4 3 Bai 13. Giai cac phuong trinh sau
Bai 16. Giai cac phuong trinh sau
1. l + 3tanx = 2sin2x 2. cot x - tan x + 4 sin 2x = sin2x Giai
1. Dieu kien: cos x 9t 0 <=> x — + k7t
Phuong trinh ô 1 + 3 ^'"^ = 4 sinx cosx <=> cosx + 3sinx - 4sinxcos^ x . cosx Day la phuong trinh dang cap bac ba nen ta chia hai ve ciia phuong trinh cho cos^ X (do cos x ^ 0 ), ta dugc phuong trinh:
1 tan — + 3 — = 4tanx <=> 1 + tan^ x + 3tanx(l + tan^ x) = 4tanx X '' ' ' '
cos X cos X I '
oStan'' X + tan^ x - tanx +1 = 0<=> tanx = -1 <=> x = -— + kTi thoa man dieu kien . 4
* Nhan xet: De giai phuang trinh nay ngay tu dau ta c6 the chia hai ve cua phuong trinh cho cos^ x hoac su dung cong thuc:
. ^ 2 sinx cosx 2 tanx sin2x = • sin^x+cos^x 1 + tan^x
phuong trinh chi chua ham tan nhu tren.
2. Dieu kifn: sin2x 5^ 0 x ^ k^ ''' ' ^ r>u - u cos X sin X ^ . - 1 1 huong trinh <=> — + 4 sin 2x = —
va chuyen phuong trinh ban dau ve
1 X':}::' sTi));'; 'i(ji)h;rj.,;.r ;
Sinx cosx sinx cosx , O cos^ X - sin^ X + 4 sin 2x. sin X cos X = 1
o cos 2x + 2sin^ 2x - 1 = 0 <=> 2cos^ 2x - cos2x - 1 = 0
ocos2x = - ^ (do sin2x^0<=>cos2x^±l )<=>x = ± ^ + k7i, k e Z .
Phdn loai va phtfangphdpgiai Dai so-Giai tick 11
* CM v:
Ta can l u u y den cong thirc: tan x + cot x = va cot x - tan x 2cot2x . sm2x
M < '
Bai 17. Giai cac p h u o n g trinh sau:
1-2>/2(sin2x + cos2x) , n 1. ^ ^ = 6 t a n 2( x - - )
sin4x 8, 2. 3cot^x + 2 V 2 s i n ^ x = (2 + 3\/2)cosx
3. 2sin2x - cos2x = 7 s i n x + 2cosx - 4 . Giai sin4x 0
1. Dieu kien:
cos(x —
'It'
"3 !f..'l'>,'} ?
Phuong trinh o — ^ = 6tan ( x - - )
sin4x 8
7t ^ , l - 4 c o s 2 t 2 . l - 4 c o s 2 t , l - c o s 2 t Dat t = x — ta co: = 6tan t <=> = 6
8 . .n cos4t l + cos2t s i n ( ^ + 4 t )
<=> (1 - 4cos2t)(l + cos2t) = 6(1 -cos2t)(2cos^ 2t - 1 )
<=> 12cos^2t- 16cos^ 2t-9cos2t + 7 = 0 o ( 2 c o s 2 t -l)(6cos^ 2t -5co,s2t - 7 ) = 0 cos2t=-
2 <
6oos^2t-5oos2t-7=0
oos2t=- 2 oos2t = 5 - ^
12
t = ± - + k7T
6
^ ^ 1 5-7T93 , t = ±-arccos + kn 2 12
; k e Z .
'S.I
Ket hop dieu kien ta c6:
X = - ± - + k7t
8 6
7t 1 5-N/i93 , X = - ± -arccos + k7t
8 2 12
la nghiem ciia phuong trinh
2. Dieu kien: x^kn
P h u o n g t r i n h ô . ^ ^ 2 ^ - ^ + 272 sin^x = (2 +3\/2)cosx . , sin X
< b 3cos^ X - 3\/2 sin^ x.cosx + 2\/2 sin"* x - 2 s i n ^ xcosx =tl"°
<=> (cosx - \/2sin^ x)(3cosx - 2sin^ x) = 0
<=> V2ccs^x + cosx-\/2=0 2co6" X + 3cosx-2 = 0
<=>
cosx = -- T + N / 3
^2 cosx= —
2
\[6-\l2 x = ±arccos + k27r
; k e Z . X = ± - + i<27t
3 3. <=> 4 s i n x c o s x - 1 + 2sin x - 7 s i n x - 2 c o s x + 4 = 0
<=> 2cos x(2 sin x - 1 ) + (2 sin x - l)(sin x - 3) = 0 '
<=>(2sinx-l)(2cosx + s i n x - 3 ) = 0 <=>
s i n x = -1 2
X = - + k27t
6
X = — + k27i
6
; k e Z .
