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P-BLTZMC06_643-726-hr 21-11-2008 12:56 Page 686

686 Chapter 6 Additional Topics in

In Exercises 18–20, convert each rectangular equation to a polar In Exercises 26–27, test for with respect to u p equation that expresses r in terms of . a. the polar axis. b. the u = . c. the pole. 18.5x - y = 7 19. y =-7 2 20. x + 1 2 + y2 = 1 26. r = 1 - 4 cos u 27. r2 = 4 cos 2u 1 2 In Exercises 21–25, convert each polar equation to a rectangular In Exercises 28–32, graph each polar equation. Be sure to test for equation. Then use your knowledge of the rectangular equation to symmetry. graph the polar equation in a polar . =- = - p 28. r 4 sin u 29. r 2 2 cos u 21.r = 6 22. u = 3 30. r = 2 - 4 cos u 31. r = 2 sin 3u =- u =- u 23.r 3 csc 24. r 10 cos 32. r2 = 16 sin 2u 25. r = 4 sin u sec2 u

S e c t i o n 6.5 Complex in Polar Form; DeMoivre’s Theorem Objectives ᕡ Plot complex numbers in the A magnification of the complex . ᕢ Find the of a complex . ne of the new frontiers of ᕣ Write complex numbers in Omathematics suggests that there is polar form. an underlying order in things that appear to be random, such as ᕤ Convert a the hiss and crackle of from polar to rectangular background noises as you form. tune a radio. Irregularities ᕥ Find products of complex in the heartbeat, some of numbers in polar form. them severe enough to ᕦ Find quotients of complex cause a heart attack, or numbers in polar form. irregularities in our ᕧ Find powers of complex sleeping patterns, such as numbers in polar form. insomnia, are examples of chaotic behavior. Chaos in the mathematical ᕨ Find roots of complex sense does not a complete lack of numbers in polar form. form or arrangement. In , chaos is used to describe something that appears to be random but is not actually random. The patterns of chaos appear in images like the one shown above, called the Mandelbrot set. Magnified portions of this image yield repetitions of the original structure, as well as new and unexpected patterns. The Mandelbrot set transforms the hidden structure of chaotic events into a source of wonder and inspiration. The Mandelbrot set is made possible by opening up graphing to include complex numbers in the form a + bi, where i = 2-1. In this section, you will learn how to graph complex numbers and write them in terms of .

ᕡ Plot complex numbers The in the complex plane. We know that a can be represented as a on a number line. By contrast, a complex number z = a + bi is represented as a point a, b in a coordi- 1 2 nate plane, as shown in Figure 6.38 at the top of the next page. The horizontal axis of the coordinate plane is called the real axis. The vertical axis is called the imaginary axis. The coordinate system is called the complex plane. Every complex number P-BLTZMC06_643-726-hr 21-11-2008 12:56 Page 687

Section 6.5 Complex Numbers in Polar Form; DeMoivre’s Theorem 687

corresponds to a point in the complex plane and every point in the complex plane corresponds to a complex number.When we represent a complex number as a point in the complex plane, we say that we are plotting the complex number.

Imaginary axis

b

z = a + bi

Real axis O a Figure 6.38 Plotting z = a + bi in the complex plane

EXAMPLE 1 Plotting Complex Numbers

Plot each complex number in the complex plane: a.z = 3 + 4i b.z =-1 - 2i c.z =-3 d. z =-4i.

Solution See Figure 6.39. Imaginary a. We plot the complex number z = 3 + 4i the same way we plot (3, 4) in the axis z = 3 + 4i rectangular coordinate system.We move three units to the right on the real axis 4 and four units up parallel to the imaginary axis. 3 b. The complex number z =-1 - 2i corresponds to the point -1, -2 in the z = −3 2 1 2 1 rectangular coordinate system. Plot the complex number by moving one unit to Real the left on the real axis and two units down parallel to the imaginary axis. −4 −3 −2 1234 axis c. Because z =-3 =-3 + 0i, this complex number corresponds to the point -3, 0 . We plot -3 by moving three units to the left on the real axis. −3 1 2 z i z = −4i = −1 − 2 −4 d. Because z =-4i = 0 - 4i, this number corresponds to the point 0, -4 . 1 2 We plot the complex number by moving four units down on the Figure 6.39 Plotting complex imaginary axis. numbers

Check Point 1 Plot each complex number in the complex plane: a. z = 2 + 3i b. z =-3 - 5i c. z =-4 d. z =-i.

ᕢ Find the absolute value Recall that the absolute value of a real number is its from 0 on a of a complex number. number line. The absolute value of the complex number z = a + bi, denoted by ƒ z ƒ , is the distance from the to the point z in the complex plane. Figure 6.40 Imaginary illustrates that we can use the to represent ƒ z ƒ in terms of axis a and b: ƒ z ƒ = 3a2 + b2 . |z| ϭ a2 ϩ b2

z = a + bi The Absolute Value of a Complex Number b The absolute value of the complex number a + bi is Real ƒ z ƒ = ƒ a + bi ƒ = 3a2 + b2 a axis .

