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Mathematics LibreTexts

8-5. Polar Form of Complex Numbers

Polar Form of Complex Numbers
In this section, you will:
  • Plot complex numbers in the complex plane.
  • Find the absolute value of a complex number.
  • Write complex numbers in polar form.
  • Convert a complex number from polar to rectangular form.
  • Find products of complex numbers in polar form.
  • Find quotients of complex numbers in polar form.
  • Find powers of complex numbers in polar form.
  • Find roots of complex numbers in polar form.

“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 asPythagoras, 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<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>a</mi><mo>+</mo><mi>b</mi><mi>i</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>is similar to plotting a real number, except that the horizontal axis represents the real part of the number,<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>a</mi><mo>,</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>and the vertical axis represents the imaginary part of the number,<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>b</mi><mi>i</mi><mo>.</mo></mrow></annotation-xml></semantics></math>

Given a complex number<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>a</mi><mo>+</mo><mi>b</mi><mi>i</mi><mo>,</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>plot it in the complex plane.

  1. Label the horizontal axis as the real axis and the vertical axis as the imaginary axis.
  2. Plot the point in the complex plane by moving<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>a</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>units in the horizontal direction and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>b</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>units in the vertical direction.
Plotting a Complex Number in the Complex Plane

Plot the complex number <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mn>2</mn><mo>−</mo><mn>3</mn><mi>i</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>in the complex plane.

From the origin, move two units in the positive horizontal direction and three units in the negative vertical direction. See[link].

<figure class="small" id="Figure_08_05_001">Plot of 2-3i in the complex plane (2 along the real axis, -3 along the imaginary axis).</figure>

Plot the point<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mn>1</mn><mo>+</mo><mn>5</mn><mi>i</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>in the complex plane.

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

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<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>|</mo></mrow></mrow></annotation-xml></semantics></math> z |. It measures the distance from the origin to a point in the plane. For example, the graph of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>z</mi><mo>=</mo><mn>2</mn><mo>+</mo><mn>4</mn><mi>i</mi><mo>,</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>in [link], shows<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>|</mo></mrow></mrow></annotation-xml></semantics></math> z |.

<figure class="small" id="Figure_08_05_003">Plot of 2+4i in the complex plane and its magnitude, |z| = rad 20.</figure>
Absolute Value of a Complex Number

Given<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>z</mi><mo>=</mo><mi>x</mi><mo>+</mo><mi>y</mi><mi>i</mi><mo>,</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>a complex number, the absolute value of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>z</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>is defined as

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mrow><mo>|</mo></mrow></mrow></annotation-xml></semantics></math> z |= x 2 + y 2

It is the distance from the origin to the point<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> x,y ).

Notice that the absolute value of a real number gives the distance of the number from 0, while the absolute value of a complex number gives the distance of the number from the origin,<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 0, 0 ).

Finding the Absolute Value of a Complex Number with a Radical

Find the absolute value of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>z</mi><mo>=</mo><msqrt/></mrow></annotation-xml></semantics></math> 5 −i.

Using the formula, we have

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mrow><mo>|</mo></mrow></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> z |= x 2 + y 2 | z |= ( 5 ) 2 + ( −1 ) 2 | z |= 5+1 | z |= 6

See [link].

<figure class="small" id="Figure_08_05_004">Plot of z=(rad5 - i) in the complex plane and its magnitude rad6.</figure>

Find the absolute value of the complex number<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>z</mi><mo>=</mo><mn>12</mn><mo>−</mo><mn>5</mn><mi>i</mi><mo>.</mo></mrow></annotation-xml></semantics></math>

13

Finding the Absolute Value of a Complex Number

Given<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>z</mi><mo>=</mo><mn>3</mn><mo>−</mo><mn>4</mn><mi>i</mi><mo>,</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>find<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>|</mo></mrow></mrow></annotation-xml></semantics></math> z |.

Using the formula, we have

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mrow><mo>|</mo></mrow></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> z |= x 2 + y 2 | z |= ( 3 ) 2 + ( −4 ) 2 | z |= 9+16 | z |= 25 | z |=5

The absolute value<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mi>z</mi></mrow><mtext> </mtext></mrow></annotation-xml></semantics></math>is 5. See [link].

<figure class="small" id="Figure_08_05_005">Plot of (3-4i) in the complex plane and its magnitude |z| =5.</figure>

Given<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>z</mi><mo>=</mo><mn>1</mn><mo>−</mo><mn>7</mn><mi>i</mi><mo>,</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>find<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>|</mo></mrow></mrow></annotation-xml></semantics></math> z |.

