Exercicios resolvidos James Stewart vol. 2 7ª ed - ingles

360 D APPENDIX H COMPLEX NUMBERS
1/ z = ~ [cos(O- ~) + i si~(O- ~ )] = ~[cos( - i) + i sin(-~) ] . For 1/z; we could al ~o use the fonnula that precedes
Example 5 to obtain 1/ z = H cos "if - i sin i).
31. For z = 2¥'3- 2i, r = J(2 v'3) 2 + (-2) 2 = 4 and tan6 = '27a = -~ => 6 = -~ =>
z = 4(cos( --jr} + isin( -i)J. Forw = - 1 + i, r = -/2, tanB = ! 1
= -1 => 6 = 3 ; =>
w = ..J2 (cos 3 4 " + isin 3 ;). Therefore, zw = 4 V2 [cos(-{f + 3 ;) + i sin(-~+ 3 ;)] = 4 ..J2 (cos i; + isin i;),
z/w- 4 [cos(-2I- 3 ")+isin(-2I - 3
71')] -
4
- 72 [cos(- 11 71')+isin(- 11 71' )] 2 f2(cos 13 "+isin 13 6 4 6 4 - 72 12 12 - v ~ 12 . . 12
")
'
and
1/z = i(cos(-{f) - isin(-{f)] = Hcos "if+ isin"lf).
33. For z = 1 + i, r = ..j2 and tan B = t = 1 => B = "i => z = ..j2 (cos "i + isin f). So by De Moivre's Tlworem,
(1 +_i) 20 = [..J2 (cos "i +isinf)J 20 = (2 1 1 2 ? 0 (cos 20 ~71' + isin 20 ~") = 2 10 (cos51l' + isin51l')
= 2 10 [-1 +i(O)] = -2 10 = - 1024
So by De Moivre's Theorem,
(2 v'3 + 2i) 5 = [4(cos "if+ isin ~)) 5 = 4 5 (cos 5 6 "+ i sin ~G11') = 1024[ -4 + ~i] = -512 v'3 + 512i.
37. 1 = 1 + Oi = 1 (cosO+ isinG). Using Equation 3 with 7' = 1, n = 8, and 6 = 0, we have
Wk- 1
- 1/8[ (0+2k1l') .. (0+2k1l')J- k1l' .. k1l' · -
cos
8
+ ~sm
8
-cos 4
+ ~sm
4
, where k- 0, 1, 2, ... , 7.
w 0 = 1(cosO+isinO) = 1, w1 = 1(cosf +isinf) = "7z + 72i,
w2 = 1 (cos ~ + i sin ~) = i, w 3 = 1 (cos 3 ; + i sin a;.) = - "7z + 72 i,
•
1m
•
w4 = 1(cos1l'+isinir) = -1, w5 = 1(cos 5 ;
+isin 5 ;) = -72 -72i,
0
Re
w 6 = 1(cos a;+ isin a;) = ...:i, w 7 = 1(cos 7 ; + isin 7 ;) = 72 - 72i
•
•
39. i = 0 + i = 1 (cos~ + i sin~). Using Equation 3 with r = 1, n = 3, and 6 = %, we have
Wk = 1 1 / 3 COS
[ (
J!. 2 + 2k1l') + i sin ( ~ - + q 2k1l')] , where ' k = 0, 1, 2.
3
wo = (cos i + i sin i) = 4 + ~i
WI = (cos· r,,. + i sin 5 ") - - v'3 + l i
6 6 - 2 2
w2 = ,(cos 0 6 " + i sin 9 ;) == - i
1m
•
0
-i
•
Re
41 . Using Euler'sfonnula (6) withy=~. we have ei71'/ 2 =cos~+ isin ~ = 0 + 1i = i.
© 2012 Cengnge Lenming. All Rights Reserve~ . May not be: scanncJ, copi