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Algebra and Trigonometry

10.5 Polar Form of Complex Numbers

Algebra and Trigonometry10.5 Polar Form of Complex Numbers
  1. Preface
  2. 1 Prerequisites
    1. Introduction to Prerequisites
    2. 1.1 Real Numbers: Algebra Essentials
    3. 1.2 Exponents and Scientific Notation
    4. 1.3 Radicals and Rational Exponents
    5. 1.4 Polynomials
    6. 1.5 Factoring Polynomials
    7. 1.6 Rational Expressions
    8. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  3. 2 Equations and Inequalities
    1. Introduction to Equations and Inequalities
    2. 2.1 The Rectangular Coordinate Systems and Graphs
    3. 2.2 Linear Equations in One Variable
    4. 2.3 Models and Applications
    5. 2.4 Complex Numbers
    6. 2.5 Quadratic Equations
    7. 2.6 Other Types of Equations
    8. 2.7 Linear Inequalities and Absolute Value Inequalities
    9. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
    10. Exercises
      1. Review Exercises
      2. Practice Test
  4. 3 Functions
    1. Introduction to Functions
    2. 3.1 Functions and Function Notation
    3. 3.2 Domain and Range
    4. 3.3 Rates of Change and Behavior of Graphs
    5. 3.4 Composition of Functions
    6. 3.5 Transformation of Functions
    7. 3.6 Absolute Value Functions
    8. 3.7 Inverse Functions
    9. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
    10. Exercises
      1. Review Exercises
      2. Practice Test
  5. 4 Linear Functions
    1. Introduction to Linear Functions
    2. 4.1 Linear Functions
    3. 4.2 Modeling with Linear Functions
    4. 4.3 Fitting Linear Models to Data
    5. Chapter Review
      1. Key Terms
      2. Key Concepts
    6. Exercises
      1. Review Exercises
      2. Practice Test
  6. 5 Polynomial and Rational Functions
    1. Introduction to Polynomial and Rational Functions
    2. 5.1 Quadratic Functions
    3. 5.2 Power Functions and Polynomial Functions
    4. 5.3 Graphs of Polynomial Functions
    5. 5.4 Dividing Polynomials
    6. 5.5 Zeros of Polynomial Functions
    7. 5.6 Rational Functions
    8. 5.7 Inverses and Radical Functions
    9. 5.8 Modeling Using Variation
    10. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  7. 6 Exponential and Logarithmic Functions
    1. Introduction to Exponential and Logarithmic Functions
    2. 6.1 Exponential Functions
    3. 6.2 Graphs of Exponential Functions
    4. 6.3 Logarithmic Functions
    5. 6.4 Graphs of Logarithmic Functions
    6. 6.5 Logarithmic Properties
    7. 6.6 Exponential and Logarithmic Equations
    8. 6.7 Exponential and Logarithmic Models
    9. 6.8 Fitting Exponential Models to Data
    10. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  8. 7 The Unit Circle: Sine and Cosine Functions
    1. Introduction to The Unit Circle: Sine and Cosine Functions
    2. 7.1 Angles
    3. 7.2 Right Triangle Trigonometry
    4. 7.3 Unit Circle
    5. 7.4 The Other Trigonometric Functions
    6. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
    7. Exercises
      1. Review Exercises
      2. Practice Test
  9. 8 Periodic Functions
    1. Introduction to Periodic Functions
    2. 8.1 Graphs of the Sine and Cosine Functions
    3. 8.2 Graphs of the Other Trigonometric Functions
    4. 8.3 Inverse Trigonometric Functions
    5. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
    6. Exercises
      1. Review Exercises
      2. Practice Test
  10. 9 Trigonometric Identities and Equations
    1. Introduction to Trigonometric Identities and Equations
    2. 9.1 Solving Trigonometric Equations with Identities
    3. 9.2 Sum and Difference Identities
    4. 9.3 Double-Angle, Half-Angle, and Reduction Formulas
    5. 9.4 Sum-to-Product and Product-to-Sum Formulas
    6. 9.5 Solving Trigonometric Equations
    7. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
    8. Exercises
      1. Review Exercises
      2. Practice Test
  11. 10 Further Applications of Trigonometry
    1. Introduction to Further Applications of Trigonometry
    2. 10.1 Non-right Triangles: Law of Sines
    3. 10.2 Non-right Triangles: Law of Cosines
    4. 10.3 Polar Coordinates
    5. 10.4 Polar Coordinates: Graphs
    6. 10.5 Polar Form of Complex Numbers
    7. 10.6 Parametric Equations
    8. 10.7 Parametric Equations: Graphs
    9. 10.8 Vectors
    10. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  12. 11 Systems of Equations and Inequalities
    1. Introduction to Systems of Equations and Inequalities
    2. 11.1 Systems of Linear Equations: Two Variables
    3. 11.2 Systems of Linear Equations: Three Variables
    4. 11.3 Systems of Nonlinear Equations and Inequalities: Two Variables
    5. 11.4 Partial Fractions
    6. 11.5 Matrices and Matrix Operations
    7. 11.6 Solving Systems with Gaussian Elimination
    8. 11.7 Solving Systems with Inverses
    9. 11.8 Solving Systems with Cramer's Rule
    10. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  13. 12 Analytic Geometry
    1. Introduction to Analytic Geometry
    2. 12.1 The Ellipse
    3. 12.2 The Hyperbola
    4. 12.3 The Parabola
    5. 12.4 Rotation of Axes
    6. 12.5 Conic Sections in Polar Coordinates
    7. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
    8. Exercises
      1. Review Exercises
      2. Practice Test
  14. 13 Sequences, Probability, and Counting Theory
    1. Introduction to Sequences, Probability and Counting Theory
    2. 13.1 Sequences and Their Notations
    3. 13.2 Arithmetic Sequences
    4. 13.3 Geometric Sequences
    5. 13.4 Series and Their Notations
    6. 13.5 Counting Principles
    7. 13.6 Binomial Theorem
    8. 13.7 Probability
    9. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
    10. Exercises
      1. Review Exercises
      2. Practice Test
  15. A | Proofs, Identities, and Toolkit Functions
  16. Answer Key
    1. Chapter 1
    2. Chapter 2
    3. Chapter 3
    4. Chapter 4
    5. Chapter 5
    6. Chapter 6
    7. Chapter 7
    8. Chapter 8
    9. Chapter 9
    10. Chapter 10
    11. Chapter 11
    12. Chapter 12
    13. Chapter 13
  17. Index