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mrow><mo>|</mo></mrow></mrow></annotation-xml></semantics></math> z |= 50 =5 2

Writing Complex Numbers in Polar Form

The polar form of a complex number expresses a number in terms of an angle<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>θ</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>and its distance from the origin<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>r</mi><mo>.</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>Given a complex number in rectangular form expressed as<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>z</mi><mo>=</mo><mi>x</mi><mo>+</mo><mi>y</mi><mi>i</mi><mo>,</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>we use the same conversion formulas as we do to write the number in trigonometric form:

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mtext> </mtext><mi>x</mi><mo>=</mo><mi>r</mi><mi>cos</mi><mtext> </mtext><mi>θ</mi></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math>  y=rsin θ   r= x 2 + y 2

We review these relationships in [link].

<figure class="small" id="Figure_08_05_006">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>

We use the term modulus to represent the absolute value of a complex number, or the distance from the origin to the point<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> x,y ). The modulus, then, is the same as<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>r</mi><mo>,</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>the radius in polar form. We use<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>θ</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>to indicate the angle of direction (just as with polar coordinates). Substituting, we have

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mi>z</mi><mo>=</mo><mi>x</mi><mo>+</mo><mi>y</mi><mi>i</mi></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> z=rcos θ+( rsin θ )i z=r( cos θ+isin θ )
Polar Form of a Complex Number

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

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow/></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow><mi>x</mi><mo>=</mo><mi>r</mi><mi>cos</mi><mtext> </mtext><mi>θ</mi></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> y=rsin θ r= x 2 + y 2

Making a direct substitution, we have

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mi>z</mi><mo>=</mo><mi>x</mi><mo>+</mo><mi>y</mi><mi>i</mi></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> z=( rcos θ )+i( rsin θ ) z=r( cos θ+isin θ )

where<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>r</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>is the modulus and <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>θ</mi></mrow></annotation-xml></semantics></math> is the argument. We often use the abbreviation<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>r</mi><mtext>cis</mtext><mtext> </mtext><mi>θ</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>to represent<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>r</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> cos θ+isin θ ).

Expressing a Complex Number Using Polar Coordinates

Express the complex number<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mn>4</mn><mi>i</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>using polar coordinates.

On the complex plane, the number<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>z</mi><mo>=</mo><mn>4</mn><mi>i</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>is the same as<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>z</mi><mo>=</mo><mn>0</mn><mo>+</mo><mn>4</mn><mi>i</mi><mo>.</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>Writing it in polar form, we have to calculate<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>r</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>first.

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mi>r</mi><mo>=</mo><msqrt/></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> x 2 + y 2 r= 0 2 + 4 2 r= 16 r=4

Next, we look at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>x</mi><mo>.</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>If<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>x</mi><mo>=</mo><mi>r</mi><mi>cos</mi><mtext> </mtext><mi>θ</mi><mo>,</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>x</mi><mo>=</mo><mn>0</mn><mo>,</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>then<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>θ</mi><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> π 2 . In polar coordinates, the complex number<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>z</mi><mo>=</mo><mn>0</mn><mo>+</mo><mn>4</mn><mi>i</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>can be written as<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>z</mi><mo>=</mo><mn>4</mn><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> cos( π 2 )+isin( π 2 ) ) or<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mn>4</mn><mtext>cis</mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math>   π 2 ). See [link].

<figure class="small" id="Figure_08_05_007">Plot of z=4i in the complex plane, also shows that the in polar coordinate it would be (4,pi/2).</figure>

Express<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>z</mi><mo>=</mo><mn>3</mn><mi>i</mi><mtext> </mtext></mrow></annotation-xml></semantics></math> as <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>r</mi><mtext> </mtext><mtext>cis</mtext><mtext> </mtext><mi>θ</mi><mtext> </mtext></mrow></annotation-xml></semantics></math> in polar form.

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>z</mi><mo>=</mo><mn>3</mn><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> cos( π 2 )+isin( π 2 ) )

Finding the Polar Form of a Complex Number

Find the polar form of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mo>−</mo><mn>4</mn><mo>+</mo><mn>4</mn><mi>i</mi><mo>.</mo></mrow></annotation-xml></semantics></math>

First, find the value of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>r</mi><mo>.</mo></mrow></annotation-xml></semantics></math>

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mi>r</mi><mo>=</mo><msqrt/></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> x 2 + y 2 r= ( −4 ) 2 +( 4 2 ) r= 32 r=4 2

Find the angle<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>θ</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>using the formula:

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mi>cos</mi><mtext> </mtext><mi>θ</mi><mo>=</mo><mfrac/></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> x r cos θ= −4 4 2 cos θ=− 1 2          θ= cos −1 ( − 1 2 )= 3π 4

Thus, the solution is<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mn>4</mn><msqrt/></mrow></annotation-xml></semantics></math> 2 cis( 3π 4 ).

Write<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>z</mi><mo>=</mo><msqrt/></mrow></annotation-xml></semantics></math> 3 +i in polar form.