2cosx+sinx-3=0
(Ltm y: l a s i n x + bcosx I < Va^ + => 2cosx + sin x < >/5 < 3 ).
Bai 18. Giai cac p h u o n g trinh sau:
2. cotx + tanx = sinx + cosx
^ sin x + cos X 1 _ 1
1. = - c o t 2 x
5sin2x 2 8sin2x
3. 3 s i n x + 2cosx = 2 + 3 t a n X 4. sin (3n x ^ 1 . 71 3x
— = —sin 1 [\0 2j 2 l l O 2 J
1. Dieu kien: sin2x5!:0
Giai
Ta C O sin"* x + cos'* x = |sin^ x + cos^ x j - 2sin^ cos^ x = 1 - - s i n ^ 2x
9 5
Nen p h u o n g trinh c6 dang 2 - sin x = 5cos2x - -
Hay 4 c o s ^ 2 x - 2 0 c o s 2 x + 9 = 0, suy ra cos2x = - (k)ai), cos2x = i Vay x = ± — + k n v a i k e Z .
6
2. Dieu kien: sin x ^ 0, cos x ^ 0
I , , . *, , . , , cosx sin X
K h i d o p h u o n g trinh o — = sin x + cos x
A
<=>|cos^ x - s i n ^ X
sm X cos X
= sin X cos X (sin X + cos x)
o (cos X + sin x)(cos x - sin x cos x - s i n x ) = 0 cosx + sinx = 0
cosx - sin x cosx - sin x = 0
tan X = -1 ; sin^ X + cos^ X + 2cosx - 2sin x - 2sin xcosx + (1 - \/2)(1 + -Jl) = 0
7t , , ,
X = — + KTt • -.m
4
| c o s x - s i n x + ! ^y2 j cosx - sin x + 1 + \/2 = 0 •
X = + i<7t
4
cos X - sin X +1 - V2 j = 0
cos X - sin X + 1 + ^\ 0
X = - — + i<TC
4 sin X 71
V 4 , 1 - V 2
X = + i<7t X = — + arcsin 4
4
X = —71 + arcsin 4
+ 2kn keZ .
+ 2 k 7 r
3. Dieu kien: cosx ^ 0
Phuong trinh o 3sinx + 2cosx = 2cosx + 3sinx
o ( 3 s i n x + cosx)
tan X = — 3 0 cos X = 1
C O S X
C O S X
X = arctan
x = 2 k 7 t
= 0 o
3sinx + C O S X = 0 1
cosx = 0
+ kTT
keZ .
i | i i | .7.
4. Dat: x = — rr + u . K h i do p h u o n g trinh tro thanh: sin u i .
— 2j = - s m 2 - + —
U 2)
1 1 V3 . V3 .
<=> - cos 2 l 2 y — = — cos 2 [2j — + — s i n 2 l 2 y — 0 — s i n 2 V2J = 0 <=> u = 2 k 7 i
4 4
<=> x 71 = 2 k 7 i <=> x = — 7 1 + 2 k 7 i , k e Z .
15 15 Bai 19. Giai cac p h u o n g trinh sau:
1.2cosx + tanx = 1 + 2sin2x ,; ;>
2. cosx - 2cos3x = 1 + .sinx 46 . • • :
3. sin x + s i n X + — + sin4x = sin
/
2x 71
I 3 j ~ 3 j
4. cos4x + sin2x
cos3x + sin3x = 2N/2sin X + —
V 4 ,
1. 2cosx + tanx = 1 + 2sin2x Dieu kien: cosx ^ 0
+ 3 Giai
s i n x
Chi do p h u o n g trinh <=> 2 cos x + = 1 + 4 sin x cos x . cosx
<=> 2cos^ X + sin X = cosx + 4sinxcos^ x <=> cosx(2cosx - 1 ) = sinx^4cos^ ^
<r> (2cosx - l ) ^ c o s x - sin x(2cosx +1)) = 0 <=>
1 ••, , . • ^
2cosx - 1 = 0 ,,, J sin X - cos X + 2 sin X cos x = 0 cos x =
(cos x - sin x)^ + (cos x - sin x) - 1 = 0 x = - + 2m7t
3
x = — + 2 m 7 r
3 cosx-sinx = cosx-sinx =
V 5 - 1 2 - ^ - 1
<=>
x = - + 2 m 7 i
3
7C „ x = i-2m7r
f 71^3 ^/5-l
COS X + - = 1^
A) 2V2
^ 7t^ - V 5 - 1
COS X + -
4 2^/2
X = - + 2 m 7 t
3
x = - - + 2 r r O T
3 x = arccosx
x = arccosx
^ V 5 - l ^
V 2V2 , - - + 2 n 7 i
4
V 2V2 , - ^ + 2 m c 2. cosx - 2cos3x = 1 + \/3.sinx
Phuang trinh cosx sin x = 2cos3x + 1 <=> 2cos X + — 71
V 3 , = l + 2cos3x Vai t = X + — t h i p h u a n g trinh tro thanh: , |
3
2cost = l - 2 c o s 3 t c i > 2 c o s t + 2cos3t = l "
Vai u = cosu t h i t a c o : 2u + 2(4u^ - 3u) = 1 o (2u + l ) ( 4 u ^ - 2 u - l ) = 0 47
u = 2 u =
<=>
t = ± - 7 i + 2 k n
t = ± | 7 r +2kTc ; k e Z .