Figure 6.40 EXAMPLE 2 Finding the Absolute Value of a Complex Number

Determine the absolute value of each of the following complex numbers: a.z = 3 + 4i b. z =-1 - 2i. P-BLTZMC06_643-726-hr 21-11-2008 12:56 Page 688

688 Chapter 6 Additional Topics in Trigonometry

Solution Imaginary a. The absolute value of z = 3 + 4i is found using a = 3 and b = 4. axis z = 3 + 4i ƒ z ƒ = 332 + 42 = 29 + 16 = 225 = 5 Use z = 3a2 + b2 with 4 = = 3 a 3 and b 4. 2 Thus, the distance from the origin to the point z = 3 + 4i, shown in quadrant I 1 Real in Figure 6.41, is five units. axis −4 −3 −2 −1−1 1234 b. The absolute value of z =-1 - 2i is found using a =-1 and b =-2.

2 2 2 2 −3 ƒ z ƒ = 4 -1 + -2 = 21 + 4 = 25 Use z = 3a + b with z 1 2i 1 2 1 2 = − − −4 a =-1 and b =-2.

Figure 6.41 Thus, the distance from the origin to the point z =-1 - 2i, shown in quadrant III in Figure 6.41, is 25 units.

Check Point 2 Determine the absolute value of each of the following complex numbers: a.z = 5 + 12i b. 2 - 3i.

ᕣ Write complex numbers Polar Form of a Complex Number in polar form. A complex number in the form z = a + bi is said to be in rectangular form. Suppose that its absolute value is r. In Figure 6.42, we let u be an in standard position Imaginary whose terminal side passes through the point a, b . From the figure, we see that axis 1 2 z = a + bi r = 3a2 + b2 . (a, b) Likewise, according to the definitions of the trigonometric functions, r b a b b cos u = sin u = tan u = . u r r a Real O a axis a = r cos u b = r sin u = + Figure 6.42 By substituting the expressions for and ba into z a bi, we write the complex number in terms of trigonometric functions. z=a+bi=r cos u+(r sin u)i=r(cos u+i sin u)

a = r cos u and b = r sin u. Factor out r from each of the two previous terms.

The expression z = r cos u + i sin u is called the polar form of a complex number. 1 2 Polar Form of a Complex Number The complex number z = a + bi is written in polar form as z = r cos u + i sin u , 1 2 b where a = r cos u, b = r sin u, r = 3a2 + b2 , and tan u = . The value of r is a called the modulus (plural: moduli) of the complex number z and the angle u is called the argument of the complex number z with 0 … u 6 2p.

EXAMPLE 3 Writing a Complex Number in Polar Form

Plot z =-2 - 2i in the complex plane. Then write z in polar form. Solution The complex number z =-2 - 2i is in rectangular form z = a + bi, with a =-2 and b =-2. We plot the number by moving two units to the left on the real axis and two units down parallel to the imaginary axis, as shown in Figure 6.43 on the next page. P-BLTZMC06_643-726-hr 21-11-2008 12:56 Page 689

Section 6.5 Complex Numbers in Polar Form; DeMoivre’s Theorem 689

Imaginary By definition, the polar form of z is r cos u + i sin u . We need to determine 1 2 axis the value for r, the modulus, and the value for u, the argument. Figure 6.43 shows r and u. We use r = 3a2 + b2 with a =-2 and b =-2 to find r.

2 2 2 2 2 # u r = 3a + b = 4 -2 + -2 = 24 + 4 = 28 = 24 2 = 222 Real 1 2 1 2 axis −2 2 b r We use tan u = with a =-2 and b =-2 to find u. a −2 b -2 z = −2 − 2i tan u = = = 1 a -2 Figure 6.43 Plotting z =-2 - 2i and writing the number in polar form p We know that tan = 1. Figure 6.43 shows that the argument,u, satisfying 4 tan u = 1 lies in quadrant III. Thus,

p 4p p 5p u = p + = + = . 4 4 4 4

5p We use r = 222 and u = to write the polar form. The polar form of 4 z =-2 - 2i is

5p 5p z = r cos u + i sin u = 222 cos + i sin . 1 2 a 4 4 b

Check Point 3 Plot z =-1 - i23 in the complex plane. Then write z in polar form. Express the argument in . (We write -1 - i23, rather than -1 - 23 i, which could easily be confused with -1 - 23i.)

