Polar Form of Complex Numbers

Transcription

1 OpenStax-CNX module: m Polar Form of Complex Numbers OpenStax College This work is produced by OpenStax-CNX and licensed under the Creative Commons Attribution License 4.0 In this section, you will: Abstract 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 rst 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. 1 Plotting Complex Numbers in the Complex Plane Plotting a complex number a + bi is similar to plotting a real number, except that the horizontal axis represents the real part of the number, a, and the vertical axis represents the imaginary part of the number, bi. how to feature: Given a complex number 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..plot the point in the complex plane by moving a units in the horizontal direction and b units in the vertical direction. Example 1 Plotting a Complex Number in the Complex Plane Plot the complex number i in the complex plane. Version 1.: Jul 8, :00 am

2 OpenStax-CNX module: m49408 Solution From the origin, move two units in the positive horizontal direction and three units in the negative vertical direction. See Figure 1. Figure 1 try it feature: Exercise (Solution on p. 18. Plot the point 1 + 5i in the complex plane. Finding the Absolute Value of a Complex Number The rst step toward working with a complex number in polar form is to nd the absolute value. The absolute value of a complex number is the same as its magnitude, or z. It measures the distance from the origin to a point in the plane. For example, the graph of z = + 4i, in Figure, shows z.

3 OpenStax-CNX module: m49408 Figure a general note label: as Given z = x + yi, a complex number, the absolute value of z is dened It is the distance from the origin to the point (x, y. z = x + y (1 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. Example Finding the Absolute Value of a Complex Number with a Radical Find the absolute value of z = 5 i.

4 OpenStax-CNX module: m Solution Using the formula, we have See Figure. z = x + y ( 5 z = + ( 1 z = z = 6 ( Figure try it feature: Exercise 4 (Solution on p. 18. Find the absolute value of the complex number z = 1 5i.

5 OpenStax-CNX module: m Example Finding the Absolute Value of a Complex Number Given z = 4i, nd z. Solution Using the formula, we have The absolute value z is 5. See Figure 4. z = x + y z = ( + ( 4 z = z = 5 z = 5 ( Figure 4

6 OpenStax-CNX module: m try it feature: Exercise 6 (Solution on p. 18. Given z = 1 7i, nd z. Writing Complex Numbers in Polar Form The polar form of a complex number expresses a number in terms of an angle θ and its distance from the origin r. Given a complex number in rectangular form expressed as z = x + yi, we use the same conversion formulas as we do to write the number in trigonometric form: We review these relationships in Figure 5. x = rcos θ y = rsin θ r = (4 x + y Figure 5

7 OpenStax-CNX module: m We use the term modulus to represent the absolute value of a complex number, or the distance from the origin to the point (x, y. The modulus, then, is the same as r, the radius in polar form. We use θ to indicate the angle of direction (just as with polar coordinates. Substituting, we have z = x + yi z = rcos θ + (rsin θ i z = r (cos θ + isin θ (5 Writing a complex number in polar form involves the following conver- a general note label: sion formulas: Making a direct substitution, we have x = rcos θ y = rsin θ r = x + y (6 z = x + yi z = (rcos θ + i (rsin θ z = r (cos θ + isin θ (7 where r is the modulus and θ is the argument. We often use the abbreviation rcis θ to represent r (cos θ + isin θ. Example 4 Expressing a Complex Number Using Polar Coordinates Express the complex number 4i using polar coordinates. Solution On the complex plane, the number z = 4i is the same as z = 0 + 4i. Writing it in polar form, we have to calculate r rst. r = x + y r = r = 16 r = 4 Next, we look at x. If x = rcos θ, and x = 0, then θ = π. In polar coordinates, the complex number z = 0 + 4i can be written as z = 4 ( cos ( ( π + isin π ( or 4cis π. See Figure 6. (8

8 OpenStax-CNX module: m Figure 6 try it feature: Exercise 8 (Solution on p. 18. Express z = i as r cis θ in polar form. Example 5 Finding the Polar Form of a Complex Number Find the polar form of 4 + 4i. Solution First, nd the value of r. r = x + y r = ( 4 + (4 r = r = 4 (9

