Trigonometric Form of a Complex Number. The Complex Plane. axis. ( 2, 1) or 2 i FIGURE The absolute value of the complex number z a bi is


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1 0_0605.qxd /5/05 0:45 AM Page Chapter 6 Additioal Topics i Trigoometry 6.5 Trigoometric Form of a Complex Number What you should lear Plot complex umbers i the complex plae ad fid absolute values of complex umbers. Write the trigoometric forms of complex umbers. Multiply ad divide complex umbers writte i trigoometric form. Use DeMoivre s Theorem to fid powers of complex umbers. Fid th roots of complex umbers. Why you should lear it You ca use the trigoometric form of a complex umber to perform operatios with complex umbers. For istace, i Exercises 05 o page 480, you ca use the trigoometric forms of complex umbers to help you solve polyomial equatios. The Complex Plae Just as real umbers ca be represeted by poits o the real umber lie, you ca represet a complex umber z a bi as the poit a, b i a coordiate plae (the complex plae). The horizotal is called the real ad the vertical is called the imagiary, as show i Figure FIGURE 6.44 (, ) or i The absolute value of the complex umber a bi is defied as the distace betwee the origi 0, 0 ad the poit a, b. (, ) or + i Defiitio of the Absolute Value of a Complex Number The absolute value of the complex umber z a bi is a bi a b. (, 5) FIGURE 6.45 If the complex umber a bi is a real umber (that is, if b 0), the this defiitio agrees with that give for the absolute value of a real umber a 0i a 0 a. Example Fidig the Absolute Value of a Complex Number Plot z 5i ad fid its absolute value. Solutio The umber is plotted i Figure It has a absolute value of z 5 9. Now try Exercise.
2 0_0605.qxd /5/05 0:45 AM Page 47 Sectio 6.5 Trigoometric Form of a Complex Number 47 Trigoometric Form of a Complex Number (a,b) b a r θ I Sectio.4, you leared how to add, subtract, multiply, ad divide complex umbers. To work effectively with powers ad roots of complex umbers, it is helpful to write complex umbers i trigoometric form. I Figure 6.46, cosider the ozero complex umber a bi. By lettig be the agle from the positive real (measured couterclockwise) to the lie segmet coectig the origi ad the poit a, b, you ca write a r cos ad b r si where r a b. Cosequetly, you have a bi r cos r si i FIGURE 6.46 from which you ca obtai the trigoometric form of a complex umber. Trigoometric Form of a Complex Number The trigoometric form of the complex umber z a bi is z rcos i si where a r cos, b r si, r a b, ad ta ba. The umber r is the modulus of z, ad is called a argumet of z. The trigoometric form of a complex umber is also called the polar form. Because there are ifiitely may choices for, the trigoometric form of a complex umber is ot uique. Normally, is restricted to the iterval 0 <, although o occasio it is coveiet to use < 0. Example Writig a Complex Number i Trigoometric Form Write the complex umber z i i trigoometric form. Solutio The absolute value of z is r i 6 4 z = 4 4π ad the referece agle ta b a is give by Because ta ad because z i lies i Quadrat III, you choose to be So, the trigoometric form is 4. z rcos i si. z = i 4 4 cos 4 4 i si. See Figure FIGURE 6.47 Now try Exercise.
3 0_0605.qxd /5/05 0:45 AM Page Chapter 6 Additioal Topics i Trigoometry Example Writig a Complex Number i Stadard Form Techology A graphig utility ca be used to covert a complex umber i trigoometric (or polar) form to stadard form. For specific keystrokes, see the user s maual for your graphig utility. Write the complex umber i stadard form a bi. z 8 cos i si Solutio Because cos ad si, you ca write z 8 cos i si i 6i. Now try Exercise 5. Multiplicatio ad Divisio of Complex Numbers The trigoometric form adapts icely to multiplicatio ad divisio of complex umbers. Suppose you are give two complex umbers z ad z r cos i si r cos i si. The product of z ad is give by z z z r r cos i si cos i si r r cos cos si si isi cos cos si. Usig the sum ad differece formulas for cosie ad sie, you ca rewrite this equatio as z z r r cos i si. This establishes the first part of the followig rule. The secod part is left for you to verify (see Exercise 7). Product ad Quotiet of Two Complex Numbers Let z r cos i si ad z r cos i si be complex umbers. z z r r cos i si Product z r cos z r i si, z 0 Quotiet Note that this rule says that to multiply two complex umbers you multiply moduli ad add argumets, whereas to divide two complex umbers you divide moduli ad subtract argumets.