ᕤ Convert a complex number from EXAMPLE 4 Writing a Complex Number in Rectangular Form polar to rectangular form. Write z = 2 cos 60° + i sin 60° in rectangular form. 1 2 Solution The complex number z = 2 cos 60° + i sin 60° is in polar form, with 1 2 r = 2 and u = 60°. We use exact values for cos 60° and sin 60° to write the number in rectangular form.

1 13 2(cos 60Њ+i sin 60Њ)=2 + i =1+i13 a 2 2 b The rectangular form of z=2(cos 60Њ+i sin 60Њ) is Write i before the radical to avoid z=1+i13. confusion with 13i.

Check Point Write z = 4 cos 30° + i sin 30° in rectangular form. 4 1 2

ᕥ Find products of complex Products and Quotients in Polar Form numbers in polar form. We can multiply and divide complex numbers fairly quickly if the numbers are expressed in polar form. P-BLTZMC06_643-726-hr 21-11-2008 12:56 Page 690

690 Chapter 6 Additional Topics in Trigonometry

Product of Two Complex Numbers in Polar Form Let z = r cos u + i sin u and z = r cos u + i sin u be two complex 1 11 1 12 2 21 2 22 numbers in polar form. Their ,z1 z2 , is

z z = r r cos u + u + i sin u + u . 1 2 1 23 1 1 22 1 1 224 To multiply two complex numbers, multiply moduli and add arguments.

To prove this result, we begin by multiplying using the FOIL method. Then we simplify the product using the sum formulas for and cosine.

z1z2=[r1(cos u1+i sin u1)][r2(cos u2+i sin u2)]

=r1r2(cos u1+i sin u1)(cos u2+i sin u2) Rearrange factors.

F O I L

2 =r1r2(cos u1 cos u2+i cos u1 sin u2+i sin u1 cos u2+i sin u1 sin u2) Use the FOIL method. 2 =r1r2[cos u1 cos u2+i(cos u1 sin u2+ sin u1 cos u2)+i sin u1 sin u2] Factor i from the second and third terms. 2 =r1r2[cos u1 cos u2+i(cos u1 sin u2+ sin u1 cos u2)-sin u1 sin u2] i =-1

=r1r2[(cos u1 cos u2-sin u1 sin u2)+i(sin u1 cos u2+cos u1 sin u2)] Rearrange terms.

This is cos (u1 + u2). This is sin (u1 + u2).

=r1r2[cos (u1 +u2)+i sin (u1+ u2)]

This result gives a rule for finding the product of two complex numbers in polar form. The two parts to the rule are shown in the following voice balloons.

z1z2=r1r2[cos (u1+ u2)+i sin (u1+ u2)]

Multiply moduli. Add arguments.

EXAMPLE 5 Finding Products of Complex Numbers in Polar Form

Find the product of the complex numbers. Leave the answer in polar form. z = 4 cos 50° + i sin 50° z = 7 cos 100° + i sin 100° 1 1 2 2 1 2 Solution

z1 z2 = 4 cos 50° + i sin 50° 7 cos 100° + i sin 100° Form the product of the given 3 1 243 1 24 # numbers. = 4 7 cos 50° + 100° + i sin 50° + 100° Multiply moduli and add 1 23 1 2 1 24 arguments. = 28 cos 150° + i sin 150° Simplify. 1 2

Check Point 5 Find the product of the complex numbers. Leave the answer in polar form.

z = 6 cos 40° + i sin 40° z = 5 cos 20° + i sin 20° 1 1 2 2 1 2

ᕦ Find quotients of complex Using algebraic methods for dividing complex numbers and the difference numbers in polar form. formulas for sine and cosine, we can obtain a rule for dividing complex numbers in polar form.The proof of this rule can be found in Appendix A, or you can derive the rule on your own by working Exercise 110 in this section’s exercise set. P-BLTZMC06_643-726-hr 21-11-2008 12:56 Page 691

Section 6.5 Complex Numbers in Polar Form; DeMoivre’s Theorem 691

Quotient of Two Complex Numbers in Polar Form

Let z1 = r1 cos u1 + i sin u1 and z2 = r2 cos u2 + i sin u2 be two complex 1 2 z 1 2 numbers in polar form. Their quotient,1 , is z2

z1 r1 = cos u1 - u2 + i sin u1 - u2 . z2 r2 3 1 2 1 24 To divide two complex numbers, divide moduli and subtract arguments.

