Giai cdc phiTdng trinh sau:
B&i 12.4cos^2x + 2cos2x + 6 = 4>/3 sin x . Htfdng d i n giai Phifdng trinh da cho ti/dng difdng vdi
(4cos^2x + 4cos2x + 1) + (5 - 2cos2x - 4 > ^ s i n x ) = 0 o (2cos2x + 1)^ + (4sin^x - 4 V 3 s i n x + 3) = 0
o (2cos2x + 1)^ + (2sinx - yf3f = 0 cos2x = - i ^
r- o sm X = — o
73 2 sinx= —
2
x = —+ 2k7i 3
X = — + 2kn 3
j3. 3 + sin^2x = 2sin2x + cos2x + 2-v^sinx.
HifOng d i n giai pjjifcjng trinh da cho trfdng di/dng vdi
(sin^2x - 2sin2x + 1) + (2 - cos2x - 2>/2 sinx ) = 0. x ^ u .,
^ ( s i n 2 x -1 ) ' +( V 2 s i n x- l) ^ = 0 fsin2x = l
1 <::>x = - + 2k7t ( k e Z ) . sinx = —r= 4
72
nii 14. cosx + 3cos3xcos5x = 4 , • j • ^ • - Hi/dng d i n giai
VI |cosx| < 1, |cos3x| < 1, |cos5x| < 1 suy ra ve trdi < 4.
Dau b^ng xay ra khi chi khi ^ _ ^ | cos X = 1
. o c o s x r = l o x = k2n ( k e Z ) . cos3xcos5x = l
Biki 15. (sinx + cosx)" = 5 - sin^2x.
Hifdng d i n giai Ta c6 (sinx + cosx)"* < 4 va 5 - sin^2x ^ 4.^
Vay dau bkng xay ra khi vh chi khi (sin X + cos x)"* = 4 | sin x + cos x sin''2x = l [cos2x = 0 Biil6.sin"'x + c o s " ' x = l .
Hvtdng d i n giai
= yl2 ~ n
o x = —+ k7t ( k e Z ) . 4
Vi sin'" X < sin^ x , cos'^x < cos^x nen s i n ' ^ + cos'^x < 1.
DS'u b i n g xay ra khi va chi khi i hoSc \
cosx = 0 sinx = 0
cos x = l sin X = 0 17.sin''x + c o s " x + l = 0 .
2 o x = — ( k e Z ) . x = kjt ^ .X8o:>^+ Hrfdng d i n giai f;Tutofb Et
c6: - s i n \ s i n \ cos"x < cos^x \ ^ f ^ > , Suyra s i n \ c o s " x > - l haysin''x + cos"x + 1 > 0 .
" a u bang chi xay ra khi v^ chi khi
• I - i' ;•'./ ;
L,UJ^n gua lie UWUL Ay mi UH J mien MMU., maigi ntm ivun nyi. • iigujiKn ran znang
{C O S X = 0 ^ ,
hoac sinx = - l
sinx = 0 cosx = - l
371 ^ ,
x = — + 2kn . „ 2 ( k e Z ) . X = Jl + 2 k j t
B k i 18. 2\/sinx + Vcosx + >/cosx = 2
Hi/()ng dSn giai
T a c 6 2>/sinx = 2^/sin^ x > 2 s i n ^ x , Vcosx + ^cosx = \fcos^x + \/cos^x >2cos^
Suy ra trdi ^ 2 . ^ , Isinx = 1
X,
Dau b^ng chi xay ra khi chi khi: [cosx = 0
i
Jsinx = 0 cosx = 1
x = - + 2k7t ^, ^ 2 ( k e Z ) . x = 2 k j r
B ^ i 19. 2 sinx cosx + cos X = 2
T a c 6 2 sinx > 2 s i n ^ x .
Htfdng dSn giai cosx + cos^ X > 2cos^ X . Suy ra ve trdi ^ 2.
D a u b^ng chi xay ra k h i chi k h i :
B a i 20. 2sin"x - cos'^x = 2.
Jsinx = ±1
cosx = 0 I T o x = k - ( k e Z ) . Jsinx = 0 2
I cosx = ±1
Hifdng d i n giai
PhiTdng trinh da cho ttfdng diTdng vdi 2 s i n " x = 2 + cos'"x.