Learning Objectives

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 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+bi a+bi is similar to plotting a real number, except that the horizontal axis represents the real part of the number, a, a, and the vertical axis represents the imaginary part of the number, bi. bi.

Given a complex number a+bi, a+bi, 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 a a units in the horizontal direction and b b units in the vertical direction.

Example 1

Plotting a Complex Number in the Complex Plane

Plot the complex number 23i 23i in the complex plane.

Try It #1

Plot the point 1+5i 1+5i in the complex plane.

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 |. | z |. It measures the distance from the origin to a point in the plane. For example, the graph of z=2+4i, z=2+4i, in Figure 2, shows | z |. | z |.

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

Absolute Value of a Complex Number

Given z=x+yi, z=x+yi, a complex number, the absolute value of z z is defined as

| z |= x 2 + y 2 | z |= x 2 + y 2

It is the distance from the origin to the point ( x,y ). ( 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, ( 0,0 ). ( 0,0 ).

Example 2

Finding the Absolute Value of a Complex Number with a Radical

Find the absolute value of z= 5 i. z= 5 i.

Try It #2

Find the absolute value of the complex number z=125i. z=125i.

Example 3

Finding the Absolute Value of a Complex Number

Given z=34i, z=34i, find | z |. | z |.

Try It #3

Given z=17i, z=17i, find | z |. | z |.

Writing Complex Numbers in Polar Form

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

x=rcosθ y=rsinθ r= x 2 + y 2 x=rcosθ y=rsinθ r= x 2 + y 2

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 ). ( x,y ). The modulus, then, is the same as r, r, the radius in polar form. We use θ θ to indicate the angle of direction (just as with polar coordinates). Substituting, we have

z=x+yi z=rcosθ+( rsinθ )i z=r( cosθ+isinθ ) z=x+yi z=rcosθ+( rsinθ )i z=r( cosθ+isinθ )

Polar Form of a Complex Number

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

x=rcosθ y=rsinθ r= x 2 + y 2 x=rcosθ y=rsinθ r= x 2 + y 2

Making a direct substitution, we have

z=x+yi z=( rcosθ )+i( rsinθ ) z=r( cosθ+isinθ ) z=x+yi z=( rcosθ )+i( rsinθ ) z=r( cosθ+isinθ )

where r r is the modulus and θ θ is the argument. We often use the abbreviation rcisθ rcisθ to represent r( cosθ+isinθ ). r( cosθ+isinθ ).