9 OpenStax-CNX module: m Find the angle θ using the formula: cos θ = x r cos θ = 4 4 cos θ = 1 θ = cos 1 ( 1 = π 4 (10 Thus, the solution is 4 cis ( π 4. try it feature: Exercise 10 (Solution on p. 18. Write z = + i in polar form. 4 Converting a Complex Number from Polar to Rectangular Form Converting a complex number from polar form to rectangular form is a matter of evaluating what is given and using the distributive property. In other words, given z = r (cos θ + isin θ, rst evaluate the trigonometric functions cos θ and sin θ. Then, multiply through by r. Example 6 Converting from Polar to Rectangular Form Convert the polar form of the given complex number to rectangular form: ( ( π ( π z = 1 cos + isin 6 6 Solution We begin by evaluating the trigonometric expressions. ( π cos = 6 After substitution, the complex number is We apply the distributive property: z = 1 ( π and sin = 1 6 ( + 1 i z = 1 ( + 1 i = (1 + (1 1 i = 6 + 6i The rectangular form of the given point in complex form is 6 + 6i. (11 (1 (1 (14

10 OpenStax-CNX module: m Example 7 Finding the Rectangular Form of a Complex Number Find the rectangular form of the complex number given r = 1 and tan θ = 5 1. Solution If tan θ = 5 1, and tan θ = y x, we rst determine r = x + y = = 1. We then nd cos θ = x r and sin θ = y r. z = 1 (cos θ + isin θ = 1 ( i = 1 + 5i The rectangular form of the given number in complex form is 1 + 5i. (15 try it feature: Exercise 1 (Solution on p. 18. Convert the complex number to rectangular form: ( z = 4 cos 11π 6 + isin11π 6 (16 5 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 ( 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. a general note label: If z 1 = r 1 (cos θ 1 + isin θ 1 and z = r (cos θ + isin θ,then the product of these numbers is given as: z 1 z = r 1 r [cos (θ 1 + θ + isin (θ 1 + θ ] z 1 z = r 1 r cis (θ 1 + θ Notice that the product calls for multiplying the moduli and adding the angles. (17 Example 8 Finding the Product of Two Complex Numbers in Polar Form Find the product of z 1 z, given z 1 = 4 (cos (80 + isin (80 and z = (cos (145 + isin (145.

11 OpenStax-CNX module: m Solution Follow the formula z 1 z = 4 [cos ( isin ( ] z 1 z = 8 [cos (5 + isin (5 ] z 1 z = 8 [ cos ( ( 5π 4 + isin 5π ] [ 4 z 1 z = 8 ( + i ] z 1 z = 4 4i (18 6 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 dierence of the two arguments. If z 1 = r 1 (cos θ 1 + isin θ 1 and z = r (cos θ + isin θ,then the quo- a general note label: tient of these numbers is z 1 z = r1 z 1 z r [cos (θ 1 θ + isin (θ 1 θ ], z 0 = r1 r cis (θ 1 θ, z 0 Notice that the moduli are divided, and the angles are subtracted. (19 how to feature: Given two complex numbers in polar form, nd the quotient. 1.Divide r1 r..find θ 1 θ..substitute the results into the formula: z = r (cos θ + isin θ. Replace r with r1 r, and replace θ with θ 1 θ. 4.Calculate the new trigonometric expressions and multiply through by r. Example 9 Finding the Quotient of Two Complex Numbers Find the quotient of z 1 = (cos (1 + isin (1 and z = 4 (cos ( + isin (. Solution Using the formula, we have z 1 z = 4 [cos (1 + isin (1 ] z 1 z = 1 [cos (180 + isin (180 ] z 1 [ 1 + 0i] z = 1 z 1 z = 1 + 0i z 1 z = 1 (0