EXAMPLE 6 Finding Quotients of Complex Numbers in Polar Form

z Find the quotient 1 of the complex numbers. Leave the answer in polar form. z2 3p 3p p p z = 12 cos + i sin z = 4 cos + i sin 1 a 4 4 b 2 a 4 4 b Solution 3p 3p 12 cos + i sin z1 a 4 4 b = Form the quotient of the given numbers. z p p 2 4 cos + i sin a 4 4 b 12 3p p 3p p = cos - + i sin - Divide moduli and subtract arguments. 4 a 4 4 b a 4 4 b p p 3p p 2p p = 3 cos + i sin Simplify: - = = . a 2 2 b 4 4 4 2 B R z Check Point Find the quotient 1 of the complex numbers. Leave the answer 6 z in polar form. 2 4p 4p p p z = 50 cos + i sin z = 5 cos + i sin 1 a 3 3 b 2 a 3 3 b

ᕧ Find powers of complex Powers of Complex Numbers in Polar Form numbers in polar form. We can use a formula to find powers of complex numbers if the complex numbers are expressed in polar form. This formula can be illustrated by repeatedly multiplying by r cos u + i sin u . 1 2 z = r cos u + i sin u Start with z. # 1 2 z z = r cos u + i sin u r cos u + i sin u Multiply z by z = r cos u + i sin u . 1 2 1 2 1 2 2 2 # z = r cos 2u + i sin 2u Multiply moduli: r r = r2. Add 1 2 arguments: u + u = 2u. 2 # 2 z z = r cos 2u + i sin 2u r cos u + i sin u Multiply z2 by z = r cos u + i sin u . 1 2 1 2 1 2 3 3 # z = r cos 3u + i sin 3u Multiply moduli: r2 r = r3. Add 1 2 arguments: 2u + u = 3u. 3 # 3 z z = r cos 3u + i sin 3u r cos u + i sin u Multiply z3 by z = r cos u + i sin u . 1 2 1 2 1 2 4 4 # z = r cos 4u + i sin 4u Multiply moduli: r3 r = r4. Add 1 2 arguments: 3u + u = 4u. Do you see a pattern forming? If n is a positive , it appears that zn is obtained by raising the modulus to the nth power and multiplying the argument by n. The formula for the nth power of a complex number is known as DeMoivre’s Theorem in honor of the French mathematician Abraham DeMoivre (1667–1754). P-BLTZMC06_643-726-hr 21-11-2008 12:56 Page 692

692 Chapter 6 Additional Topics in Trigonometry

DeMoivre’s Theorem Let z = r cos u + i sin u be a complex number in polar form. If n is a positive 1 2 integer, then z to the nth power,zn, is zn = r cos u + i sin u n = rn cos nu + i sin nu . 3 1 24 1 2

EXAMPLE 7 Finding the Power of a Complex Number

Find 2 cos 20° + i sin 20° 6. Write the answer in rectangular form, a + bi. 3 1 24 Solution We begin by applying DeMoivre’s Theorem. 2 cos 20° + i sin 20° 6 3 1 24 6 # # = 2 cos 6 20° + i sin 6 20° Raise the modulus to the 6th power 3 1 2 1 24 and multiply the argument by 6. = 64 cos 120° + i sin 120° Simplify. 1 2 1 23 = 64 - + i Write the answer in rectangular form. 2 2

=-32 + 32i23 Multiply and express the answer in a + bi form. ¢ ≤ Check Point Find 2 cos 30° + i sin 30° 5. Write the answer in rectangular 7 3 1 24 form.

EXAMPLE 8 Finding the Power of a Complex Number

Find 1 + i 8 using DeMoivre’s Theorem. Write the answer in rectangular form, 1 2 a + bi. Imaginary Solution DeMoivre’s Theorem applies to complex numbers in polar form. axis Thus, we must first write 1 + i in r cos u + i sin u form. Then we can use 1 2 DeMoivre’s Theorem. The complex number 1 + i is plotted in Figure 6.44. From the figure, we obtain values for r and u. 2 z = 1 + i = 3 2 + 2 = 3 2 + 2 = 2 r r a b 1 1 2 u Real b 1 p −2 2 axis tan u = = = 1 and u = because u lies in quadrant I. a 1 4

−2 Using these values, p p 1 + i = r cos u + i sin u = 22 cos + i sin . Figure 6.44 Plotting 1 + i and 1 2 a 4 4 b writing the number in polar form Now we use DeMoivre’s Theorem to raise 1 + i to the 8th power. 1 + i 8 1 2 p p 8 = 22 cos + i sin Work with the polar form of 1 + i. a 4 4 b p p Apply DeMoivre’s Theorem. Raise the = 22 8 cos 8 # + i sin 8 # A B a 4 b a 4 b modulus to the 8th power and B R multiply the argument by 8. = 16 cos 2p + i sin 2p Simplify: 22 8 = 21/2 8 = 24 = 16. 1 2 A B 1 2 = 16 1 +B0i R cos 2p = 1 and sin 2p = 0. 1 2 = 16 or 16 + 0i Simplify. P-BLTZMC06_643-726-hr 21-11-2008 12:56 Page 693

Section 6.5 Complex Numbers in Polar Form; DeMoivre’s Theorem 693

Check Point Find 1 + i 4 using DeMoivre’s Theorem. Write the answer in 8 1 2 rectangular form.

