COMPLEX NUMBERS AND DE MOIVRE S THEOREM

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Sibyl Mosley
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1 COMPLEX NUMBERS AND DE MOIVRE S THEOREM OBJECTIVE PROBLEMS. s equl to b d. 9 9 b 9 9 d. The mgr prt of s 5 5 b 5. If m, the the lest tegrl vlue of m s b 8 5. The vlue of 5... s f s eve, f s odd b f s eve, f s odd f s odd, f s eve d f s eve, f s odd 6. The rel vlues of d for whch the equto s s stsfed, re 5 8, b 8 5, 5,

2 7. sθ sθ wll be rel, f θ b 8. If cosθ sθ, the cot θ b cot θ cot θ θ d t 9. The multplcto verse of umber s the umber tself, the ts tl vlue s b d 0. If d, the Im b d. If / b, the b s equl to b b b b. The vlues of d stsfg the equto re, b, 0, d, 0. sθ sθ wll be purel mgr, f θ ± b ±. If, the b 0

3 5. The smllest postve teger for whch s b d 6. The rel prt of cosθ sθ s 5 cos θ b 5 cos θ 5 cosθ d 5 cos θ 7. If, the the multplctve verse of s where 8. If b / d /, cosθ sθ the s equl to b 9. If b c d b c, the b c b b c c b 0. If p q, the p q, b p, q q p,. If 6 0, the, s, b, 0, d 0, If b, the, b b, b 0 0, b d, b b b. If b, the b b d b

4 . If, the s equl to 5 b 6 5 d 5. The cojugte of, the form of b, s b d If d be comple umbers such tht hs egtve mgr prt, the d m be. If hs postve rel prt d Purel mgr Rel d egtve brel d postve 7. The mmum vlue of where stsfes the codto s b d 8. If s comple umber, the b 0 9. If 5, the 98 5 b 5 5 d 5 0. The umber of solutos of the equto 0 s b d. If, re two comple umbers, the s equl to b d

5 . If d re two o-ero comple umbers such tht, the rg rg s equl to b d 0. If cos α sα, the mp α b α. If α d cos s 6 6 the α, rg b, rg 6 5, rg d, rg t 5. If d mp mp 0, the b 0 d 6. If rg θ, the rg θ b θ θ d θ 7. If be the cojugte of the comple umber, the whch of the followg reltos s flse b. d rg rg 8. If..., the the vlue of b......

6 9. The cojugte of comple umber, s b d 0. If d re two comple umbers stsfg the equto, the s umber whch s Postve rel Zero or purel mgr bnegtve rel. If, re two comple umbers such tht d k, where k R, the the gle betwee d s t k k b t k k t k d t k. If d ω where, the 0 b Reω s. d. If be the cojugte of the comple umber, the whch of the followg reltos s flse b. d rg rg. Let be comple umber wth d be comple umber, the 0 b d

7 5. For two comple umbers, we hve the Re 0 b Im 0 Re 0 d Im 0 6. If, the the dfferece the mpltudes of d s b d 0 7. If d ω re two o ero comple umbers such tht ω d rg rg ω, the ω s equl to b d 8. rg 0 s equl to b d 9. If...., the rg rg... rg d rg dffer b Multple of b Multple of Greter th dless th 50. Whch of the followg re correct for two comple umbers d b rg rg rg d 5. If d 5, d the Re s equl to 7 7 b 7 d If 8 b the b b 8 9 d 8 s

8 5. If d r g, the s equl to 0 Purel rel b Purel mgr 5. The rel prt of s / e cos log b / e s log / / e cos log d e s log 55. If b, the possble vlue of b s b d 56. log s equl to t b t t d t cos cos s cos s b s dnoe of these / 58. The vlue of s b d - b c d 59. If, the b c d b b c d c d b d b c d