4 0_0605.qxd /5/05 0:45 AM Page 47 Sectio 6.5 Trigoometric Form of a Complex Number 47 Example 4 Multiplyig Complex Numbers Techology Some graphig utilities ca multiply ad divide complex umbers i trigoometric form. If you have access to such a graphig utility, use it to fid z z ad z z i Examples 4 ad 5. Fid the product z z of the complex umbers. z cos i si Solutio z z cos i si 6 cos 6 i si 6 6 cos 5 5 i si 6 cos i si 60 i 6i You ca check this result by first covertig the complex umbers to the stadard forms z i ad z 4 4i ad the multiplyig algebraically, as i Sectio.4. z z i4 4i 4 4i i 4 6i Now try Exercise 47. z 8 cos i si cos 6 i si 6 Multiply moduli ad add argumets. Example 5 Dividig Complex Numbers Fid the quotiet z z of the complex umbers. z 4cos 00 i si 00 Solutio z 4cos 00 i si 00 z 8cos 75 i si 75 4 cos00 75 i si cos 5 i si 5 i i Now try Exercise 5. z 8cos 75 i si 75 Divide moduli ad subtract argumets.
5 0_0605.qxd /5/05 0:45 AM Page Chapter 6 Additioal Topics i Trigoometry Powers of Complex Numbers The trigoometric form of a complex umber is used to raise a complex umber to a power. To accomplish this, cosider repeated use of the multiplicatio rule. z rcos i si z rcos i si rcos i si r cos i si z r cos i si rcos i si r cos i si z 4 r 4 cos 4 i si 4 z 5 r 5 cos 5 i si 5. This patter leads to DeMoivre s Theorem, which is amed after the Frech mathematicia Abraham DeMoivre ( ). The Grager Collectio Historical Note Abraham DeMoivre ( ) is remembered for his work i probability theory ad DeMoivre s Theorem. His book The Doctrie of Chaces (published i 78) icludes the theory of recurrig series ad the theory of partial fractios. DeMoivre s Theorem If z rcos i si is a complex umber ad is a positive iteger, the z rcos i si Example 6 r cos i si. Fidig Powers of a Complex Number Use DeMoivre s Theorem to fid i. Solutio First covert the complex umber to trigoometric form usig r So, the trigoometric form is ad z i cos i si. The, by DeMoivre s Theorem, you have i cos i si cos i si 4096cos 8 i si Now try Exercise 75. arcta.
6 0_0605.qxd /5/05 0:45 AM Page 475 Roots of Complex Numbers Recall that a cosequece of the Fudametal Theorem of Algebra is that a polyomial equatio of degree has solutios i the complex umber system. So, the equatio x 6 has six solutios, ad i this particular case you ca fid the six solutios by factorig ad usig the Quadratic Formula. x 6 x x Cosequetly, the solutios are x ±, Sectio 6.5 Trigoometric Form of a Complex Number 475 x x x x x x 0 x ± i, Each of these umbers is a sixth root of. I geeral, the th root of a complex umber is defied as follows. ad x ± i. Defiitio of the th Root of a Complex Number The complex umber u a bi is a th root of the complex umber z if z u a bi. Exploratio The th roots of a complex umber are useful for solvig some polyomial equatios. For istace, explai how you ca use DeMoivre s Theorem to solve the polyomial equatio x [Hit: Write 6 as 6cos i si. ] To fid a formula for a th root of a complex umber, let u be a th root of z, where u scos i si ad z rcos i si. By DeMoivre s Theorem ad the fact that u z, you have s cos i si rcos i si. Takig the absolute value of each side of this equatio, it follows that s r. Substitutig back ito the previous equatio ad dividig by r, you get cos i si cos i si. So, it follows that cos cos ad si si. Because both sie ad cosie have a period of, these last two equatios have solutios if ad oly if the agles differ by a multiple of. Cosequetly, there must exist a iteger k such that k k. By substitutig this value of ito the trigoometric form of u, you get the result stated o the followig page.