ᕨ Find roots of complex Roots of Complex Numbers in Polar Form numbers in polar form. In Example 7, we showed that 2 cos 20° + i sin 20° 6 = 64 cos 120° + i sin 120° . 3 1 24 1 2 We say that 2 cos 20° + i sin 20° is a complex sixth root of 64 cos 120° + i sin 120° . 1 2 1 2 It is one of six distinct complex sixth roots of 64 cos 120° + i sin 120° . 1 2 In general, if a complex number z satisfies the equation zn = w, we say that z is a complex of w . It is one of n distinct nth complex roots that can be found using the following theorem:

DeMoivre’s Theorem for Finding Complex Roots Let w = r cos u + i sin u be a complex number in polar form. If w Z 0, w has 1 2 n distinct complex nth roots given by the formula

u + 2pk u + 2pk z = 1n r cos + i sin radians k a n b a n b 1 2

n u + 360°k u + 360°k or z = 1r cos + i sin degrees , k B a n b a n R b 1 2 where k = 0, 1, 2, Á , n - 1. B R

By raising the or formula for zk to the nth power, you can use n DeMoivre’s Theorem for powers to show that zk = w. Thus, each zk is a complex nth root of w. DeMoivre’s Theorem for finding complex roots states that every complex number has two distinct complex roots, three distinct complex cube roots, four distinct complex fourth roots, and so on. Each root has the same modulus, 1n r. 2p 360° Successive roots have arguments that differ by the same amount, or . This n n that if you plot all the complex roots of any number, they will be equally spaced on a centered at the origin, with 1n r.

EXAMPLE 9 Finding the Roots of a Complex Number

Find all the complex fourth roots of 16 cos 120° + i sin 120° . Write roots in polar 1 2 form, with u in degrees. Solution There are exactly four fourth roots of the given complex number. From DeMoivre’s Theorem for finding complex roots, the fourth roots of 16 cos 120° + i sin 120° are 1 2

4 120Њ+360Њk 120Њ+360Њk z =16 cos +i sin , k=0, 1, 2, 3. k a 4 b a 4 b

n u k u k Use z = r cos + 360° + i sin + 360° . BRk QRn QRn In 16(cos 120° + i sin 120°), r = 16 and u = 120°. Because we are finding fourth roots, n = 4. P-BLTZMC06_643-726-hr 21-11-2008 12:56 Page 694

694 Chapter 6 Additional Topics in Trigonometry

The four fourth roots are found by substituting 0, 1, 2, and 3 for k in the expression for zk , repeated in the margin. Thus, the four complex fourth roots are as follows:

4 120 +360 k 120 +360 k + # + # z =16 BRa   b+i a   b 120° 360° 0 120° 360° 0 k q cos sin z = 24 16 cos + i sin 4 4 0 a 4 b a 4 b The formula for the four fourth roots of 16 cos 120° + i sin 120° , k = 0, 1, 2, 3 1 2 120° 120° (repeated) = 24 16 cos + i sin = 2 cos 30° + i sin 30° aB 4 4 b 1 2R

120° + 360° # 1 120° + 360° # 1 z = 24 16 cos + i sin 1 a 4 b a 4 b

480° 480° = 24 16 cos + i sin = 2 cos 120° + i sin 120° aB 4 4 b 1 R2

120° + 360° # 2 120° + 360° # 2 z = 24 16 cos + i sin 2 a 4 b a 4 b

840° 840° = 24 16 cos + i sin = 2 cos 210° + i sin 210° aB 4 4 b 1 R2

120° + 360° # 3 120° + 360° # 3 z = 24 16 cos + i sin 3 a 4 b a 4 b

1200° 1200° = 24 16 cos + i sin = 2 cos 300° + i sin 300° . aB 4 4 b 1 R 2

In Figure 6.45, we have plotted each of the four fourth roots of 16 cos 120° + i sin 120° . Notice that they are equally spaced at 90° intervals on a 1 2 circle with radius 2.

Imaginary axis

2(cos 120° + i sin 120°) 2(cos 30° + i sin 30°) 3

Real −3 3 axis

Figure 6.45 Plotting the four fourth 2(cos 210° + i sin 210°) −3 2(cos 300° + i sin 300°) roots of 16 cos 120° + i sin 120° 1 2

Check Point Find all the complex fourth roots of 16 cos 60° + i sin 60° . 9 1 2 Write roots in polar form, with u in degrees.

EXAMPLE 10 Finding the Roots of a Complex Number

Find all the cube roots of 8. Write roots in rectangular form.