t = ± - 7 l + 2k7I
; 5 Vay nghiem cua p h u o n g trinh da cho:.
n 4 14
X = 71 + 2 k n , X = — + 2 k 7 r , x = — n + 2kn, x = rr + 2 k 7 r ,
3 5 15 2
X = 71 + 2 k 7 r
15
X = T : + 2 k 7 r
15
k e Z .
3, sin x + s i n X + - + sin4x = sin 2x — 3 Phuong trinh o sin x + sin
<=> 2 sin
71 X + —
3J + sin4x - sin 2x — 3 = 0
r 7t> 7t^
x + — cos — + sin X + — cos 3x
V 6j k6v 6) - 0
<=> sin
<=> sin
7t X + —
6 cos 3 x - -
6, + cos = 0
71
Sin cos
X +
n
X + —
V 6J
M x ^
C O S
2 ;
rsx 7t^
I 2 6J 2 C O S
= 0
= 0 - 0
3xA
V 2 J C O S ' 3 X _ 7 I
. 2 6
7t
= 0
X = — + k7t
6
x = - + - k T C , k e Z . 3 3
2,
X = 7C + - k7I
3 4. cos4x + sin2x = 272 sin X + — 7t
cos3x + sin3x 4
Dieu kien: cos 3x + s in3x ^ 0 Ta c6: cos4x + sin2x = cos4x + cos
+ 3
2x — 2 = 2 cos cos 3x + s in3x = cos 3x + cos 3x —
3x — cos X + — 7t 4 j
2J ••••;.5 j ; , ) ' /
CtyTNHH MTV DWH Khans Viet"
Khi do p h u o n g trinh tro thanh: 2cos^ X + — TC = 272 s i n
/ \ 7t X + —
I 4 , I 4 J
<r>2sin^ ^ 71 X + —
V 4/ + 272 sin X + — 71
V 4/ + 1 = O o s i n x + - 4j
V2
+ 3 X = — + 2k7i
2 k e Z .
X = 71 + 2k7:
Bai 20. Giai cac p h u o n g trinh sau: ' 1
1. sin^ x.cos3x + cos^ xsin3x = — ,>
4
2. 2sin2x + (273 - 3)sin x + (2 - 373)cosx = 6 - ^3 3. sin^" x + cos'° x sin^ x + cos^ x
4cos^ 2x + sin^ 2x 4. cos3x + 72-cos^ 3x = 2^1 + sin^ 2x
Giai 1 A p d u n g cong thuc ha bac ta c6:
Phuong trinh ci> (?, . 1 . ^
—sin X — sin 3x cos3x +
^4 4 - c o s x + — cos3x
4 4 sin3x = —
4
<=> —(sinxcos3x + sin3xcosx) = --<=> sin4x = - 1 3 3
Vay ta c6 duoc x = - — + —, ke Z la nghiem cua p h u o n g trinh da cho.
8 2
2. Phuong trinh <=> 2sin2x + ^273 - sjsinx + ^2 - 373)cosx = 6 - 73
•ằ 2sin 2x + 2173 sin x + cos X j - 3^sin X + 73 cos x j = 6 - 73
<=>2sin2x + 4sin
771
^ 7t^
x + —
6 - 6 s i n X + —
3 = 6- 7 3 Dat x = t + — . Taco: 2sin
6 2t + - 7 : ^
3 + 4sin t + - 7 t
6 - 6sin t + 971 = 6 - 7 3
<=> sin2t + 7 3 c o s 2 t - 2 s i n t - 2 7 3 c o s t + 6 c o s t - 6 + 73 = 0 o s i n 2 t - 2 s i n t + 6 ( c o s t - l ) + 7 3( c o s 2 t - 2 c o s t + l ) = 0
<=> 2sin t(cost -1 ) + 6(cost -1 ) + 73|2cos^ t - 2 c o stj = 0
'cost = l •^"•••^
sint + 7 3 c o s t = - 3 (vn)
< = > ( c o s t - l ) ( 2 s i n t + 2N/^cost + 6] = 0 o ô t = k27t
Suy ra nghiem cua p h u o n g trinh: x = — + k27t.