Ta c6 2 s i n " x ^ 2, 2 + cos'^x ^ 2. /
D a u d^ng thtfc xay ra k h i chi k h i : " ° o x = - + 2kJi (k € Z ) . sinx = l 2
B&i 2 1 . 3Vsinx - Vcosx = 3
5^ HiTidng d3n giai
Phifdng trtnh d5 cho^iTdc vi^t lai \h SVsinx = 3 + Vcosx . Ta c6 3Vsinx ^ 3 , 3 + Vcosx ^ 3
Suy ra dau d i n g thil'c chi xay ra khi va chi k h i :
" ' " " ^ ^ o x = ^ + 2k;i ( k e Z ) . cosx = 0 2
2 +cos X . 10 10
S.S 22. ^— = sin X + cos x
Ta c6: 2 + cos x 1 + s i n ^ X
HiTdng dSn giai
> 1 o 2 + c o s " X ^ 1 + sin^ x <=> 1 + cos'' x > s i n ^ x ( d u n g ) . Ta c6 sin"'x + cos"'x < 1. Suy ra dau d i n g ihtfc xay ra khi chi khi
cos X = 0 j r
- ô > x = - + k j i ( k € Z ) .
sin X = 1 2 . . S i ; ft I,, l i r li • .V
2 + sinx
3 + cosx s m x cosx
H i f d n g d3n giai T a c 6 : 2 + s m x
3 + Ta lai c6
cosx < l o 2 + |sinx < 3 + cosx sinx ^ 1 + c o s x (dung).
smx + < cosx ^ 1
Suy ra dau d i n g thil'c xay ra khi vh chi khi
i i iii'-th. aferti .salt* Jôd sa>
sinx = 1 cosx Bki 24. cot^x + 4tan^x + col^x = 6.
Ht/dngdSngiai
o x = —+ k n ( k e Z ) .
= 0 2
= TV
Ta c6 cot^ X + tan^ x > 2 , ' ^
3tan^x + cot^x = cot'^x + tan^x + tan^x + tan^x ^ 4>/cotSaan'' x = 4 Suyra ve? trdi ^ 6.
Suy ra dau d i n g thtfc xay ra k h i \k chi k h i : lan^x = l o x = — + — ( k e Z ) . 4 2
B&i25. V l + tanx + V c o t x - c o t ^ x = 1 .
Hi/dng dSn giai
Tfif d i ^ u k i ^ n c6 nghia ciia phifdng trtnh suy ra tanx ^ - 1 , 0 < cotx S 1. K h i d6 trdi cua phUdng trtnh Idn hdn hoSc b i n g y/l. V a y phiTdng trtnh v d nghi^m
^ ^ i 2 6 . s i n W x = 1 72
6
K5)
HiTdng dSn giai Ta c6 F = c o s V l - cos^x)^ = l ' ( l - 0^
1
72 .2t.2t.2t(3 - 3t)(3 - 3t) vdi 0 ^ t = cos' ^ 1.
295
A p dyn g ba't d i n g thuTc Co-si cho 5 so hang, suy ra F <
72 \5)
Dau d i n g thtfc \iy ra k h i chl k h i : 2t = 3 - 3 t < = > t = -|<=> cos^x = - . B a i 27. sin'^x + cos'^x = _ 1 _
3 2
Hifdng d§n giai a + b
A p dung bat d i n g thiJc >
2 >
khi va chi k h i a = b, ta c6 V T = (sin^x)'' + (cos'x)' > 2
'• ."••* t,,-,..f . -
v d i a, b ^ 0 va dau d i n g thuTc
sin X + cos 1 1
Da'u b^ng cua bat d i n g ihuTc xay ra khi va chi khi sin^x = cos^x = 2^ 32
khi sin^x
B a i 28.
s i n ' " x c o s ' " x = 6 4 .
Hifdng dSn giai
^ , s i n " ' x + cos"-'x . ,() 10 • 1 sin X cos X
sin^xcos^x <,
f • 2 2 Sin x + cos X _1_
4
2
sin^ xcos^ X
> 4
s i n ' ằ x c o s ' %
> 4 ' .
Suy r a V T > — . 4 ^ = 4 - ^ = 6 4 . . s.
Dau bkng cua bat d i n g thtfc xay ra k h i v^ chi khi sin^x = cos^x = ^ • mat Hi xm am i
(Id Ui^tf:
2
,Mq am iJrs.?