Solution DeMoivre’s Theorem for roots applies to complex numbers in polar form. Thus, we will first write 8, or 8 + 0i, in polar form. We express u in radians, although degrees could also be used. 8 = r cos u + i sin u = 8 cos 0 + i sin 0 1 2 1 2 P-BLTZMC06_643-726-hr 21-11-2008 12:56 Page 695

Section 6.5 Complex Numbers in Polar Form; DeMoivre’s Theorem 695

There are exactly three cube roots of 8. From DeMoivre’s Theorem for finding complex roots, the cube roots of 8 are

3 0+2pk 0+2pk z =8 cos +i sin , k=0, 1, 2. k a 3 b a 3 b

n u pk u pk Use z = r cos + 2 + i sin + 2 . BRk QRn QRn In 8(cos 0 + i sin 0), r = 8 and u = 0. Because we are finding cube roots, n = 3.

The three cube roots of 8 are found by substituting 0, 1, and 2 for k in the expression for zk above the voice balloon. Thus, the three cube roots of 8 are

Imaginary 0 + 2p # 0 0 + 2p # 0 axis z = 23 8 cos + i sin 0 a 3 b a 3 b −1 + i3 3 = 2 cos 0 + i sin 0 = 2 1 + i # 0 = 2 1 2 1 2 2 B R + p # + p # Real 3 0 2 1 0 2 1 z1 = 28 cos + i sin −3 3 axis a 3 b a 3 b

2p 2p 1 # 23 = 2 cos + i sin = 2 - + i =-1 + i23 −3 a b −1 − i3 B 3 3 2 2 R

Figure 6.46 The three cube roots 0 + 2p # 2 0 + 2p # 2 z = 23 8 cos + i sin of 8 are equally spaced at intervals of 2 a 3 b a 3 b 2p ¢ ≤ on a circle with radius 2. 3 4p 4p 1 # 23 = 2 cos + i sin = 2 - + i - =-1 - i23. a B 3 3 b 2 2R Discovery Use DeMoivre’s Theorem to cube The three cube roots of 8 are plotted in Figure 6.46. -1 + i23 or -1 - i23 and ¢ ¢ ≤≤ obtain 8. Check Point 10 Find all the cube roots of 27. Write roots in rectangular form.

The Mandelbrot Set

Imaginary axis The set of all complex numbers for which the z, z2 + z, z2 + z 2 + z, z2 + z 2 + z 2 + z, Á 1 2 31 2 4 is bounded is called the Mandelbrot set. Plotting these complex numbers in the complex plane results in a graph that is “buglike” in shape, shown in Figure 6.47. Colors can be added to the boundary of the graph. At the boundary, color choices Real axis depend on how quickly the numbers in the boundary approach infinity when substituted into the sequence shown. The magnified boundary is shown in the introduction to this section. It includes the original buglike structure, as well as new and interesting patterns. With each level of magnification, repetition and unpre- dictable formations interact to create what has been called the most complicated mathematical object ever known. Figure 6.47 P-BLTZMC06_643-726-hr 21-11-2008 12:56 Page 696

