I Vay CO hai can bac hai (dang dai so) cu aw la:
1 v aM la diem bieu dien ciia so
phuc z, ta co: O M = O I + I M
I M = 1 va (I X , I M ) = 2a + k2n Vay tap hop cac diem M la duong tron tarn I ban kinh R = 1.
2
O
M
•2a
L3:a) z,= ( l i i H I i i l . O l O l . i b) z,= (2i ^ Dd - 00 ^ 3i) ^ i ' ( l - i ) ( l + i) ' ( l - i ) ( l + i) ^c) z , = V2 (l + i ) ( l - 3 1 ) ( l - i ) ( l + 3i) 2 3n . . 3n c o s — + ism — 4 4 = - l + i n '6 n '6' / \ 71 71 cos + isin — — I 3 ; 3 j STI . . 571 47: 47t
, z , = cos — + isin — ; z. = c o s — + ism —
^ 6 6 " 3 3
C '/y TNHH MTV D VVH Khang ViC-i S = Z| + 7,2 + Z 3 + Z^ = i - i N/3 = (1 - V3 )i P = Z,Z2Z3Z,= 1 z^ = l ( c o s 7 : + isin TT) = - 1 b) Z 3 = l(cos3n + isin571 = - 1 Zj = 1(CO(-27T) + i sin(-27t)) = 1 z'l = l(cos87t + isinSTX = 1
Bai 5: Ta c6 + b^ = (a + ib)(a - ib), do do:
P = [(x2 - X + 1) + i(x2 - 2x + 3)] . [(x2 - X + 1) - i(x2 - 2x + 3)] Thua so thiV 1 c6 hai nghiem: x = 1 - i va x =
l - 3 i
Thua so thti 2 co hai nghiem: x = 1 + i va x = , do do: P = 2( x - 1 + i ) ( x - l - i ) x - • 1 + 3 i X - • l - 3 i P = 2 I ( x - 1)= X — 1 V 2 , 9 + — 4 = (x2 - 2x + 3)(2x2 - 2x + 5)
Bai 6: Ta co: ăl + i) la nghiem ciia phuong trinh nen:
(1 + i)[aXl + i)^ - 2a2(l + i)(cosa + sina + isina) ,
+ ăl - 4sinacosa + 4isinacosa) - 2sina] = 0 D o n g nhat 0 ca phan thyc va phan ao ta co he:
-2a" cosa + ă1 + 4 s i n a c o s a ) - 2 s i n a = 0 (1) 2a a " - a ( c o s a + 2 s i n a ) + 2 s i n a c o s a = 0 (2)
Giai p h u o n g trinh (2) ta co nghiem: a = 0; a = cosa, a = 2sina
Chi C O a = 2sina thoa man phuong trinh (1), do do zi = 2(1 + i)sina la 1
nghiem, p h u o n g trinh tro thanh: [z - 2(1 + i)sina] {z? - 2cosaz + 1] = 0.
Giai z^ - 2cosa + z + 1 = 0
Ta C O nghiem zi = C O S T: + isinn, z.i = cosTt - isinTt
Goi M l la diem bieu dien ciia zi = 2(1 + i)sin7t, khi n thay d o i tren [0, 27t] thi
M l d i dong tren doan AB voi A(2, 2), va B(-2, -2).
Goi M2, M l Ian lugt la diem bieu dien cua Z2 va Z3 thi tap hop cac diem b\ê
dien M2, M 3 la d u o n g tron tarn O ban kinh R = 1.
Cam nang hiyen I hi DH - Nguyen ham - Tich phan - S6 phírc - Tran Bd Ha
iiJT: a) A = (x2 + 1 )2 - x2 = (x2 + X + 1) (x2 - x + 1) ')B = (x2 +1)2-2x2(1 - c o s a ) = ( x 2- l) 2 - 4 x 2 s i n y
B = (x2 + 2 x s i n - + I) ( x 2- 2 x s i n - +1) ' ' • C = (x2 + l) 2 - 2 x 2( l + c o s a ) = (x2 + l) 2 - 4 x 2 c o s 2 j
C = (x2 + 2xcos I + 1 )(x2 - 2xcos I + 1)
ki 8: a) Ta co: 1 + ia va 1 - ia la hai so phuc lien hop nen co cung m o d u n do do ti so i J l i i CO m o d u n b5ng 1 (phep chia so phuc dang l u g n g giac).