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>z</mi><mo>=</mo><mn>2</mn><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> cos( π 6 )+isin( π 6 ) )

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<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>z</mi><mo>=</mo><mi>r</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> cos θ+isin θ ), first evaluate the trigonometric functions<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>cos</mi><mtext> </mtext><mi>θ</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>sin</mi><mtext> </mtext><mi>θ</mi><mo>.</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>Then, multiply through by<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>r</mi><mo>.</mo></mrow></annotation-xml></semantics></math>

Converting from Polar to Rectangular Form

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

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>z</mi><mo>=</mo><mn>12</mn><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> cos( π 6 )+isin( π 6 ) )

We begin by evaluating the trigonometric expressions.

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>cos</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> π 6 )= 3 2  and sin( π 6 )= 1 2  

After substitution, the complex number is

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>z</mi><mo>=</mo><mn>12</mn><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 3 2 + 1 2 i )

We apply the distributive property:

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mi>z</mi><mo>=</mo><mn>12</mn><mrow><mo>(</mo></mrow></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> 3 2 + 1 2 i )   =( 12 ) 3 2 +( 12 ) 1 2 i   =6 3 +6i

The rectangular form of the given point in complex form is<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mn>6</mn><msqrt/></mrow></annotation-xml></semantics></math> 3 +6i.

Finding the Rectangular Form of a Complex Number

Find the rectangular form of the complex number given<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>r</mi><mo>=</mo><mn>13</mn><mtext> </mtext></mrow></annotation-xml></semantics></math>and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>tan</mi><mtext> </mtext><mi>θ</mi><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> 5 12 .

If<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>tan</mi><mtext> </mtext><mi>θ</mi><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> 5 12 , and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>tan</mi><mtext> </mtext><mi>θ</mi><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> y x , we first determine<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>r</mi><mo>=</mo><msqrt/></mrow></annotation-xml></semantics></math> x 2 + y 2 = 12 2 + 5 2 =13. We then find<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>cos</mi><mtext> </mtext><mi>θ</mi><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> x r  and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>sin</mi><mtext> </mtext><mi>θ</mi><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> y r .

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mi>z</mi><mo>=</mo><mn>13</mn><mo stretchy="false">(</mo><mi>cos</mi><mtext> </mtext><mi>θ</mi><mo>+</mo><mi>i</mi><mi>sin</mi><mtext> </mtext><mi>θ</mi><mo stretchy="false">)</mo></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math>    =13( 12 13 + 5 13 i )    =12+5i

The rectangular form of the given number in complex form is<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mn>12</mn><mo>+</mo><mn>5</mn><mi>i</mi><mo>.</mo></mrow></annotation-xml></semantics></math>

Convert the complex number to rectangular form:

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>z</mi><mo>=</mo><mn>4</mn><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> cos 11π 6 +isin 11π 6 )

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>z</mi><mo>=</mo><mn>2</mn><msqrt/></mrow></annotation-xml></semantics></math> 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.

Products of Complex Numbers in Polar Form

If<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><msub/></mrow></annotation-xml></semantics></math> z 1 = r 1 (cos  θ 1 +isin  θ 1 ) and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><msub/></mrow></annotation-xml></semantics></math> z 2 = r 2 (cos  θ 2 +isin  θ 2 ), then the product of these numbers is given as:

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow/></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><msub><mi>z</mi></msub></mrow></mtd></mtr></mtable></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> 1 z 2 = r 1 r 2 [ cos( θ 1 + θ 2 )+isin( θ 1 + θ 2 ) ] z 1 z 2 = r 1 r 2 cis( θ 1 + θ 2 )

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

Finding the Product of Two Complex Numbers in Polar Form

Find the product of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><msub/></mrow></annotation-xml></semantics></math> z 1 z 2 , given<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><msub/></mrow></annotation-xml></semantics></math> z 1 =4(cos(80°)+isin(80°)) and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><msub/></mrow></annotation-xml></semantics></math> z 2 =2(cos(145°)+isin(145°)).

Follow the formula

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><msub><mi>z</mi></msub></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> 1 z 2 =4⋅2[cos(80°+145°)+isin(80°+145°)] z 1 z 2 =8[cos(225°)+isin(225°)] z 1 z 2 =8[ cos( 5π 4 )+isin( 5π 4 ) ] z 1 z 2=8[ − 2 2 +i( − 2 2 ) ] z 1 z 2 =−4 2 −4i 2

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<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><msub/></mrow></annotation-xml></semantics></math> z 1 = r 1 (cos  θ 1 +isin  θ 1 ) and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><msub/></mrow></annotation-xml></semantics></math> z 2 = r 2 (cos  θ 2 +isin  θ 2 ), then the quotient of these numbers is

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtable columnalign="left"><mtr><mtd><mfrac><mrow><msub><mi>z</mi></msub></mrow></mfrac></mtd></mtr></mtable></annotation-xml></semantics></math> 1 z 2 = r 1 r 2 [ cos( θ 1 − θ 2 )+isin( θ 1 − θ 2 ) ],   z 2 ≠0 z 1 z 2 = r 1 r 2 cis( θ 1 − θ 2 ),   z 2 ≠0 

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

Given two complex numbers in polar form, find the quotient.