9 9 60. If b, the b s equl to b 56 0 d 9 6. The umber of o-ero tegrl solutos of the equto s Ifte b 5 6. The mgr prt of t s 0 b log d log 6. If cosθ sθ,the the vlue of s cos θ b sθ cosecθ d t θ 6. s equl to cos s s b cos cos s 65. If, the les o A ellpse A crcle b The mgr s d The rel s 66. The equto b 0, b R represets crcle f b b > b < b 67. Let be comple umber such tht < d,... be vertces of polgo such tht, k... k. The the vertces of the polgo le wth crcle b d

10 68. Let the comple umbers, d be the vertces of equlterl trgle. Let 0 be the crcum cetre of the trgle, the 0 b 0 0 d The comple umbers,, re the vertces of trgle. The the comple umbers whch mke the trgle to prllelogrm s b d All the bove 70. Let,, be three vertces of equlterl trgle crcumscrbg the crcle. If d,, re tclockwse sese the s b d 7. If s comple umber the Argd ple, the the equto 8 represets Prbol Hperbol b Ellpse d Crcle 7. If,,, re the ffes of four pots the Argd ple d s the ff of pot such tht, the,,, re Cocclc b Vertces of prllelogrm Vertces of rhombus d I strght le 7. If, the re of the trgle whose vertces re pots, d b d s 7. If, Strght le Prbol the the locus descrbed b the pot the Argd dgrm s b Crcle

11 75. If the re of the trgle formed b the pots, d o the comple ple s 8, the the vlue of s 6 b 9 d 76. The rego of Argd ple defed b s Iteror of ellpse b Eteror of crcle Iteror d boudr of ellpse 77. Let d be two comple umbers such tht, re coller b, d the org form rght gled trgle, d the org form equlterl trgle 78. The locus represeted b s A crcle of rdus b A ellpse wth foc t, 0 d 0, A strght le through the org d A crcle o the le jog,0,0, s dmeter 79. If d, the locus of s 5 0 b d 5 0. The 80. cosϕ sϕ cosϕ sϕ cos φ s φ b cos φ s φ s φ cos φ d s φ cos φ

12 8. If s postve teger, the s equl to cos b cos s s d 8. cos / 8 s / 8 cos / 8 s / 8 8 s equl to b 0 d 8. If ω s cube root of ut, the ω ω 0 ω b ω 8. If cosθ sθcos θ s θ... cos θ s θ m b m, the the vlue of θ s m d m where s equl to 5 b 5 5 d 5 cos α sα 86. s β cos β 5 cos α 5β sα 5β b cos α 5β sα 5β s α 5β cosα 5β 87. sθ cos θ sθ cos θ θ s θ θ cos b cos θ s θ cos θ θ s d cos θ s

13 88. The product of ll the roots of cos s / s b d 89. If cos θ, the s equl to cos θ b s θ cos θ d s θ cos s 90. The vlue of 0 0 cos s b d 9. If cos α cos β cos γ 0 0 s α s β s γ cos α β γ b cos α β γ 0 d the cos α cos β cos γ equls 9. If ω s cube root of ut, the ω ω ω ω b 0 d If ω s cube root of ut, the the vlue of ω ω ω ω 6 b 8 d 9. If ω s comple cube root of ut, the b d ω ω

14 bω cω 95. The vlue of b cω ω b bω cω c ω bω wll be d 96. If,,... re th, roots of ut, the for k,,..., k b k k k k k d k k k k k s equl to 6 b 6 d b 0 9 d If, the 00. If s equl to b I d s root of equto 0 the ts rel roots re, b,, d, 0. If ω s comple cube root of ut, the 5 ω 8 ω ω 8 ω 7 b 9 00 d 8 0. sh s s b s s d s

15 0. If, ω, ω re the cube roots of ut, the ω ω ω ω ω ω 0 b ω d ω

16 COMPLEX NUMBERS AND DE MOIVRE S THEOREM HINTS AND SOLUTIONS.. d We hve b 5. Let m m [ 5... ] Clerl seres s A.P. wth commo dfferece d T T sθ sθ s θ 8 sθ sθ sθ s θ s θ Now, sce t s rel, therefore Im 0 8 sθ s θ 8. cosθ sθ. 0 s 0 cosθ sθ. cosθ sθ θ, θ Rtolto of deomtor, we get cosθ sθ cosθ sθ cosθ sθ cosθ sθ 9. b Verfcto 0 d If d The Im.