12 OpenStax-CNX module: m try it feature: Exercise 16 (Solution on p. 18. Find the product and the quotient of z 1 = (cos (150 + isin (150 and z = (cos (0 + isin (0. 7 Finding Powers of Complex Numbers in Polar Form Finding powers of complex numbers is greatly simplied using De Moivre's Theorem. It states that, for a positive integer n, z n is found by raising the modulus to the nth power and multiplying the argument by n. It is the standard method used in modern mathematics. a general note label: If z = r (cos θ + isin θ is a complex number, then where n is a positive integer. z n = r n [cos (nθ + isin (nθ] z n = r n cis (nθ (1 Example 10 Evaluating an Expression Using De Moivre's Theorem Evaluate the expression (1 + i 5 using De Moivre's Theorem. Solution Since De Moivre's Theorem applies to complex numbers written in polar form, we must rst write (1 + i in polar form. Let us nd r. r = x + y r = (1 + (1 r = Then we nd θ. Using the formula tan θ = y x gives ( tan θ = 1 1 tan θ = 1 ( θ = π 4 Use De Moivre's Theorem to evaluate the expression. (a + bi n = r n [cos (nθ + isin (nθ] (1 + i 5 = ( 5 [ ( ( ] cos 5 π 4 + isin 5 π 4 (1 + i 5 = 4 [ cos ( ( 5π 4 + isin 5π ] 4 (1 + i 5 = 4 [ ( + i ] (1 + i 5 = 4 4i (4

13 OpenStax-CNX module: m Finding Roots of Complex Numbers in Polar Form To nd the nth root of a complex number in polar form, we use the 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 nding nth roots of complex numbers in polar form. a general note label: given as where k = 0, 1,,,..., n 1. We add kπ n To nd the nth root of a complex number in polar form, use the formula [ ( z 1 1 θ n = r n cos n + kπ ( θ + isin n n + kπ ] n to θ n Example 11 Finding the nth Root of a Complex Number Evaluate the cube roots of z = 8 ( cos ( ( π + isin π. Solution We have [ ( z 1 = 8 1 π ( cos + kπ π + isin z 1 = [ cos ( π 9 + ( kπ + isin π 9 + kπ in order to obtain the periodic roots. + kπ There will be three roots: k = 0, 1,. When k = 0, we have ( ( z 1 π π = (cos + isin 9 9 When k = 1, we have z 1 z 1 When k =, we have z 1 z 1 ] (5 ] (6 = [ cos ( π 9 + ( 6π 9 + isin π 9 + ] 6π 9 Add (1π to each angle. = ( cos ( ( 8π 9 + isin 8π (8 9 = [ cos ( π = ( cos ( 14π π 9 (7 ( + isin π 9 + ] 1π 9 Add (π to each angle. (9 + isin ( 14π 9 Remember to nd the common denominator to simplify fractions in situations like this one. For k = 1, the angle simplication is π + (1π = π = π 9 + 6π 9 = 8π 9 ( 1 + (1π ( (0

14 OpenStax-CNX module: m try it feature: Exercise 19 (Solution on p. 18. Find the four fourth roots of 16 (cos (10 + isin (10. media feature label: Access these online resources for additional instruction and practice with polar forms of complex numbers. The Product and Quotient of Complex Numbers in Trigonometric Form 1 De Moivre's Theorem 9 Key Concepts Complex numbers in the form a + bi are plotted in the complex plane similar to the way rectangular coordinates are plotted in the rectangular plane. Label the x-axis as the real axis and the y-axis as the imaginary axis. See Example 1. The absolute value of a complex number is the same as its magnitude. It is the distance from the origin to the point: z = a + b. See Example and Example. To write complex numbers in polar form, we use the formulas x = rcos θ, y = rsin θ, and r = x + y. Then, z = r (cos θ + isin θ. See Example 4 and Example 5. To convert from polar form to rectangular form, rst evaluate the trigonometric functions. Then, multiply through by r. See Example 6 and Example 7. To nd the product of two complex numbers, multiply the two moduli and add the two angles. Evaluate the trigonometric functions, and multiply using the distributive property. See Example 8. To nd the quotient of two complex numbers in polar form, nd the quotient of the two moduli and the dierence of the two angles. See Example 9. To nd the power of a complex number z n, raise r to the power n,and multiply θ by n. See Example 10. Finding the roots of a complex number is the same as raising a complex number to a power, but using a rational exponent. See Example Section Exercises 10.1 Verbal Exercise 0 (Solution on p. 18. A complex number is a + bi. Explain each part. Exercise 1 What does the absolute value of a complex number represent? Exercise (Solution on p. 18. How is a complex number converted to polar form? Exercise How do we nd the product of two complex numbers? Exercise 4 (Solution on p. 19. What is De Moivre's Theorem and what is it used for? 1