I'
2Qfi
^iuj/Sn M IS:
ckc B A i T O A N T I M £>I6U K I ^ N V 6 T A M O I A C
VA PHLfdNG T R l N H Ll/ONG O l A c C 6 T H A M s 6
J T 6 M T A T L i T H U Y E T
^ H§ thufc li/fling t r o n g t a m gi^c:
pjnh l i h^m so sin: — b = 2R
S ! fIfii'U ;. ; sin A sinB sinC
I, Djnh l i h^m so cos: a^ = b^ + c^ - 2bc.cosA b^ = a^ + c^ - 2ac.cosB
= a^ + b^ - 2abcosC
' 8 0 3 . — mv
2 2 2 c. Dinh h' diTdng trung tuyen = ^ ~^
d. Djnh h' dirdng phan gidc 1^ = •
2bc.cos- b + c
803
— nii' A
e. Dipn tich tam gidc:
1 1 abc
S = 2 ô h a = - ab. sin C = = pr = (p - a)r, = V P ( P - a)(p - b)(p - c) f. Bdn kinh diTdng tr6n npi tiep
A B C
r = ( p - a ) t a n — = ( p - b ) t a n — = ( p - c ) t a n —
2 2 2
... . A ^
— nôl am
a
*".ô>) +I Of.
g. Bdn kinh dirdng tron b^ng tiep: = p . t a n y , , , s''"^^
n. B A I T A P M I N H H Q A
84i 1. Cho tam gidc A B C khong tu thoa man dieu k i e n sau cos2A + 2>y^cosB + 2%/! cosC = 3
Tim so' do 3 goc cua tam giac A B C . '^'"^
Hifdng d i n giai T a c 6 : C O S 2 A + 2 N ^ C O S B + 2 N ^ C O S C - 3 = 0
O cos^ A + 2 V 2 c o s ^ ^ c o s ^ ^ - 2 = 0 2 2
<^ (cos^ A - cos A ) + cos A + l^Jl sin—cos
2 2
f /r . A B - C ^ ' v 2 s m cos
< ^ c o s A ( c o s A - l ) - - s i n - 2 = 0
2 B - C
= 0 (1)
n^ngimd^mot xy im vn 3 mien OUL. ttung, „u.„-^
Do tarn gidc A B C kh6ng tu ntn cosA ^ 0 cosA - 1< 0. K h i d6 VT(1) s 0
^cosA = 0
D^ng thtfc x&y ra k h i vk chi k h i r A B - C [ A = 90"
>/2sin — = cos——— <=>-^
2 2 [ B = C = 45"
B&i 2, Chtfng minh r i n g tam gidc can neu thoa man di^u ki?n A i B . B 3 A
s i n y . c o s ^ - = s m - c o s — HiTdng d i n giai A T B . B 3 A
T a c 6 : s i n — . c o s ^ - = s m - c o s — 0 2
' . A^
s m — 2_ A c o s - ^
1 2 A cos —
2
s m — 2_
B cos—
cos — 2 B
o tan- 1 + tan'' —•
2)
= t a n — B 2
1 + tan^ B 2)
T A 3 B A B „
ô t a n ^ y - t a n ' - + t a n y - t a n - = 0 A
tan t a n —
.„ . 2 2)
A . B o t a n Y = t a n —
B B^
1 + tan^ — + tan'' - + t a n — . t a n - ^ 2 " 2
> rtfjl •
= 0
(do l + t a n 2 y + t a n 2 | + t a n 2 y t a n | > 0 ) A Q H o A = B o tam gidc A B C can tai C. ' '
BJki 3. Churng minh r i n g tam gidc A B C c6 It nhat mOt g6c j k h i chi k h i sinA + sinB + sinC
cosA + cosB + cosC Hifdng d i n giai _ , s i n A + sinB + sinC rr
Ta c6: = >/3 cosA + cosB + cosC
o (sin A - V3 cos A) + (sinB - >/3 COSB) + (sinC - >/3 cosCJ = 0
o s i n + sm + sin = 0
( . A + B 7t> r A- B ]
sm cos + sin C -
I 2 3J I 2 J V 3 . = 0
sm cos-
C _ 7 t
12 6J
A - B
= 0
• = cos 71 A + B^
= cos
12 6J 2 j
£-1=0
2 6
A - B _ 7 t a - b 2 ~ 3 2 A - B 7t A + B
" 2 ~ 3 2
A = ^ ' 3
B = i ^ 3
3