1 — ia
) Ta CO moi so phuc co m o d u n bang 1 deu co the viet dang: z = coscp + isincp (1) ^ , 1 + ia (1 + ia)^ l - a ^ . 2a
Ta co: = i— = + 1
1 - i a 1+â 1 + á 1+â r^-. , (t) 1-â . 2 a
Dat a = tan—=> - = cos^, ;- = sm(j)
2 1 + a l + a"
(2) Do do: z = coscp + isin(p = 1 + ia
1 - i a
I acgumen cua z xac d j n h boi: ^ = arctana + k2Tt
>ai 9: Theo cong thuc M o i v e ta co: (cos5x + isinSx) = (cosx + isinx)^ = cos^x +
+ Sicos-'xsinx - 10cos^xsin2x - 10icos2xsin''x + Scosxsin^x + isin'^x Dong nhat phan thuc va phan ao 6 hai ve ta co:
cos5x = cos^'x - 10cos^xsin2x + 5cosxsin''x = 16cos''x - 20cos^x + 5cosx
sinSx = 16sin'^x - 20sin^x + 5sin x
K h i X = — thi sinSx = 0 => u = sin — la nghiem cua p h u o n g trinh:
16x^ - 20x-^ + 5x = 0 c:> x l l 6x^ - 20x2 + 5] = 0
K h i cos5x = 0 =:> v = cos^^ la nghiem cua p h u o n g trinh: 16x^ - 20x^ = 5x = 0 , tuong t u ta co: ' f • * - ; '
71 7^0 + 275 . 71 „ . 7 1 Vl0-2>y5 '
cos — = ; sin — > 0 => sm — =
10 4 5 5 4
Cty TNHH MTV D VVH KhangVi^,
So P la tích cua ba s o p h u c c6 m o d u n bang 1 va acgumen Ian lugt a, b, c nê
P CO m o d u n bang 1 va acgumen la (a + b + c)
D o do P = cos(a + b + c) + isin(a + b + c).
D o n g nhat phan thyc va phan ao, ta co: cos(a + b + c) la phan thuc va
sin(a + b + c) la phan aọ .,
Bai 11: a) Giai he ta co: cos (p = , sm{t) =
2 2i , u " =cosn(j) + isin(j) u " + b) Ta co: cos nij) =
v " =cosn())-isinn(j) 2 , s i n n(t) = u " - v " c) cos"* (j) = sin"* (j) = cos^ (j) = ' U + V ^ , ..-5 V 2 , u" + v + 3 u v ( u + v) cos3(1) + 3 c o s * , , V T T ( v i u . v = l ) 8 4 ^
u - v ^ l ^ u"^ - v ' - 3 u v ( u - v) . 1^ - sin 3(1) + 3 sin d)
sm' (j) = V 2i , 8i^ u + V ..S , ,,5 V 2 , _ u^ + v^ +5uv(u^ + v^) + 10u-'v^(u + v ) 32 . 2 . , 2 ,
5 cos 5^ + 5 cos 3(j) + 10 cos (j)
n > cos (1) - —
, • c sin5(t)-5sin3(t) + 10sin(i)
T u o n g t u : sm^cp =
16
Bai 12: T) Ta co V = cosAn + isin47t = -1 -'\~ = ] (lien hgp)
= 1 do do: Neu n = 3P thi J" = 1; n = 3P + 1 thi J" = J; n = 3P + 2 thi J" = f
2) 1 +J = c o s - + i s i n P i = - P = 5 l +J + J2 = 0
Sn = 1 + J + P + ... + J" = (J2 + J + l)n 2 + (J2 + J + 1)"-? + .
vi P + J + 1 nen: Neu n = 3P thi Sn = 1 ; n = 3P + 1 thi S,, = 1 + J = -f
n = 3P + 2 thi Sn = 0
3) Neu n = 3k thi J" = J^" = l ; l + Jn + j-<n = 3
Neu u = 3 k ± 1 thi 1 + J" + J2" = 0
4) A p d u n g khai trien N e w t o n (1 + Ja)" = 1 + C\]a +... + Cf^J^'ấ +... + Cl]"a"
(1 + J'a)" = 1 + c;j-a +... +C,i;'j''^â^ +... + c;;j-"a"
5) V o i n = 100, a = 1 , ta co: (1 + ])"'» = f va (1 + }2)iw' = J, ta co S = 1 ( 2 " » + + J)
3
= > S = i[ 2 ' » " - 1 ]
Cdm nang luyen thi DH - Nguyen ham - Tich phan - So phíic - Trdn Bd Ha
Bai 13: Ta co: 1 + i = V2 (cos - + isin - ) = > ( ! + i ) " = (-Jl )"(cosn - + isinn - )
4 4 4 4
(1 - i ) " = ( V2 )"(cosn ^ - isinn ^ )
Theo khai trien N e w t o n ta co: (1 + i)" = 1 + C^^i + C^j' + C^i^ + ... + Cj^i" ( l - i ) - l - C ; , i + C ; i ^ - C ^ J ^ + . . . + ( - l ) " C ^ i " V i ( l +i)"va (1 - i ) " l a s o p h u c l i e n h g p n e n : (>/2)"oo6n^=l-Cf^+C;',-. .+(-l)"CfP+... V o i n = 100, ta co Suw =-2-^" x + i y + 1 ' x - i y + 1 (x + l ) ^ + y ^ = k ^ ( x - l ) 2 + y2 = k2[(x + l)2 + y2] — x + i y- 1 x - y i- 1 ( x - l ) ^ + y^ Bai 14: a) Ta co: w . w = ^ ^ ~ Do do: w <::> x2(l - k^) + y 2 ( l + k^) - 2x(l + k^) + 1 - k^ = 0 (1) Neil k = 1 thi (1) ^ X = 0: Tap hgp cac diem M la true O y
Neu k;^ 1 thi (1) » x^ + y2 - 2 1 + k
1-k^
2 ^
X +1 = 0 : Tap hgp cac diem M la d u o n g tron tarn I 1 + k^
1 - k ^ ,0 , R = 2 k
b) Ta CO I z - 1 I = M I va i z + 1 I = MJ do do I w I = M I MJ
w = k <r> M I = KMJ: tap hgp cac diem M la d u o n g tron (k 1) hay duong
t h ^ n g l j ( k h i k = l )
Bai 15: a) Theo v i - e t cac so phuc phai t i m la nghiem cua p h u o n g t i i n h :
Z2 + Z + 1 = 0 (Z + - )2 = - 4i2
2 4
Do do cac nghiem la:
1 s. z + - = — 1 z + - = — 1 2 2 1 N/3. z + - = 1 2 2 1 S z = — + — 2 2 1 N/3 . z = 1 2 2 b) Dat z' = 1 + z ta co phuong trinh tro thanh:
z" + Iz' + z'2 + 2 = 0 <:> (z'^ + l).(z'2 + 2) = 0 (1)
Neu z'o la nghiem thi -z'o cung la nghiem cua phuong trinh do do cac diem
bieu dien doi x u n g qua I .