15 OpenStax-CNX module: m Algebraic For the following exercises, nd the absolute value of the given complex number. Exercise i Exercise 6 (Solution on p i Exercise 7 i Exercise 8 6i (Solution on p. 19. Exercise 9 i Exercise 0 (Solution on p i For the following exercises, write the complex number in polar form. Exercise 1 + i Exercise (Solution on p i Exercise 1 1 i Exercise 4 + i (Solution on p. 19. Exercise 5 i For the following exercises, convert the complex number from polar to rectangular form. Exercise 6 (Solution on p. 19. z = 7cis ( π 6 Exercise 7 z = cis ( π Exercise 8 (Solution on p. 19. z = 4cis ( 7π 6 Exercise 9 z = 7cis (5 Exercise 40 (Solution on p. 19. z = cis (40 Exercise 41 z = cis (100 For the following exercises, nd z 1 z in polar form. Exercise 4 (Solution on p. 19. z 1 = cis (116 ; z = cis (8 Exercise 4 z 1 = cis (05 ; z = cis (118

16 OpenStax-CNX module: m Exercise 44 (Solution on p. 19. z 1 = cis (10 ; z = 1 4 cis (60 Exercise 45 ; z = 5cis ( π 6 z 1 = cis ( π 4 Exercise 46 (Solution on p. 19. z 1 = 5cis ( 5π 8 Exercise 47 z 1 = 4cis ( π ; z = 15cis ( π 1 ; z = cis ( π 4 For the following exercises, nd z1 z in polar form. Exercise 48 (Solution on p. 19. z 1 = 1cis (15 ; z = cis (65 Exercise 49 z 1 = cis (90 ; z = cis (60 Exercise 50 (Solution on p. 19. z 1 = 15cis (10 ; z = cis (40 Exercise 51 ; z = cis ( π 4 z 1 = 6cis ( π Exercise 5 (Solution on p. 19. z 1 = 5 cis (π ; z = cis ( π Exercise 5 z 1 = cis ( π 5 ; z = cis ( π 4 For the following exercises, nd the powers of each complex number in polar form. Exercise 54 (Solution on p. 19. Find z when z = 5cis (45. Exercise 55 Find z 4 when z = cis (70. Exercise 56 (Solution on p. 19. Find z when z = cis (10. Exercise 57 Find z when z = 4cis ( π 4. Exercise 58 (Solution on p. 19. Find z 4 when z = cis ( π 16. Exercise 59 Find z when z = cis ( 5π. For the following exercises, evaluate each root. Exercise 60 (Solution on p. 19. Evaluate the cube root of z when z = 7cis (40. Exercise 61 Evaluate the square root of z when z = 16cis (100. Exercise 6 (Solution on p. 19. Evaluate the cube root of z when z = cis ( π. Exercise 6 Evaluate the square root of z when z = cis (π. Exercise 64 (Solution on p. 19. Evaluate the cube root of z when z = 8cis ( 7π 4.

