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Hence, by eqs. (168), (143), and (144), we have p 3 j2 7 13 7 cos 2284:178 j sin 2284:178 7921:40 cos 124:178 j sin 124:178 4449:06 j6553:97 ; answer: 104. We begin by writing the problem in the form 6000 3 j4 5 . Thus n 5 in eq. (168). Next, since 3 j4 lies in the rst quadrant, p have (Fig. 110) we  h arctan 4=3 53:13018. Also, by eq. (143), A 25 5. Thus, using these values and setting n 5 in the right-hand side of eq. (168), and noting that cos x cos x and sin x sin x, we have 6000 5 5 cos 265:658 j sin 265:658 0:1457 j1:9144 approx:; answer: 105. The four roots are the values of 3 j7 1=4 . Thus, for use in eq. (176), we have, for this problem, a 3; b 7; and n 4. We can proceed as follows. First, the magnitude A of the complex number 3 j7 is, by eq. (143), equal to p A 58 58 1=2 Thus, for this problem n 4 , we have that A1=4 58 1=8 1:6612 approx:; by calculator: Next, noting that 3 j7 lies in the rst quadrant of the complex plane, the value of  is, by eq. (144), equal to  arctan 7=3 66:80148 approx:; thus  66:80148 16:70048 16:78 approx: n 4 Let us now denote the four roots by r1 ; r2 ; r3 , and r4 . Since, in this problem, n 4, the four values of k to be substituted into eq. (176) are k 0; 1; 2, and 3, and upon doing this we nd that for k 0: for k 1: for k 2: for k 3: r1 1:6612 cos 16:78 j sin 16:78 1:591 j0:477; answer: r2 1:6612 cos 106:78 j sin 106:78 0:477 j1:591; answer: r3 1:6612 cos 196:78 j sin 196:78 1:591 j0:477; answer: r4 1:6612 cos 286:78 j sin 286:78 0:477 j1:591; answer:
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The locations of the four roots on the complex plane are shown in the gure below. In the gure, the LENGTH of a line drawn from the origin to any point represents the MAGNITUDE of the complex number associated with that point. Reference to eq. (176) shows that all roots will have the SAME MAGNITUDE A1=n , and this fact is evident, geometrically, from inspection of the following gure. Next, the ANGULAR factors associated with each root can be clearly seen by writing eq. (176) in the exponential form, thus a jb 1=n A1=n  j =n 360k=n 8
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Note that the angle for the FIRST ROOT (the root for k 0) is equal to =n, where  (eq. (144)) is relative to the real axis; here, in this problem (where n 4), we have =n =4 16:78, as shown in the gure. Next notice that 360 k=n 360=n k is the ANGULAR SEPARATION, in degrees, between the lines drawn from the origin to the points at the root locations. Since n is a given constant in any given problem, the angular separation is the SAME for all roots in any given problem. Thus, in the present problem, where n 4, the angular separation is 90k 8, that is, 90 degrees, as inspection of the gure shows. 106. Here a 19; b 33; n 5. Thus, rst, we have A1=n 1450 1=10 2:07 approx: Next, since the point 19; 33 lies in the fourth quadrant (Fig. 110), we have h arctan 33=19 60:078 thus,  360 h 299:938 approx: therefore, =n 59:998, and hence eq. (176) becomes 19 j33 1=5 2:07 cos 59:99 72k 8 j sin 59:99 72k 8 2:07= 59:99 72k 8 see eqs: 157 and 158 ; section 6:5 (see eqs. (157) and (158), section 6.5) Now, setting, successively, k 0; 1; 2; 3, and 4 into the last expressions gives for k 0: for k 1: for k 2: for k 3: for k 4: 2:07=59:998 1:04 j1:79; answer: 2:07=131:998 1:39 j1:54; answer: 2:07=203:998 1:89 j0:84; answer: 2:07=275:998 0:22 j2:06; answer: 2:07=347:998 2:03 j0:43; answer:
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