  1. Divide<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mfrac/></mrow></annotation-xml></semantics></math> r 1 r 2 .
  2. Find<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><msub/></mrow></annotation-xml></semantics></math> θ 1 − θ 2 .
  3. Substitute the results into the formula:<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>z</mi><mo>=</mo><mi>r</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> cos θ+isin θ ). Replace<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>r</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>with<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mfrac/></mrow></annotation-xml></semantics></math> r 1 r 2 , and replace<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>θ</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>with<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><msub/></mrow></annotation-xml></semantics></math> θ 1 − θ 2 .
  4. Calculate the new trigonometric expressions and multiply through by<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>r</mi><mo>.</mo></mrow></annotation-xml></semantics></math>
Finding the Quotient of Two Complex Numbers

Find the quotient of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><msub/></mrow></annotation-xml></semantics></math> z 1 =2(cos(213°)+isin(213°)) and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><msub/></mrow></annotation-xml></semantics></math> z 2 =4(cos(33°)+isin(33°)).

Using the formula, we have

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mfrac><mrow><msub><mi>z</mi></msub></mrow></mfrac></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> 1 z 2 = 2 4 [cos(213°−33°)+isin(213°−33°)] z 1 z 2 = 1 2 [cos(180°)+isin(180°)] z 1 z 2 = 1 2 [−1+0i] z 1 z 2 =− 1 2 +0i z 1z 2 =− 1 2

Find the product and the quotient of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><msub/></mrow></annotation-xml></semantics></math> z 1 =2 3 (cos(150°)+isin(150°)) and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><msub/></mrow></annotation-xml></semantics></math> z 2 =2(cos(30°)+isin(30°)).

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><msub/></mrow></annotation-xml></semantics></math> z 1 z 2 =−4 3 ; z 1 z 2 =− 3 2 + 3 2 i 

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<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>n</mi><mo>,</mo><msup/></mrow></annotation-xml></semantics></math> z n  is found by raising the modulus to the<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>n</mi><mtext>th</mtext><mtext> </mtext></mrow></annotation-xml></semantics></math>power and multiplying the argument by<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>n</mi><mo>.</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>It is the standard method used in modern mathematics.

De Moivre’s Theorem

If<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>z</mi><mo>=</mo><mi>r</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> cos θ+isin θ ) is a complex number, then

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtable columnalign="left"><mtr><mtd><msup><mi>z</mi></msup></mtd></mtr></mtable></annotation-xml></semantics></math> n = r n [ cos( nθ )+isin( nθ ) ] z n = r n cis( nθ )

where<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>n</mi><mtext> </mtext></mrow></annotation-xml></semantics></math> is a positive integer.

Evaluating an Expression Using De Moivre’s Theorem

Evaluate the expression<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><msup/></mrow></annotation-xml></semantics></math> ( 1+i ) 5  using De Moivre’s Theorem.

Since De Moivre’s Theorem applies to complex numbers written in polar form, we must first write<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 1+i ) in polar form. Let us find<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>r</mi><mo>.</mo></mrow></annotation-xml></semantics></math>

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mi>r</mi><mo>=</mo><msqrt/></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> x 2 + y 2 r= ( 1 ) 2 + ( 1 ) 2 r= 2

Then we find<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>θ</mi><mo>.</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>Using the formula<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>tan</mi><mtext> </mtext><mi>θ</mi><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> y x  gives

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mi>tan</mi><mtext> </mtext><mi>θ</mi><mo>=</mo><mfrac/></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> 1 1 tan θ=1         θ= π 4

Use De Moivre’s Theorem to evaluate the expression.

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><msup><mrow><mo stretchy="false">(</mo><mi>a</mi><mo>+</mo><mi>b</mi><mi>i</mi><mo stretchy="false">)</mo></mrow></msup></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> n = r n [cos(nθ)+isin(nθ)]      (1+i) 5 = ( 2 ) 5 [ cos( 5⋅ π 4 )+isin( 5⋅ π 4 ) ]      (1+i) 5 =4 2 [ cos( 5π 4 )+isin( 5π 4 ) ]     (1+i) 5 =4 2 [ − 2 2 +i( − 2 2 ) ]      (1+i) 5 =−4−4i

Finding Roots of Complex Numbers in Polar Form

To find the nth root of a complex number in polar form, we use the<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>n</mi><mtext>th</mtext><mtext> </mtext></mrow></annotation-xml></semantics></math>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<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>n</mi><mtext>th</mtext><mtext> </mtext></mrow></annotation-xml></semantics></math>roots of complex numbers in polar form.