6
• :•• . (K' • 49
3. D i e u kien: 4cos2 2x + sin^ 2x ?i: 0 Vx.
Ta c6: sin^ x + cos^ x = 1 - - s i n ^ 2x = - ( 4 - 3sin^ 2 x )
4 " 4V / Suy ra sin^ x + cos^ x 1
= —. N e n p h u o n g t r i n h <:> sin^'' x + cos^° x = 1 (*) 4cos^2x + sin^2x 4
,10,
Do • 0 < sin'' x < 1 0 < cos^ x < 1
N e n p h u o n g t r i n h ằ
sin^" x < sin^ x . I Q 10 • 2 2
=> sin x + cos x < sin x + cos x = 1 10 2
cos X < cos x •
s i n ' " X = sin^ x cos^°x = cos^x
sinx = 1 cosx = 0 sinx = 0 cosx = 1 4 . A p d u n g b d t a + b < /2(a^+b^^ ta co:
= 2
<=> X = k 7 t , k e Z . ;
cos3x + V2-cos^ 3x < ^2^cos^ 3x + 2 - c o s ^ 3x
Ma 2 f l + s i n 2 2 x ) > 2 nen phuong trinh J c o s 3 x= 7 2 ^ ^ ^jcx>63x=l
o i n^ Y^ n sinxcosx=0
V i cosx = 0 => cos3x = 0 ; sinx = 0 :
sin2x=0
cos X = 1 => cos 3x = 1 cos X = - 1 => cos 3x = - 1 D o do (*) o cosx = 1 <=> X = k27i, k e Z .
Bai 21. Giai cac p h u o n g trinh sau:
1. 1+ sin^ X cosx + ^1+ cos^ x j s i n x = 1+ sin2x
.r"\ ,j;
2. sin —+ cos— X X
x 2 + V S c o s x = 2
- sin2x + 2 c o s x - s i n x - l „
3. 7 = = 0 4.
sin 3x + -~
4J
t a n x sinx + cosx = N/2cot x + -
, ' . ^ * j H W w Giai
1. P h u o n g t r i n h <=>(sinx + cosx)(sinx.cosx + l ) - ( s i n x + cosx) = 0
<=> (sinx + cosx)(sinx.cosx + 1 - s i n x - c o s x ) = 0 . ti ' • o (sin X + cos x ) ( l - sin x ) ( l - cos x) = 0
Giai ra ta dugc cac nghiem: x = - - ^ + k 7 i , x = ^ + k 2 T t , x = k 2 7 i , k e Z .
50 ; .
2 Phuong t r i n h <=> 1 + sinx + Vscosx = 2 <=>—sinx + — c o s x = i
<=> cos X — 71
6 = — o 2
X = — + k27i \ x = — + k27r
6
3. Dieu kien: tan x —J3 <=> x ?t - — + k n 3 ;|
Phuong t r i n h ằ sin 2x + 2 cos x - sin x - 1 = 0 • y
<r>(sinx + l ) ( 2 c o s x - l ) = 0 giai p h u o n g t r i n h nay va d o i chieu dieu kien ta dugc nghiem cua p h u o n g t r i n h la: x = — + k 2 K , k e Z .
3 1 i
4. Dieu kien:
sinx +cosx ^0 sin X + — 7t
4
X ^ — + kn
^ 0 4 f
sin P h u o n g t r i n h <=>
3x + cos x + •
V 4 sin 3x + - 2 cos sin X + cosx
sin X + — 4
X + - V 4 , sin x + c o s x cos x + s i n x
sin 3x + —
4 = 2cos sin
^ 7 1 ^ X + —
V 4 , . Dat t = X + — ta CO p h u o n g t r i n h j ^ ' i- H " ?
3 t - ^
2 = 2cos t o - cos3t = 2cos t <=> 4cos^ t - 5cos t = 0
<=> COS t - 0 < = > t = — + k 7 i= > x = — + k7r
Bai 22. Giai cac p h u o n g trinh sau:
V l + C O S X + 7 l - cosx 1. 4 s i n x = •
C O S X
3. sin^ X + —sin^ 3x = s i n x s i n ^ 3x 4
4. ^ 2cos 1 ^ x +
COS xj
Vay nghiem cua p h u o n g t r i n h la: x = — + k::.
2. tan^ x + tan^ y + cot^ ( x + y) = 1
sin^ x + -—\
sin X
= 12 + —siny . 2
51