7 0_0605.qxd /5/05 0:45 AM Page Chapter 6 Additioal Topics i Trigoometry Fidig th Roots of a Complex Number For a positive iteger, the complex umber z rcos i si has exactly distict th roots give by r cos k i si where k 0,,,...,. k r FIGURE 6.48 π π Whe k exceeds, the roots begi to repeat. For istace, if k, the agle is cotermial with, which is also obtaied whe k 0. The formula for the th roots of a complex umber z has a ice geometrical iterpretatio, as show i Figure Note that because the th roots of z all have the same magitude r, they all lie o a circle of radius r with ceter at the origi. Furthermore, because successive th roots have argumets that differ by, the roots are equally spaced aroud the circle. You have already foud the sixth roots of by factorig ad by usig the Quadratic Formula. Example 7 shows how you ca solve the same problem with the formula for th roots. Example 7 Fidig the th Roots of a Number + i + 0i FIGURE 6.49 i + + 0i i i Fid all the sixth roots of. Solutio First write i the trigoometric form cos 0 i si 0. The, by the th root formula, with 6 ad r, the roots have the form 6 cos 0 k 6 So, for k 0,,,, 4, ad 5, the sixth roots are as follows. (See Figure 6.49.) cos 0 i si 0 cos i si i cos i si i cos i si cos 4 4 i si i cos 5 5 i si i i si 0 k 6 cos k i si k. Now try Exercise 97. Icremet by 6
8 0_0605.qxd /5/05 0:45 AM Page 477 Sectio 6.5 Trigoometric Form of a Complex Number 477 Activities. Use DeMoivre s Theorem to fid i. Aswer: 64cos 4 i si Fid the cube roots of 8 i 8 cos i si. Aswer: cos i si 9 9 cos 8 8 i si 9 9 cos 4 4 i si 9 9. Fid all of the solutios of the equatio x 4 0. Aswer: cos 4 i si 4 i cos i si 4 4 i cos 5 5 i si 4 4 i cos 7 7 i si 4 4 i i FIGURE i i Note i Example 8 that the absolute value of z is r i 8 ad the agle is give by ta b a. I Figure 8.49, otice that the roots obtaied i Example 7 all have a magitude of ad are equally spaced aroud the uit circle. Also otice that the complex roots occur i cojugate pairs, as discussed i Sectio.5. The distict th roots of are called the th roots of uity. Example 8 Fidig the th Roots of a Complex Number Fid the three cube roots of z i. Solutio Because z lies i Quadrat II, the trigoometric form of z is z i 8 cos 5 i si 5. By the formula for th roots, the cube roots have the form 68 cos Fially, for k 0,, ad, you obtai the roots 68 cos 5600 i si 5600 cos 45i si cos 560 i si 560 cos 65i si cos 560 i si 560 cos 85i si 85 See Figure k i si Now try Exercise 0. 5º 60k. W RITING ABOUT MATHEMATICS A Famous Mathematical Formula The famous formula e a bi e a cos b i si b arcta 5 i i i. is called Euler s Formula, after the Swiss mathematicia Leohard Euler (707 78). Although the iterpretatio of this formula is beyod the scope of this text, we decided to iclude it because it gives rise to oe of the most woderful equatios i mathematics. e i 0 This elegat equatio relates the five most famous umbers i mathematics 0,,, e, ad i i a sigle equatio. Show how Euler s Formula ca be used to derive this equatio.
9 0_0605.qxd /8/05 0:09 AM Page Chapter 6 Additioal Topics i Trigoometry 6.5 Exercises VOCABULARY CHECK: Fill i the blaks.. The of a complex umber a bi is the distace betwee the origi 0, 0 ad the poit a, b.. The of a complex umber z a bi is give by z rcos i si, where r is the of z ad is the of z.. Theorem states that if z rcos i si is a complex umber ad is a positive iteger, the z r cos i si. 4. The complex umber u a bi is a of the complex umber z if z u a bi. PREREQUISITE SKILLS REVIEW: Practice ad review algebra skills eeded for this sectio at I Exercises 6, plot the complex umber ad fid its absolute value.. 7i i 4. 5 i i 6. 8 i I Exercises 7 0, write the complex umber i trigoometric form z = i z = i 4 z = 6 4 z = + i I Exercises 0, represet the complex umber graphically, ad fid the trigoometric form of the umber.. i. i. i i 5. i 5 6. i 7. 5i 8. 4i i 0. i i 4. i 4 5. i 6. i 7. 5 i 8. 8 i i i I Exercises 40, represet the complex umber graphically, ad fid the stadard form of the umber.. cos 0 i si 0. 5cos 5 i si 5. cos 00 i si cos 5 i si cos i si cos 5 5 i si 7. 8 cos i si 8. 7cos 0 i si 0 9. cos8 45 i si cos0º 0 i si0º 0 I Exercises 4 44, use a graphig utility to represet the complex umber i stadard form cos 4. 9 i si 4. cos 65.5 i si cos 58º i si 58º 9 0 cos i si 5 5 I Exercises 45 ad 46, represet the powers z, z, z, ad graphically. Describe the patter. 45. z i 46. z i z 4