The nth Root Theorem

To find the<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>n</mi><mtext>th</mtext><mtext> </mtext></mrow></annotation-xml></semantics></math>root of a complex number in polar form, use the formula given as

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msup><mi>z</mi></msup></mrow></annotation-xml></semantics></math> 1 n = r 1 n [ cos( θ n + 2kπ n )+isin( θ n + 2kπ n ) ]

where<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>k</mi><mo>=</mo><mn>0</mn><mo>,</mo><mtext> </mtext><mtext> </mtext><mn>1</mn><mo>,</mo><mtext> </mtext><mtext> </mtext><mn>2</mn><mo>,</mo><mtext> </mtext><mtext> </mtext><mn>3</mn><mo>,</mo><mtext> </mtext><mo>.</mo><mtext> </mtext><mtext> </mtext><mo>.</mo><mtext> </mtext><mtext> </mtext><mo>.</mo><mtext> </mtext><mtext> </mtext><mo>,</mo><mtext> </mtext><mtext> </mtext><mi>n</mi><mo>−</mo><mn>1.</mn><mtext> </mtext></mrow></annotation-xml></semantics></math>We add <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mfrac/></mrow></annotation-xml></semantics></math> 2kπ n   to<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mfrac/></mrow></annotation-xml></semantics></math> θ n  in order to obtain the periodic roots.

Finding the nth Root of a Complex Number

Evaluate the cube roots of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>z</mi><mo>=</mo><mn>8</mn><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> cos( 2π 3 )+isin( 2π 3 ) ).

We have

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><msup><mi>z</mi></msup></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> 1 3 = 8 1 3 [ cos( 2π 3 3 + 2kπ 3 )+isin( 2π 3 3 + 2kπ 3 ) ] z 1 3 =2[ cos( 2π 9 + 2kπ 3 )+isin( 2π 9 + 2kπ 3 ) ]

There will be three roots:<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>k</mi><mo>=</mo><mn>0</mn><mo>,</mo><mtext> </mtext><mtext> </mtext><mn>1</mn><mo>,</mo><mtext> </mtext><mtext> </mtext><mn>2.</mn><mtext> </mtext></mrow></annotation-xml></semantics></math>When<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>k</mi><mo>=</mo><mn>0</mn><mo>,</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>we have

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msup><mi>z</mi></msup></mrow></annotation-xml></semantics></math> 1 3 =2( cos( 2π 9 )+isin( 2π 9 ) )

When<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>k</mi><mo>=</mo><mn>1</mn><mo>,</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>we have

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><msup><mi>z</mi></msup></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> 1 3 =2[ cos( 2π 9 + 6π 9 )+isin( 2π 9 + 6π 9 ) ]     Add  2(1)π 3  to each angle. z 1 3 =2( cos( 8π 9 )+isin( 8π 9 ) )

When<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>k</mi><mo>=</mo><mn>2</mn><mo>,</mo><mtext> </mtext></mrow></annotation-xml></semantics></math> we have

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><msup><mi>z</mi></msup></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> 1 3 =2[ cos( 2π 9 + 12π 9 )+isin( 2π 9 + 12π 9 ) ] Add  2(2)π 3  to each angle. z 1 3 =2( cos( 14π 9 )+isin( 14π 9 ) )

Remember to find the common denominator to simplify fractions in situations like this one. For<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>k</mi><mo>=</mo><mn>1</mn><mo>,</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>the angle simplification is

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mfrac><mrow><mfrac><mrow><mn>2</mn><mi>π</mi></mrow></mfrac></mrow></mfrac></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> 3 3 + 2(1)π 3 = 2π 3 ( 1 3 )+ 2(1)π 3 ( 3 3 )                           = 2π 9 + 6π 9                           = 8π 9

Find the four fourth roots of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mn>16</mn><mo stretchy="false">(</mo><mi>cos</mi><mo stretchy="false">(</mo><mn>120°</mn><mo stretchy="false">)</mo><mo>+</mo><mi>i</mi><mi>sin</mi><mo stretchy="false">(</mo><mn>120°</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>.</mo></mrow></annotation-xml></semantics></math>

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msub><mi>z</mi></msub></mrow></annotation-xml></semantics></math> 0 =2(cos(30°)+isin(30°))

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msub><mi>z</mi></msub></mrow></annotation-xml></semantics></math> 1 =2(cos(120°)+isin(120°))

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msub><mi>z</mi></msub></mrow></annotation-xml></semantics></math> 2 =2(cos(210°)+isin(210°))

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msub><mi>z</mi></msub></mrow></annotation-xml></semantics></math> 3 =2(cos(300°)+isin(300°))