17 . b / b b. b. b b b b b Equtg rel d mgr prts, we get b b d b b b. b Equtg rel d mgr prts, we get 7 d 9. Hece d.. sθ sθ wll be purel mgr, f the rel prt vshes,.e., 0 s θ s θ s θ 0. b If... We kow tht f two comple umbers re equl, ther modul must lso be equl, therefore from, we hve, > 0 / 0 5. b We hve 6. d { cosθ. sθ}. θ s. s θ cos θ θ s θ θ s. cos θ s θ θ s. cos θ θ θ θ s. cos s. cos

18 7. Gve d. Squrg both sdes, we get or. Sce t s multplctve dett, therefore multplctve verse of. 8. b If cos θ sθ cosθ sθ 6 cosθ sθ cosθ s θ cos θ cosθ s θ 6 cos θ sθ 5 cos θ 5 cos θ 9. b c/ c b b c/ c b c b c b c b 0. p q p q q p p q 0, q p d p q q p. d b Gve, b ; b b 5 b [ ] 0,. Gve tht b, therefore b b b b b b b b b b b b b b b b b b b

19 . Gve tht 7, 7 d Cojugte Assume two comple umbers stsfg both codtos.e., Let,, Hece the result. d 0 7. b ± 8 ±. Hece m. vlue of s 8. Let, d ; 9. 5, Hece, d Let, the. d Check b puttg 0 d 0. d, les o sme strght le. rg rg rg rg 0. d mp t sα cos α t α cot t α t α.. b cos s 6 6

20 d rg t t / / t rg t t b Let r θ s cos θ The r d rg 0 rg rg rg θ r [cos θ s θ ] r cosθ s θ. 6. b cocept. 7. d Let, Sce rg θ t rg θ t Thus rg rg. 8. We hve k, k,,... k k k k k Therefore b Cojugte Gve cos θ sθ cos θ sθ cot cos θ sθ θ s

21 Whch s ero, f, I θ d s otherwse purel mgr.. α cos α s α α α α s cos s cos Applg compoedo d dvdedo. ω. b. b We hve d be comple umber. ; ; Gve tht. 5. We hve cos θ θ Where, rg rg θ θ 0 cos θ θ θ θ 0 cos Re rg 6. Squrg the gve reltos mples tht 0 Now t t mp mp t t t.

22 7. d ω... d rg ω ω ω... From equto d ω ω ω d 0 ; ω ω 0 ω ω ω ω ; ω ω.. ω rg rg 5 0 rg We kow tht the prcpl vlue of θ les betwee d. 50. Cocept 5. d Gve, 5 d The Re b Tkg modulus d squrg o both sdes, we get b b 98 b b We hve rg rg rg rg rg Let rg θ, the rg θ [cos θ s ] θ

23 cosθ sθ d cosθ s cosθ sθ θ Hece Let. Tkg log o both sdes, log log log cos s / e log / log log e log log / / log e e. Tkg rel prt ol 55. d b b, b d hece b 56. b Let log log log log log θ log b log re log r θ... log b t b / Hece, [b eq. ] log t log t log t 0 t t t t Let

24 r cosθ d sθ r θ d r Thus 7 cos s Sce cos s d cos s 6 6 cos s 6 6 d cos s. 59. Hece the result. b c d b c d Also b c d c b d / 60. cos s e 9 / 9 9 e. e 9 cos s 9 9 b ; 0 9 b. 6. d Sce cos s, / 0 6. t 5 t 5 5 log 5 5 Im t log. log log.