Example 4

Expressing a Complex Number Using Polar Coordinates

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

Try It #4

Express z=3i z=3i as rcisθ rcisθ in polar form.

Example 5

Finding the Polar Form of a Complex Number

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

Try It #5

Write z= 3 +i z= 3 +i in polar form.

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( cosθ+isinθ ), z=r( cosθ+isinθ ), first evaluate the trigonometric functions cosθ cosθ and sinθ. sinθ. Then, multiply through by r. r.

Example 6

Converting from Polar to Rectangular Form

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

z=12( cos( π 6 )+isin( π 6 ) ) z=12( cos( π 6 )+isin( π 6 ) )

Example 7

Finding the Rectangular Form of a Complex Number

Find the rectangular form of the complex number given r=13 r=13 and tanθ= 5 12 . tanθ= 5 12 .

Try It #6

Convert the complex number to rectangular form:

z=4( cos 11π 6 +isin 11π 6 ) z=4( cos 11π 6 +isin 11π 6 )

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 +isin θ 1 ) z 1 = r 1 (cos θ 1 +isin θ 1 ) and z 2 = r 2 (cos θ 2 +isin θ 2 ), z 2 = r 2 (cos θ 2 +isin θ 2 ), then the product of these numbers is given as:

z 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 ) z 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.

Example 8

Finding the Product of Two Complex Numbers in Polar Form

Find the product of z 1 z 2 , z 1 z 2 , given z 1 =4(cos(80°)+isin(80°)) z 1 =4(cos(80°)+isin(80°)) and z 2 =2(cos(145°)+isin(145°)). z 2 =2(cos(145°)+isin(145°)).

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 +isin θ 1 ) z 1 = r 1 (cos θ 1 +isin θ 1 ) and z 2 = r 2 (cos θ 2 +isin θ 2 ), z 2 = r 2 (cos θ 2 +isin θ 2 ), then the quotient of these numbers is

z 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 z 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 r 1 r 2 . r 1 r 2 .
  2. Find θ 1 θ 2 . θ 1 θ 2 .
  3. Substitute the results into the formula: z=r( cosθ+isinθ ). z=r( cosθ+isinθ ). Replace r r with r 1 r 2 , r 1 r 2 , and replace θ θ with θ 1 θ 2 . θ 1 θ 2 .
  4. Calculate the new trigonometric expressions and multiply through by r. r.

Example 9

Finding the Quotient of Two Complex Numbers

Find the quotient of z 1 =2(cos(213°)+isin(213°)) z 1 =2(cos(213°)+isin(213°)) and z 2 =4(cos(33°)+isin(33°)). z 2 =4(cos(33°)+isin(33°)).

Try It #7

Find the product and the quotient of z 1 =2 3 (cos(150°)+isin(150°)) z 1 =2 3 (cos(150°)+isin(150°)) and z 2 =2(cos(30°)+isin(30°)). z 2 =2(cos(30°)+isin(30°)).

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 n, z n is found by raising the modulus to the nth nth power and multiplying the argument by n. n. It is the standard method used in modern mathematics.

De Moivre’s Theorem

If z=r( cosθ+isinθ ) z=r( cosθ+isinθ ) is a complex number, then

z n = r n [ cos( nθ )+isin( nθ ) ] z n = r n cis( nθ ) z n = r n [ cos( nθ )+isin( nθ ) ] z n = r n cis( nθ )

where n n is a positive integer.

Example 10

Evaluating an Expression Using De Moivre’s Theorem

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

Finding Roots of Complex Numbers in Polar Form

To find the nth root of a complex number in polar form, we use the nth nth 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 nth nth roots of complex numbers in polar form.

The nth Root Theorem

To find the nth nth root of a complex number in polar form, use the formula given as

z 1 n = r 1 n [ cos( θ n + 2kπ n )+isin( θ n + 2kπ n ) ] z 1 n = r 1 n [ cos( θ n + 2kπ n )+isin( θ n + 2kπ n ) ]

where k=0,1,2,3,...,n1. k=0,1,2,3,...,n1. We add 2kπ n 2kπ n to θ n θ n in order to obtain the periodic roots.

Example 11

Finding the nth Root of a Complex Number

Evaluate the cube roots of z=8( cos( 2π 3 )+isin( 2π 3 ) ). z=8( cos( 2π 3 )+isin( 2π 3 ) ).

Try It #8

Find the four fourth roots of 16(cos(120°)+isin(120°)). 16(cos(120°)+isin(120°)).

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

10.5 Section Exercises

Verbal

1.