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

Key Concepts

  • Complex numbers in the form<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>a</mi><mo>+</mo><mi>b</mi><mi>i</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>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:<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>|</mo></mrow></mrow></annotation-xml></semantics></math> z |= a 2 + b 2 . See [link] and [link].
  • To write complex numbers in polar form, we use the formulas<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>x</mi><mo>=</mo><mi>r</mi><mi>cos</mi><mtext> </mtext><mi>θ</mi><mo>,</mo><mi>y</mi><mo>=</mo><mi>r</mi><mi>sin</mi><mtext> </mtext><mi>θ</mi><mo>,</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>and <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>r</mi><mo>=</mo><msqrt/></mrow></annotation-xml></semantics></math> x 2 + y 2 . Then,<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>z</mi><mo>=</mo><mi>r</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> cos θ+isin θ ). See [link] and [link].
  • To convert from polar form to rectangular form, first evaluate the trigonometric functions. Then, multiply through by<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>r</mi><mo>.</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>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<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><msup/></mrow></annotation-xml></semantics></math> z n , raise <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>r</mi><mtext> </mtext></mrow></annotation-xml></semantics></math> to the power <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>n</mi><mo>,</mo></mrow></annotation-xml></semantics></math> and multiply <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>θ</mi><mtext> </mtext></mrow></annotation-xml></semantics></math> by <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>n</mi><mo>.</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>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].

Section Exercises

Verbal

A complex number is<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>a</mi><mo>+</mo><mi>b</mi><mi>i</mi><mo>.</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>Explain each part.

a is the real part, b is the imaginary part, and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>i</mi><mo>=</mo><msqrt/></mrow></annotation-xml></semantics></math> −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<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>x</mi><mo>=</mo><mi>r</mi><mi>cos</mi><mi>θ</mi><mtext> </mtext></mrow></annotation-xml></semantics></math> and <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>y</mi><mo>=</mo><mi>r</mi><mi>sin</mi><mi>θ</mi><mo>.</mo></mrow></annotation-xml></semantics></math>

How do we find the product of two complex numbers?

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

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msup><mi>z</mi></msup></mrow></annotation-xml></semantics></math> n = r n ( cos( nθ )+isin( nθ ) ) 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.

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>5</mn><mo>+</mo><mtext>​</mtext><mn>3</mn><mi>i</mi></mrow></annotation-xml></semantics></math>

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mo>−</mo><mn>7</mn><mo>+</mo><mtext>​</mtext><mi>i</mi></mrow></annotation-xml></semantics></math>

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>5</mn><msqrt/></mrow></annotation-xml></semantics></math> 2

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mo>−</mo><mn>3</mn><mo>−</mo><mn>3</mn><mi>i</mi></mrow></annotation-xml></semantics></math>

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msqrt><mn>2</mn></msqrt></mrow></annotation-xml></semantics></math> −6i

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msqrt><mrow><mn>38</mn></mrow></msqrt></mrow></annotation-xml></semantics></math>

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>2</mn><mi>i</mi></mrow></annotation-xml></semantics></math>

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>2.2</mn><mo>−</mo><mn>3.1</mn><mi>i</mi></mrow></annotation-xml></semantics></math>

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msqrt><mrow><mn>14.45</mn></mrow></msqrt></mrow></annotation-xml></semantics></math>

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

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>2</mn><mo>+</mo><mn>2</mn><mi>i</mi></mrow></annotation-xml></semantics></math>

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>8</mn><mo>−</mo><mn>4</mn><mi>i</mi></mrow></annotation-xml></semantics></math>

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>4</mn><msqrt/></mrow></annotation-xml></semantics></math> 5 cis( 333.4° )

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mo>−</mo><mfrac/></mrow></annotation-xml></semantics></math> 1 2 − 1 2 ​i

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msqrt><mn>3</mn></msqrt></mrow></annotation-xml></semantics></math> +i

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>2</mn><mi mathvariant="normal">cis</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> π 6 )

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>3</mn><mi>i</mi></mrow></annotation-xml></semantics></math>

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

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>z</mi><mo>=</mo><mn>7</mn><mi mathvariant="normal">cis</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> π 6 )

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mrow><mn>7</mn><msqrt/></mrow></mfrac></mrow></annotation-xml></semantics></math> 3 2 +i 7 2

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>z</mi><mo>=</mo><mn>2</mn><mi mathvariant="normal">cis</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> π 3 )

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>z</mi><mo>=</mo><mn>4</mn><mi mathvariant="normal">cis</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 7π 6 )

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mo>−</mo><mn>2</mn><msqrt/></mrow></annotation-xml></semantics></math> 3 −2i

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>z</mi><mo>=</mo><mn>7</mn><mi mathvariant="normal">cis</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 25° )

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>z</mi><mo>=</mo><mn>3</mn><mi mathvariant="normal">cis</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 240° )

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mo>−</mo><mn>1.5</mn><mo>−</mo><mi>i</mi><mfrac/></mrow></annotation-xml></semantics></math> 3 3 2

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>z</mi><mo>=</mo><msqrt/></mrow></annotation-xml></semantics></math> 2 cis( 100° )

For the following exercises, find<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><msub/></mrow></annotation-xml></semantics></math> z 1 z 2  in polar form.

