_{1} = 21 cis (135°); z_{2} = 3 cis (65°)$

Solution

$7 cis (70°)$

Exercise 30

$z_{1} = \sqrt{2} cis (90°); z_{2} = 2 cis (60°)$

Exercise 31

$z_{1} = 15 cis (120°); z_{2} = 3 cis (40°)$

Solution

$5 cis (80°)$

Exercise 32

$z_{1} = 6 cis (\frac{π}{3}); z_{2} = 2 cis (\frac{π}{4})$

Exercise 33

$z_{1} = 5 \sqrt{2} cis (π); z_{2} = \sqrt{2} cis (\frac{2 π}{3})$

Solution

$5 cis (\frac{π}{3})$

Exercise 34

$z_{1} = 2 cis (\frac{3 π}{5}); z_{2} = 3 cis (\frac{π}{4})$

For the following exercises, find the powers of each complex number in polar form.

Exercise 35

Find $z^{3}$ when $z = 5 cis (45°) .$

Solution

$125 cis (135°)$

Exercise 36

Find $z^{4}$ when $z = 2 cis (70°) .$

Exercise 37

Find $z^{2}$ when $z = 3 cis (120°) .$

Solution

$9 cis (240°)$

Exercise 38

Find $z^{2}$ when $z = 4 cis (\frac{π}{4}) .$

Exercise 39

Find $z^{4}$ when $z = cis (\frac{3 π}{16}) .$

Solution

$cis (\frac{3 π}{4})$

Exercise 40

Find $z^{3}$ when $z = 3 cis (\frac{5 π}{3}) .$

For the following exercises, evaluate each root.

Exercise 41

Evaluate the cube root of $z$ when $z = 27 cis (240°) .$

Solution

$3 cis (80°), 3 cis (200°), 3 cis (320°)$

Exercise 42

Evaluate the square root of $z$ when $z = 16 cis (100°) .$

Exercise 43

Evaluate the cube root of $z$ when $z = 32 cis (\frac{2 π}{3}) .$

Solution

$2 \sqrt[3]{4} cis (\frac{2 π}{9}), 2 \sqrt[3]{4} cis (\frac{8 π}{9}), 2 \sqrt[3]{4} cis (\frac{14 π}{9})$

Exercise 44

Evaluate the square root of $z$ when $z = 32 cis (π) .$

Exercise 45

Evaluate the square root of $z$ when $z = 8 cis (\frac{7 π}{4}) .$

Solution

$2 \sqrt{2} cis (\frac{7 π}{8}), 2 \sqrt{2} cis (\frac{15 π}{8})$

Graphical

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

Exercise 46

$2 + 4 i$

Exercise 47

$- 3 - 3 i$

Solution

Plot of -3 -3i in the complex plane (-3 along real axis, -3 along imaginary axis).

Exercise 48

$5 - 4 i$

Exercise 49

$- 1 - 5 i$

Solution

Plot of -1 -5i in the complex plane (-1 along real axis, -5 along imaginary axis).

Exercise 50

$3 + 2 i$

Exercise 51

$2 i$

Solution

Plot of 2i in the complex plane (0 along the real axis, 2 along the imaginary axis).

Exercise 52

$- 4$

Exercise 53

$6 - 2 i$

Solution

Plot of 6-2i in the complex plane (6 along the real axis, -2 along the imaginary axis).

Exercise 54

$- 2 + i$

Exercise 55

$1 - 4 i$

Solution

Plot of 1-4i in the complex plane (1 along the real axis, -4 along the imaginary axis).

Technology

For the following exercises, find all answers rounded to the nearest hundredth.

Exercise 56

Use the rectangular to polar feature on the graphing calculator to change $5 + 5 i$ to polar form.

Exercise 57

Use the rectangular to polar feature on the graphing calculator to change $3 - 2 i$ to polar form.

Solution

$3.61 e^{- 0.59 i}$

Exercise 58

Use the rectangular to polar feature on the graphing calculator to change $- 3 - 8 i$ to polar form.

Exercise 59

Use the polar to rectangular feature on the graphing calculator to change $4 cis (120°)$ to rectangular form.

Solution

$- 2 + 3.46 i$

Exercise 60

Use the polar to rectangular feature on the graphing calculator to change $2 cis (45°)$ to rectangular form.

Exercise 61

Use the polar to rectangular feature on the graphing calculator to change $5 cis (210°)$ to rectangular form.

Solution

$- 4.33 - 2.50 i$

argument

the angle associated with a complex number; the angle between the line from the origin to the point and the positive real axis

De Moivre’s Theorem

formula used to find the

n th

power or nth roots of a complex number; states that, for a positive integer

n, z^{n}

is found by raising the modulus to the

n th

power and multiplying the angles by

n

modulus

the absolute value of a complex number, or the distance from the origin to the point

(x, y);

also called the amplitude

polar form of a complex number

a complex number expressed in terms of an angle

θ

and its distance from the origin

r;

can be found by using conversion formulas

x = r \cos θ, y = r \sin θ,

and

r = \sqrt{x^{2} + y^{2}}