A complex number is a+bi. a+bi. Explain each part.

2.

What does the absolute value of a complex number represent?

3.

How is a complex number converted to polar form?

4.

How do we find the product of two complex numbers?

5.

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

Algebraic

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

6.

5+3i 5+3i

7.

7+i 7+i

8.

33i 33i

9.

2 6i 2 6i

10.

2i 2i

11.

2.23.1i 2.23.1i

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

12.

2+2i 2+2i

13.

84i 84i

14.

1 2 1 2 i 1 2 1 2 i

15.

3 +i 3 +i

16.

3i 3i

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

17.

z=7cis( π 6 ) z=7cis( π 6 )

18.

z=2cis( π 3 ) z=2cis( π 3 )

19.

z=4cis( 7π 6 ) z=4cis( 7π 6 )

20.

z=7cis( 25° ) z=7cis( 25° )

21.

z=3cis( 240° ) z=3cis( 240° )

22.

z= 2 cis( 100° ) z= 2 cis( 100° )

For the following exercises, find z 1 z 2 z 1 z 2 in polar form.

23.

z 1 =2 3 cis( 116° ); z 2 =2cis( 82° ) z 1 =2 3 cis( 116° ); z 2 =2cis( 82° )

24.

z 1 = 2 cis( 205° ); z 2 =2 2 cis( 118° ) z 1 = 2 cis( 205° ); z 2 =2 2 cis( 118° )

25.

z 1 =3cis( 120° ); z 2 = 1 4 cis( 60° ) z 1 =3cis( 120° ); z 2 = 1 4 cis( 60° )

26.

z 1 =3cis( π 4 ); z 2 =5cis( π 6 ) z 1 =3cis( π 4 ); z 2 =5cis( π 6 )

27.

z 1 = 5 cis( 5π 8 ); z 2 = 15 cis( π 12 ) z 1 = 5 cis( 5π 8 ); z 2 = 15 cis( π 12 )

28.

z 1 =4cis( π 2 ); z 2 =2cis( π 4 ) z 1 =4cis( π 2 ); z 2 =2cis( π 4 )

For the following exercises, find z 1 z 2 z 1 z 2 in polar form.

29.

z 1 =21cis( 135° ); z 2 =3cis( 65° ) z 1 =21cis( 135° ); z 2 =3cis( 65° )

30.

z 1 = 2 cis( 90° ); z 2 =2cis( 60° ) z 1 = 2 cis( 90° ); z 2 =2cis( 60° )

31.

z 1 =15cis( 120° ); z 2 =3cis( 40° ) z 1 =15cis( 120° ); z 2 =3cis( 40° )

32.

z 1 =6cis( π 3 ); z 2 =2cis( π 4 ) z 1 =6cis( π 3 ); z 2 =2cis( π 4 )

33.

z 1 =5 2 cis( π ); z 2 = 2 cis( 2π 3 ) z 1 =5 2 cis( π ); z 2 = 2 cis( 2π 3 )

34.

z 1 =2cis( 3π 5 ); z 2 =3cis( π 4 ) z 1 =2cis( 3π 5 ); z 2 =3cis( π 4 )

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

35.

Find z 3 z 3 when z=5cis( 45° ). z=5cis( 45° ).

36.

Find z 4 z 4 when z=2cis( 70° ). z=2cis( 70° ).

37.

Find z 2 z 2 when z=3cis( 120° ). z=3cis( 120° ).

38.

Find z 2 z 2 when z=4cis( π 4 ). z=4cis( π 4 ).

39.

Find z 4 z 4 when z=cis( 3π 16 ). z=cis( 3π 16 ).

40.

Find z 3 z 3 when z=3cis( 5π 3 ). z=3cis( 5π 3 ).

For the following exercises, evaluate each root.

41.

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

42.

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

43.

Evaluate the cube root of z z when z=32cis( 2π 3 ). z=32cis( 2π 3 ).

44.

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

45.

Evaluate the cube root of z z when z=8cis( 7π 4 ). z=8cis( 7π 4 ).

Graphical

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

46.

2+4i 2+4i

47.

33i 33i

48.

54i 54i

49.

15i 15i

50.

3+2i 3+2i

51.

2i 2i

52.

4 4

53.

62i 62i

54.

2+i 2+i

55.

14i 14i

Technology

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

56.

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

57.

Use the rectangular to polar feature on the graphing calculator to change 32i 32i to polar form.

58.

Use the rectangular to polar feature on the graphing calculator to change 38i 38i to polar form.

59.

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

60.

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

61.

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

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