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msub><mi>z</mi></msub></mrow></annotation-xml></semantics></math> 1 =2 3 cis( 116° );   z 2 =2cis( 82° )

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>4</mn><msqrt/></mrow></annotation-xml></semantics></math> 3 cis( 198° )

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msub><mi>z</mi></msub></mrow></annotation-xml></semantics></math> 1 = 2 cis( 205° );  z 2 =2 2 cis( 118° )

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msub><mi>z</mi></msub></mrow></annotation-xml></semantics></math> 1 =3cis( 120° );  z 2 = 1 4 cis( 60° )

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mn>3</mn></mfrac></mrow></annotation-xml></semantics></math> 4 cis( 180° )

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msub><mi>z</mi></msub></mrow></annotation-xml></semantics></math> 1 =3cis( π 4 );  z 2 =5cis( π 6 )

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msub><mi>z</mi></msub></mrow></annotation-xml></semantics></math> 1 = 5 cis( 5π 8 );  z 2 = 15 cis( π 12 )

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>5</mn><msqrt/></mrow></annotation-xml></semantics></math> 3 cis( 17π 24 )

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msub><mi>z</mi></msub></mrow></annotation-xml></semantics></math> 1 =4cis( π 2 );  z 2 =2cis( π 4 )

For the following exercises, find<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mfrac/></mrow></annotation-xml></semantics></math> z 1 z 2  in polar form.

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msub><mi>z</mi></msub></mrow></annotation-xml></semantics></math> 1 =21cis( 135° );  z 2 =3cis( 65° )

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>7</mn><mi mathvariant="normal">cis</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 70° )

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msub><mi>z</mi></msub></mrow></annotation-xml></semantics></math> 1 = 2 cis( 90° );  z 2 =2cis( 60° )

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msub><mi>z</mi></msub></mrow></annotation-xml></semantics></math> 1 =15cis( 120° );  z 2 =3cis( 40° )

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>5</mn><mi mathvariant="normal">cis</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 80° )

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msub><mi>z</mi></msub></mrow></annotation-xml></semantics></math> 1 =6cis( π 3 );  z 2 =2cis( π 4 )

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msub><mi>z</mi></msub></mrow></annotation-xml></semantics></math> 1 =5 2 cis( π );  z 2 = 2 cis( 2π 3 )

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>5</mn><mi mathvariant="normal">cis</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> π 3 )

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msub><mi>z</mi></msub></mrow></annotation-xml></semantics></math> 1 =2cis( 3π 5 );  z 2 =3cis( π 4 )

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

Find<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><msup/></mrow></annotation-xml></semantics></math> z 3  when<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>z</mi><mo>=</mo><mn>5</mn><mi mathvariant="normal">cis</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 45° ).

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>125</mn><mi mathvariant="normal">cis</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 135° )

Find<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><msup/></mrow></annotation-xml></semantics></math> z 4  when<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>z</mi><mo>=</mo><mn>2</mn><mi mathvariant="normal">cis</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 70° ).

Find<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><msup/></mrow></annotation-xml></semantics></math> z 2  when<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>z</mi><mo>=</mo><mn>3</mn><mi mathvariant="normal">cis</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 120° ).

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>9</mn><mi mathvariant="normal">cis</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 240° )

Find<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><msup/></mrow></annotation-xml></semantics></math> z 2  when<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>z</mi><mo>=</mo><mn>4</mn><mi mathvariant="normal">cis</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> π 4 ).

Find<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><msup/></mrow></annotation-xml></semantics></math> z 4  when<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>z</mi><mo>=</mo><mi mathvariant="normal">cis</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 3π 16 ).

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi mathvariant="normal">cis</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 3π 4 )

Find<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><msup/></mrow></annotation-xml></semantics></math> z 3  when<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>z</mi><mo>=</mo><mn>3</mn><mi mathvariant="normal">cis</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 5π 3 ).

For the following exercises, evaluate each root.

Evaluate the cube root of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>z</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>when<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>z</mi><mo>=</mo><mn>27</mn><mi mathvariant="normal">cis</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 240° ).

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mn>3</mn><mi mathvariant="normal">cis</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 80° ),3cis( 200° ),3cis( 320° ) 

Evaluate the square root of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>z</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>when<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>z</mi><mo>=</mo><mn>16</mn><mi mathvariant="normal">cis</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 100° ).