696 Chapter 6 Additional Topics in Trigonometry Exercise Set 6.5

= + = + Practice Exercises 43. z1 1 i 44. z1 1 i z =-1 + i z = 2 + 2i In Exercises 1–10, plot each complex number and find its 2 2 absolute value. z1 In Exercises 45–52, find the quotient of the complex numbers. 1.z = 4i 2. z = 3i z2 Leave answers in polar form. In Exercises 49–50, express the 3.z = 3 4. z = 4 argument as an angle between 0° and 360°. 5.z = 3 + 2i 6. z = 2 + 5i 45. z = 20 cos 75° + i sin 75° 1 1 2 7.z = 3 - i 8. z = 4 - i z 2 = 4 cos 25° + i sin 25° =- + =- - 1 2 9.z 3 4i 10. z 3 4i 46. z = 50 cos 80° + i sin 80° 1 1 2 In Exercises 11–26, plot each complex number. Then write the z 2 = 10 cos 20° + i sin 20° complex number in polar form. You may express the argument in 1 2 p p degrees or radians. 47. z = 3 cos + i sin 1 a 5 5 b 11.2 + 2i 12. 1 + i23 p p = + - - - z 2 4 cos i sin 13.1 i 14. 2 2i a 10 10 b 15.-4i 16. -3i 5p 5p 48. z1 = 3 cos + i sin 17.223 - 2i 18. -2 + 2i23 a 18 18 b - - p p 19.3 20. 4 z = 10 cos + i sin 2 a 16 16 b 21.-322 - 3i23 22. 322 - 3i22 49. z1 = cos 80° + i sin 80° 23.-3 + 4i 24. -2 + 3i z 2 = cos 200° + i sin 200° 25.2 - i23 26. 1 - i25 50. z1 = cos 70° + i sin 70° In Exercises 27–36, write each complex number in rectangular z 2 = cos 230° + i sin 230° form. If necessary, round to the nearest tenth. 51. z1 = 2 + 2i 52. z1 = 2 - 2i 27.6 cos 30° + i sin 30° 28. 12 cos 60° + i sin 60° 1 2 1 2 z 2 = 1 + i z2 = 1 - i 29.4 cos 240° + i sin 240° 30. 10 cos 210° + i sin 210° 1 2 1 2 7p 7p 5p 5p 31.8 cos + i sin 32. 4 cos + i sin In Exercises 53–64, use DeMoivre’s Theorem to find the indicated a 4 4 b a 6 6 b power of the complex number. Write answers in rectangular form. p p 3p 3p 3 3 33.5 cos + i sin 34. 7 cos + i sin 53.4 cos 15° + i sin 15° 54. 2 cos 10° + i sin 10° a 2 2 b a 2 2 b 3 1 24 3 1 24 55.2 cos 80° + i sin 80° 3 56. 2 cos 40° + i sin 40° 3 35.20 cos 205° + i sin 205° 36. 30 cos 2.3 + i sin 2.3 3 1 24 3 1 24 1 2 1 2 1 p p 6 1 p p 5 57. cos + i sin 58. cos + i sin In Exercises 37–44, find the product of the complex numbers. 2 a 12 12 b 2 a 10 10 b Leave answers in polar form. p p 4 5 + 5 37. z = 6 cos 20° + i sin 20° 59. 22 cos i sin 1 1 2 a 6 6 b = + B R B R z 2 5 cos 50° i sin 50° p p 6 1 2 5 + 5 = + 60. 23 cos i sin 38. z1 4 cos 15° i sin 15° a 18 18 b 1 2 B R = + 5 5 z 2 7 cos 25° i sin 25° 61. 1 + i 62. 1 - i 1 2 1 2 1 2 p p 6 4 = + 63. 23 - i 64. 22 - i 39. z1 3 cos i sin B R a 5 5 b A B A B In Exercises 65–68, find all the complex roots. Write roots in polar p p z = 4 cos + i sin form with u in degrees. 2 a 10 10 b 65. The complex square roots of 9 cos 30° + i sin 30° 5p 5p 1 2 40. z = 3 cos + i sin 66. The complex square roots of 25 cos 210° + i sin 210° 1 a 8 8 b 1 2 67. The complex cube roots of 8 cos 210° + i sin 210° p p 1 2 = + z 2 10 cos i sin 68. The complex cube roots of 27 cos 306° + i sin 306° a 16 16 b 1 2 p p p p In Exercises 69–76, find all the complex roots. Write roots in 41. z = cos + i sin 42. z = cos + i sin 1 4 4 1 6 6 rectangular form. If necessary, round to the nearest tenth. p p p p 4p 4p z = cos + i sin z = cos + i sin 69. The complex fourth roots of 81 cos + i sin 2 3 3 2 4 4 a 3 3 b P-BLTZMC06_643-726-hr 21-11-2008 12:56 Page 697

Section 6.5 Complex Numbers in Polar Form; DeMoivre’s Theorem 697

5p 5p 70. The complex fifth roots of 32 cos + i sin Writing in Mathematics a 3 3 b 93. Explain how to plot a complex number in the complex 71. The complex fifth roots of 32 plane. Provide an example with your explanation. 72. The complex sixth roots of 64 94. How do you determine the absolute value of a complex number? 73. The complex cube roots of 1 95. What is the polar form of a complex number? 74. The complex cube roots of i 96. If you are given a complex number in rectangular form, how + 75. The complex fourth roots of 1 i do you write it in polar form? - + 76. The complex fifth roots of 1 i 97. If you are given a complex number in polar form, how do you write it in rectangular form? Practice Plus 98. Explain how to find the product of two complex numbers in polar form. In Exercises 77–80, convert to polar form and then perform the 99. Explain how to find the quotient of two complex numbers indicated operations. Express answers in polar and rectangular form. in polar form. 77. i 2 + 2i - 23 + i 1 2A B 100. Explain how to find the power of a complex number in 78. 1 + i 1 - i23 - 23 + i polar form. 1 2A BA B 1 + i23 1 - i -1 + i23 2 - 2i23 101. Explain how to use DeMoivre’s Theorem for finding 79.A B1 2 80. A BA B complex roots to find the two square roots of 9. 223 - 2i 423 - 4i 102. Describe the graph of all complex numbers with an In Exercises 81–86, solve each equation in the complex number absolute value of 6. system. Express solutions in polar and rectangular form. 103. The image of the Mandelbrot set in the section opener 81.x6 - 1 = 0 82. x6 + 1 = 0 exhibits self-similarity: Magnified portions repeat much of 83.x4 + 16i = 0 84. x5 - 32i = 0 the pattern of the whole structure, as well as new and unexpected patterns. Describe an object in nature that 85.x3 - 1 + i23 = 0 86. x3 - 1 - i23 = 0 A B A B exhibits self-similarity. In , it can be shown that

