P-BLTZMC06_643-726-hr 21-11-2008 12:56 Page 686 686 Chapter 6 Additional Topics in Trigonometry In Exercises 18–20, convert each rectangular equation to a polar In Exercises 26–27, test for symmetry with respect to u p equation that expresses r in terms of . a. the polar axis. b. the line 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 coordinate system. =- = - 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 Numbers in Polar Form; DeMoivre’s Theorem Objectives ᕡ Plot complex numbers in the A magnification of the Mandelbrot set complex plane. ᕢ Find the absolute value of a complex number. 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 complex number 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 mean a complete lack of numbers in polar form. form or arrangement. In mathematics, 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 trigonometric functions. ᕡ Plot complex numbers The Complex Plane in the complex plane. We know that a real number can be represented as a point 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 distance 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 origin to the point z in the complex plane. Figure 6.40 Imaginary illustrates that we can use the Pythagorean Theorem 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 angle 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 .