17 OpenStax-CNX module: m Graphical For the following exercises, plot the complex number in the complex plane. Exercise i Exercise 66 (Solution on p. 19. i Exercise i Exercise 68 (Solution on p i Exercise 69 + i Exercise 70 (Solution on p. 1. i Exercise 71 4 Exercise 7 (Solution on p.. 6 i Exercise 7 + i Exercise 74 (Solution on p.. 1 4i 10.4 Technology For the following exercises, nd all answers rounded to the nearest hundredth. Exercise 75 Use the rectangular to polar feature on the graphing calculator to change 5 + 5i to polar form. Exercise 76 (Solution on p. 4. Use the rectangular to polar feature on the graphing calculator to change i to polar form. Exercise 77 Use the rectangular to polar feature on the graphing calculator to change 8i to polar form. Exercise 78 (Solution on p. 4. Use the polar to rectangular feature on the graphing calculator to change 4cis (10 to rectangular form. Exercise 79 Use the polar to rectangular feature on the graphing calculator to change cis (45 to rectangular form. Exercise 80 (Solution on p. 4. Use the polar to rectangular feature on the graphing calculator to change 5cis (10 to rectangular form.

18 OpenStax-CNX module: m Solutions to Exercises in this Module Solution to Exercise (p. Solution to Exercise (p. 4 1 Solution to Exercise (p. 6 z = 50 = 5 Solution to Exercise (p. 8 z = ( cos ( ( π + isin π Solution to Exercise (p. 9 z = ( cos ( ( π 6 + isin π 6 Solution to Exercise (p. 10 z = i Solution to Exercise (p. 1 z 1 z = 4 ; z1 z = + i Solution to Exercise (p. 14 z 0 = (cos (0 + isin (0 z 1 = (cos (10 + isin (10 z = (cos (10 + isin (10 z = (cos (00 + isin (00 Solution to Exercise (p. 14 a is the real part, b is the imaginary part, and i = 1

19 OpenStax-CNX module: m Solution to Exercise (p. 14 Polar form converts the real and imaginary part of the complex number in polar form using x = rcosθ and y = rsinθ. Solution to Exercise (p. 14 z n = r n (cos (nθ + isin (nθ It is used to simplify polar form when a number has been raised to a power. Solution to Exercise (p Solution to Exercise (p Solution to Exercise (p Solution to Exercise (p cis (.4 Solution to Exercise (p. 15 cis ( π 6 Solution to Exercise (p i 7 Solution to Exercise (p. 15 i Solution to Exercise (p i Solution to Exercise (p cis (198 Solution to Exercise (p cis (180 Solution to Exercise (p cis ( 17π 4 Solution to Exercise (p. 16 7cis (70 Solution to Exercise (p. 16 5cis (80 Solution to Exercise (p. 16 5cis ( π Solution to Exercise (p cis (15 Solution to Exercise (p. 16 9cis (40 Solution to Exercise (p. 16 cis ( π 4 Solution to Exercise (p. 16 cis (80, cis (00, cis (0 Solution to Exercise (p. 16 4cis ( π 9, 4cis ( 8π 9, 4cis ( 14π 9 Solution to Exercise (p. 16 cis ( ( 7π 8, cis 15π 8 Solution to Exercise (p. 17

20 OpenStax-CNX module: m Figure 7 Solution to Exercise (p. 17

21 OpenStax-CNX module: m Figure 8 Solution to Exercise (p. 17

22 OpenStax-CNX module: m49408 Figure 9 Solution to Exercise (p. 17

23 OpenStax-CNX module: m49408 Figure 10 Solution to Exercise (p. 17

24 OpenStax-CNX module: m Figure 11 Solution to Exercise (p e 0.59i Solution to Exercise (p i Solution to Exercise (p i Glossary Denition 1: argument the angle associated with a complex number; the angle between the line from the origin to the point and the positive real axis

25 OpenStax-CNX module: m Denition : De Moivre's Theorem formula used to nd the nth power or nth roots of a complex number; states that, for a positive integer n, z n is found by raising the modulus to the nth power and multiplying the angles by n Denition : modulus the absolute value of a complex number, or the distance from the origin to the point (x, y ; also called the amplitude Denition 4: polar form of a complex number a complex number expressed in terms of an angle θ and its distance from the origin r; can be found by using conversion formulas x = rcos θ, y = rsin θ, and r = x + y