10 0_0605.qxd /8/05 0:09 AM Page 479 Sectio 6.5 Trigoometric Form of a Complex Number 479 I Exercises 47 58, perform the operatio ad leave the result i trigoometric form i si 4 6 cos i si cos i si 4 cos i si cos 40i si 40 cos 60i si cos 00i si cos 00 i si cos 0i si cos 00 i si cos 5 i si 5cos 0 i si 0 cos 50 i si cos 0 i si 0 cos 0 i si cos 40 i si 40 cos5 i si5 55. cos i si 5cos 4. i si cos. i si. cos 5 i si cos 0 i si 0 6cos 40 i si cos 00 i si 00 I Exercises 59 66, (a) write the trigoometric forms of the complex umbers, (b) perform the idicated operatio usig the trigoometric forms, ad (c) perform the idicated operatio usig the stadard forms, ad check your result with that of part (b). 59. i i 60. i i 6. i i 6. 4 i 6. 4i i 64. i 6 i i 4i 4 i I Exercises 67 70, sketch the graph of all complex umbers z satisfyig the give coditio cos z z I Exercises 7 88, use DeMoivre s Theorem to fid the idicated power of the complex umber.write the result i stadard form. 7. i 5 7. i 6 7. i i i i 77. 5cos 0 i si cos 50 i si cos 4 i si 80. i si 8. 5cos. i si cos 0 i si i i 85. cos 5 i si cos 0 i si cos 0 i si cos cos 8 i si I Exercises 89 04, (a) use the theorem o page 476 to fid the idicated roots of the complex umber, (b) represet each of the roots graphically, ad (c) write each of the roots i stadard form. 89. Square roots of 5cos 0 i si Square roots of 6cos 60 i si Cube roots of 9. Fifth roots of 8 cos i si cos 5 5 i si Square roots of 5i 94. Fourth roots of 65i 95. Cube roots of 5 i 96. Cube roots of 4 i 97. Fourth roots of Fourth roots of i 99. Fifth roots of 00. Cube roots of Cube roots of 5 0. Fourth roots of 4 0. Fifth roots of 8 i 04. Sixth roots of 64i
11 0_0605.qxd /8/05 0:09 AM Page Chapter 6 Additioal Topics i Trigoometry I Exercises 05, use the theorem o page 476 to fid all the solutios of the equatio ad represet the solutios graphically. 05. x 4 i x x x x 4 6i 0 0. x 6 64i 0. x i 0. x 4 i 0 Sythesis True or False? I Exercises 6, determie whether the statemet is true or false. Justify your aswer.. Although the square of the complex umber bi is give by bi b, the absolute value of the complex umber z a bi is defied as a bi a b. 4. Geometrically, the th roots of ay complex umber z are all equally spaced aroud the uit circle cetered at the origi. 5. The product of two complex umbers z r cos i si ad z r cos i si. is zero oly whe r 0 ad/or r By DeMoivre s Theorem, 4 6i 8 cos i si Give two complex umbers z r cos i si ad z r cos i si, z 0, show that z r cos z r i si. 8. Show that z rcos i si is the complex cojugate of z rcos i si. 9. Use the trigoometric forms of z ad z i Exercise 8 to fid (a) zz ad (b) zz, z Show that the egative of z rcos i si is z rcos i si.. Show that i is a sixth root of.. Show that 4 i is a fourth root of. Graphical Reasoig I Exercises ad 4, use the graph of the roots of a complex umber. (a) Write each of the roots i trigoometric form. (b) Idetify the complex umber whose roots are give. (c) Use a graphig utility to verify the results of part (b) Skills Review I Exercises 5 0, solve the right triagle show i the figure. Roud your aswers to two decimal places. B a C 5. A, a A 0, b A 4 5, c. 0. Harmoic Motio I Exercises 4, for the simple harmoic motio described by the trigoometric fuctio, fid the maximum displacemet ad the least positive value of t for which d d 6 cos. 4 t. d 4. 6 si 5 4t B 66, a.5 B 6, b. B 8 0, c 6.8 d cos t 8 d si 60t I Exercises 5 ad 6, write the product as a sum or differece si 8 cos 6. cos 5 si c b A
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