u Technology Exercises ei = cos u + i sin u. 104. Use the rectangular-to-polar feature on a graphing utility to In Exercises 87–90, use this result to plot each complex number. verify any four of your answers in Exercises 11–26. Be pi - - 87.e 4 88.e 6 89.-e pi 90. -2e 2pi aware that you may have to adjust the angle for the correct quadrant. Application Exercises 105. Use the polar-to-rectangular feature on a graphing utility to verify any four of your answers in Exercises 27–36. In Exercises 91–92, show that the given complex number z plots as a point in the Mandelbrot set. Critical Thinking Exercises a. Write the first six terms of the sequence Make Sense? In Exercises 106–109, determine whether Á z1 , z2 , z3 , z4 , z5 , z6 , each statement makes sense or does not make sense, and explain where your reasoning. 106. This stamp, honoring the work done by the German mathe- z = z: Write the given number. 1 matician (1777–1855) with complex = 2 + z2 z z: Square z1 and add the given number. numbers, illustrates that a complex number a + bi can be z = z2 + z 2 + z: Square z and add the given number. interpreted geometrically as the point a, b in the xy-plane. 3 1 2 2 1 2 z = z2 + z 2 + z 2 + z: Square z and add the given 4 31 2 4 3 number.

z5 : Square z4 and add the given number.

z6 : Square z5 and add the given number. b. If the sequence that you began writing in part (a) is bounded, the given complex number belongs to the Mandelbrot set. Show that the sequence is bounded by writing two complex numbers. One complex number should be greater in absolute value than the absolute values of the terms in the sequence. The second complex number should be less in absolute value than the absolute values of the terms in the sequence. 91.z = i 92. z =-i P-BLTZMC06_643-726-hr 21-11-2008 12:56 Page 698

698 Chapter 6 Additional Topics in Trigonometry

107. I multiplied two complex numbers in polar form by first Exercise multiplying the moduli and then multiplying the arguments. 112. Group members should prepare and present a seminar on 108. The proof of the formula for the product of two complex mathematical chaos. Include one or more of the following numbers in polar form uses the sum formulas for cosines topics in your presentation: images, the role of and that I studied in the previous chapter. complex numbers in generating fractal images, , iterations, iteration number, and in nature. Be sure 109. My work with complex numbers verified that the only to include visual images that will intrigue your audience. possible cube root of 8 is 2. 110. Prove the rule for finding the quotient of two complex Preview Exercises numbers in polar form. Begin the proof as follows, using the Exercises 113–115 will help you prepare for the material covered conjugate of the denominator’s second factor: in the next section. + + - 113. Use the distance formula to determine if the line segment r1 cos u1 i sin u1 r1 cos u1 i sin u1 # cos u2 i sin u2 1 2 = 1 2 1 2 . with endpoints -3, -3 and (0, 3) has the same length as r2 cos u2 + i sin u2 r2 cos u2 + i sin u2 cos u2 - i sin u2 1 2 1 2 1 2 1 2 the line segment with endpoints (0, 0) and (3, 6). Perform the indicated . Then use the 114. Use to determine if the line through -3, -3 and 1 2 difference formulas for sine and cosine. (0, 3) is parallel to the line through (0, 0) and (3, 6). 111. Plot each of the complex fourth roots of 1. 115. Simplify: 4 5x + 4y - 2 6x - 9y . 1 2 1 2

S e c t i o n 6.6 Vectors Objectives

ᕡ Use magnitude and direction F3 to show vectors are equal. ᕢ Visualize scalar , F2 vector addition, and vector F3 subtraction as geometric F vectors. 2 ᕣ Represent vectors in the rectangular coordinate Fg F F system. g 1 ᕤ Perform operations with f vectors in terms of i and j. f ᕥ Find the in the F direction of v. 1 ᕦ Write a vector in terms of its magnitude and direction. ᕧ Solve applied problems It’s been a dynamic lecture, but now that it’s over it’s obvious that my professor is exhausted. involving vectors. She’s slouching motionless against the board and—what’s that? The acting against her body, including the pull of , are appearing as arrows. I know that mathematics reveals the hidden patterns of the universe, but this is ridiculous. Does the arrangement of the arrows on the right have anything to do with the fact that my wiped-out professor is not sliding down the wall?

urs is a world of pushes and pulls. For example, suppose you are pulling a cart up Oa 30° incline, requiring an effort of 100 pounds. This quantity is described by giving its magnitude (a number indicating size, including a unit of measure) and also its direction. The magnitude is 100 pounds and the direction is 30° from the horizontal. Quantities that involve both a magnitude and a direction are called vector quantities, or vectors for short. Here is another example of a vector: You are driving due north at 50 miles per hour.The magnitude is the speed, 50 miles per hour. The direction of motion is due north.