Evaluate the cube root of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>z</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>when<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>z</mi><mo>=</mo><mn>32</mn><mi mathvariant="normal">cis</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 2π 3 ).

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mn>2</mn><mroot/></mrow></annotation-xml></semantics></math> 4 3 cis( 2π 9 ),2 4 3 cis( 8π 9 ),2 4 3 cis( 14π 9 )

Evaluate the square root of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>z</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>when<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>z</mi><mo>=</mo><mn>32</mn><mtext>cis</mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> π ).

Evaluate the cube root of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>z</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>when<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>z</mi><mo>=</mo><mn>8</mn><mi mathvariant="normal">cis</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 7π 4 ).

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>2</mn><msqrt/></mrow></annotation-xml></semantics></math> 2 cis( 7π 8 ),2 2 cis( 15π 8 )

Graphical

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

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>2</mn><mo>+</mo><mn>4</mn><mi>i</mi></mrow></annotation-xml></semantics></math>

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mo>−</mo><mn>3</mn><mo>−</mo><mn>3</mn><mi>i</mi></mrow></annotation-xml></semantics></math>

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

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>5</mn><mo>−</mo><mn>4</mn><mi>i</mi></mrow></annotation-xml></semantics></math>

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mo>−</mo><mn>1</mn><mo>−</mo><mn>5</mn><mi>i</mi></mrow></annotation-xml></semantics></math>

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

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>3</mn><mo>+</mo><mn>2</mn><mi>i</mi></mrow></annotation-xml></semantics></math>

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>2</mn><mi>i</mi></mrow></annotation-xml></semantics></math>

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

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mo>−</mo><mn>4</mn></mrow></annotation-xml></semantics></math>

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>6</mn><mo>−</mo><mn>2</mn><mi>i</mi></mrow></annotation-xml></semantics></math>

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

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mo>−</mo><mn>2</mn><mo>+</mo><mi>i</mi></mrow></annotation-xml></semantics></math>

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>1</mn><mo>−</mo><mn>4</mn><mi>i</mi></mrow></annotation-xml></semantics></math>

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.

Use the rectangular to polar feature on the graphing calculator to change<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mn>5</mn><mo>+</mo><mn>5</mn><mi>i</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>to polar form.

Use the rectangular to polar feature on the graphing calculator to change<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mn>3</mn><mo>−</mo><mn>2</mn><mi>i</mi><mtext> </mtext></mrow></annotation-xml></semantics></math> to polar form.

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mn>3.61</mn><msup/></mrow></annotation-xml></semantics></math> e −0.59i  

Use the rectangular to polar feature on the graphing calculator to change <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mo>−</mo><mn>3</mn><mo>−</mo><mn>8</mn><mi>i</mi><mtext> </mtext></mrow></annotation-xml></semantics></math> to polar form.

Use the polar to rectangular feature on the graphing calculator to change<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mn>4</mn><mi mathvariant="normal">cis</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 120° ) to rectangular form.

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mo>−</mo><mn>2</mn><mo>+</mo><mn>3.46</mn><mi>i</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>

Use the polar to rectangular feature on the graphing calculator to change<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mn>2</mn><mi mathvariant="normal">cis</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 45° ) to rectangular form.

Use the polar to rectangular feature on the graphing calculator to change<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mn>5</mn><mi mathvariant="normal">cis</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 210° ) to rectangular form.

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mo>−</mo><mn>4.33</mn><mo>−</mo><mn>2.50</mn><mi>i</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>

Glossary

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<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>n</mi><mtext>th</mtext><mtext> </mtext></mrow></annotation-xml></semantics></math>power or nth roots of a complex number; states that, for a positive integer<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>n</mi><mo>,</mo><msup/></mrow></annotation-xml></semantics></math> z n  is found by raising the modulus to the<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>n</mi><mtext>th</mtext><mtext> </mtext></mrow></annotation-xml></semantics></math>power and multiplying the angles by<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>n</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>
modulus
the absolute value of a complex number, or the distance from the origin to the point<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> x,y ); also called the amplitude
polar form of a complex number
a complex number expressed in terms of an angle <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>θ</mi></mrow></annotation-xml></semantics></math> and its distance from the origin<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>r</mi><mo>;</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>can be found by using conversion formulas<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>x</mi><mo>=</mo><mi>r</mi><mi>cos</mi><mtext> </mtext><mi>θ</mi><mo>,</mo><mtext> </mtext><mtext> </mtext><mi>y</mi><mo>=</mo><mi>r</mi><mi>sin</mi><mtext> </mtext><mi>θ</mi><mo>,</mo><mtext> </mtext><mtext> </mtext></mrow></annotation-xml></semantics></math>and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>r</mi><mo>=</mo><msqrt/></mrow></annotation-xml></semantics></math> x 2 + y 2