25 6., s cos θ θ θ e the θ θ θ s cos e cosθ. 6. b whch c be wrtte s cos s 65. b We hve 66. b B ddg o both the sdes of b we get, b },{ b Ths equto wll represet crcle wth cetre, f b e b > >.. 0, sce b represets pot crcle ol. 67. We hve k k k... k k k k k < k les wth. 68. Let r be the crcum rdus of the equlterl trgle d ω the cube root of ut. 69. dstdrd problem 70. d / e s cos. 7. b 8 8

26 [ ] Whch s ellpse. 7. We hve, Therefore the pot hvg ff s equdstt from the four pots hvg ffes,,,. Thus s the ff of ether the cetre of crcle or the pot of tersecto of dgols of squre or rectgle. Therefore,,, re ether cocclc or vertces of squre. Hece,,, re cocclc. 7. b Let ; d If A deotes the re of the trgle formed b, d, the A Applg trsformto R R R R, we get A b Puttg [ ] 6 0. Whch s the equto of crcle Are of the trgle We hve 6 6

27 We hve,, where 0 d the org 0 form equlterl trgle e., strght le through the org b L.H.S. cos cos φ / s φ / cos φ / φ / s φ / cos φ / cos φ / s φ / cos φ / s φ / φ / e φ / e e φ cos φ s φ. 8. cos s cos s /. cos cos cos. 8. cos / 8 s / 8 cos / 8 s / 8 8 cos cos [cos [cos /6 s /6 cos /6 /6 s /6 cos /6 8 /6 s /6 ] /6 s /6] 8 cos s 6 6 [cos /6 s /6] 8 cos s 6 6 cos 6 s6 cos

28 8. ω ω ω ω 6 ω ω ω ω ω 0 8. We hve cosθ sθcos θ s θ... cos θ s θ cos θ θ θ... θ s θ θ. θ cos θ s θ cos θ d s θ o o 5 cos 50 s50 o 5 [cos50 5 s50 5] 5 [cos 0 o s 0 o ] 5 [cos 0 o s 0 o ] 5. o cos α sα s β cos β cos α s α 5 5 cos β s β 5 cos α s αcos β s β [cos α s α] [cos 5β s 5β] [cos α 5β sα 5β] 87. sθ cos θ sθ cosθ cos α sα cos α sα cos cos α α α s cos α α α s cos α α cos s α α cos s 88. b Gve tht cos s / / [cos s ]. Sce the epresso hs ol dfferet roots, therefore o puttg 0,,, cos s d multplg them, We get cos s cos s cos s cos s

29 89. cos θ cosθ 0 cos θ ± sθ cos θ ± sθ cos θ ± sθ cos θ sθ cos θ s θ b Let cos s d cos s 0 0 Therefore, stdrd problem 9. d If ω s comple cube root of ut the ω d ω ω 0 ω ω ω ω ω ω b ω ω ω ω 5 5 ω ω ω ω ω ω ω ω 9. ω ω ω ω ω ω ω ω ω, therefore 95. b Multplg the umertor d deomtor b ω d ω respectvel I d II epressos bω cω bω cω ω bω cω ω bω cω ω ω b cω ω c ω bω bω cω cω ω th 96. d The k roots of ut re gve b e k, k,..., k e for ll k,,..., k k k for ll k,...,

30 ω 0 0 ω d As ω d ω ω 0 0 ω ω 8. ω ω 9. ω ω ω 99. d 0 ω or ω For ω, ω ω 00 ω ω ω ω For ω , ω ω ω ω ω ω. ω ω Or 0 so ts rel roots re d. 0. d 5 ω 8ω ω 8ω,,, 5 5ω 5ω ω ω b s sh. 0. ω ω ω ω ω ω ω 0 ω [ ω ω ω ]

The z-transform. LTI System description. Prof. Siripong Potisuk

The z-transform. LTI System description. Prof. Siripong Potisuk The -Trsform Prof. Srpog Potsuk LTI System descrpto Prevous bss fucto: ut smple or DT mpulse The put sequece s represeted s ler combto of shfted DT mpulses. The respose s gve by covoluto sum of the put