72
Giai (1): z'2 + 2 = 0 « z' = ±i N/2 ; z'4 + 1 = 0 z' = ± — (1 ± i)
Cty TNHHMTVDVVHKhanfi Viet
Bai 16: A p dung khai trien Newton (1 + I)" = C" + iC', + i^C^ + + i " C "
= C - •< - C ; -iC;^ .c:, -iCf, = ( l - C ^ < , -C^; ....).i(c^ -C-^
Theo cong thuc Moive: (1 + i)" = yfl" (cosn - + isinn - ) 4 4 Dong nhat phan thuc va phan ao voi n = 2010, ta c6:
"^2010 "-MIO +'-20K) ^2010 + - - ^ 2 0 1 0 +^2010 = -Sin ^2"*'^ Bai 17: Ta co: ( l + iSJ" = 2^" c o s 2 n - + isin2n —
3 3
Khai trien Newton ta c6: U + isJsf" =Y,^2ní^^t
k=()
= ( C - 3 C L - 2 7 C ; +... + (-3)"Q^„") +i(7ic^^ -37ic^„ +...^-3rC^-')
Dong nhat ta c6: S = 2^" cos
,2n 2nn 3ai 18: Ta c6: 1 - ^ 2n cos I I 6 + isin I 6 1!! 3" cos + isin 3 3 2n , / r, 2 n 1 - = T = ~ , v 6 i T = C" -ic^ +lct - V3 2n 3 2n ^ ^ 2 n "
Dong nhát phan ao ta dirge: S = Vs
v3y sin-nn
'ai 19: a) Phuong trinh tuong duong: z(z^ + l)(z - 2i) - (z^ + 1) = o
o (z2 + l ) ( z 2 - 2z.i - 1) = 0 z2 + 1 = 0 hay z2 - 2zi - 1 = 0
. Z ^ + 1 = 0 <=> Z2 = i2 <=> z = ±i
. z 2- 2 i z- l = 0 c 6 A = 0 r : > z = — = i
2
Vay p h u o n g trinh c6 nghif m z = ±i
)Z2 = [(N/6 + SY-{42 - V6)2] + 2(72 + V6)(V2 - ^/6)i
z 2 = 8>/3-8i = ] 6 ^/i 1.1
2 2 = 16 cos + isin
MUC L U C
Phan 1: Nguyen ham-tich phan va ling dung
Chuong 1: Nguyen ham 3
Chuycn del: Khai niem ngui/cn ham 4 Chiiyen del: Phuvng phdp thn nguycn ham 16
Chuycn de3: Nguyen ham cua cac ham soca ban 26
Chuong 2: Tich phan 60
Chui/cn del: Klidi niem tich phan 60 Chuycn del: Phuvng phdp tinh tich phan 87
Chuong 3: iTng dung cua tich phan 168
Chuycn difl: Dim tich hinh phdng 168 Chuycn del: Úng dung tich phan delinh the tich cua vat thẹ 189
Chuong 4: Cac van de lien quan phep tinh tich phan 210
Chuycn del: Tich phan lien ket 110 Chuycn del: Phuang trinh hephuvng trinh tich phan 219
Chuycn dc3: Tich phan quy nap giai hqn cua tich phan 225
Chuycn de4: Bat đng thuc tich phan 23S Chuycn de5: Tich phdn ham sohuu ti 251 Chuycn de6: Tich phan ham sovo ti 259 Chuycn de7: Tich phdn cac ham soluvnggidc 266 Chuycn de8: Tich phdn cUa ham so mil ham aologarit 273
Chuycn de9: Bin tap tong hap - bai tap luyen thi 281
Phan 2: Sophuc
Chuycn del: Khai nicni sophuc -phep tinh sophuc 30S Chui/cn de2: Can bdc hai cua sophuc phuang trinh bac hai 320
Chuycn de3: Dang hmiggidccua sophucz = a + bi 327 Chuycn di}'4: Bai tap long hap ve sophuc ; ^35