Vay tam gidc A B C c6 it nhat mpt g6c - HI. BAI T A P TV L U Y ^ N C 6 HUldNG D A N
Bai 1 Cho phiTdng trlnh: (cosx + l)(cos2x - mcosx) = sin^x a. Giai phiTdng tnnh (1) k h i m = - 2
2n l>m£-
(1)
b. Tim m de phtfdng trinh (1) c6 ddng 2 nghipm tren HiTdng d i n giai
0;-
a. X = 7t + k2n (k 6 Z )
b. (1) o (cosx + l)(2cos^x - 1 - m) = 0
X6l ham so f(t) = 2t^ - 1 - m tren = > - ! < m ^ - 2
.1)
B i i 2. Cho phi/dng trinh: 4cos*x.sin'x.cosx = sin^4x + m (1)
a- Bie't r i n g x = n la nghi^m cua (1). H l y giai (1) trong trtfdng hdp 66
Cho biet \ ~ l h nghi?m cua (1). Hay t i m tSft ca cdc nghi?m ciia (1) th6a x' ' - 3 x 2 + 2 < 0
Htfdng d i n giai (1) o sin^4x - sin4x + m = 0 (2)
X = Tt la nghi^m =>m = 0 ' * ^ TTiay m = 0 vao (2), ta dtfdc x = k ^ ; x = - ^ + - ^
x''-3x^ + 2 < 0 o - > ^ < x < - l h o ? c l < x < V2
( k e Z ) .
x = — = > m = - 2 8
Luy?n gidi dS trudc f^lWEflTTimmTJac, irong, num luammc-Tfgayen ran immg
Bai 3. Cho phiTdng trinh: cos^x - sin\ m (1) a. Giai phiTdng trinh (1) khi m = 1.
b. Tim m sao cho phifdng trtnh (1) c6 diing hai nghi?m x e
... ' j I
"4'4 Hi^dng din giai
a. x = k27i; x = - - + k2n ( k e Z ) b. Dat t = cosx - sinx
PhiTdng trinh (1) trd thanh 3t - - 2ni = 0
Xet ham so f(t) = - t ' + 3t - 2m lien tuc tren 0;v^
Dip so: — < m < 1 Bai 4. Cho phiTdng trinh
(4 - 6m)sin'x + 3(2m - l)sinx + 2(m - 2)sin^xcosx - (4m - 3)cosx = 0 (1) a. Giai phiTOng trinh (1) khi m = 2.
b. Tim m de phifdng trinh (1) c6 duy nhat mpt nghiem tren
<" ^ f5 Hi^ng dSn giai
'hbw
a. X = - + kn (k 6 Z) khong la nghiem , Dat t = tanx. Phifdng trinh trd thinh:
it = tanx t ^ - ( 2 m + l ) t ^ + 3 ( 2 m - l ) t - 4 m + 3 = 0 m = 2 => X = — + krt (k e Z)
4 e-3
t = tanx
( t - l ) ( t ^ - 2 m t + 4 m - 3 ) = 0
lien tuc tren [0; 1]. Vay m < - hoSc m < 1
t - 2 4 b. Xet h^m so f(t) =
1. Phi/dng phap dat dieu ki?n so siinh nghi?m thSng qua ham so liT^ng giac
2 cos^ X - cos' X - 1 Bai 1. Giai phifdng trinh: cos2x - tan x =
cos^x (1) ' / SV HtfdngdSngiai
Dieu ki?n cosx ^ 0. Khi do
(1) <=> 2cos^x - i - tan^x = 1 - cosx - 1 - tan^x i '
<:> 2cos^x + cosx - 1 = 0 cosx = - 1 cosx = —
2 1
p^i chieu vdi Dk cosx ^Ota thSfy ci hai gii tri deu thoa man. ' V|y phi/dng trinh (1) c6 cdc nghiem la x = TT + kn; x = ± - + 2k7t, (k e Z ) . NhSn x6t: trong phiTdng trinh (1). ta da bien ddi dieu ki$n nghiem, tim dtfdc thong qua h^m so y = cosx. Tif 66 chuyen vi^c doi chieu dieu kien cua X \i d6i chi6u dieu ki^n ci5a y ddn gian hdn nhieu (giong nhur trong phifdng trinh dai s6').
1 1 2
Bii 2. GiJi phifdng trinh: + = (2) cosx sin2x sin4x
Di6u ki^n
cosx s i n 2 x 9 i 0 o sin4x;t0
Htftfng dSn giai sinx 5*11
sinx ^0
s i n x? t ± —
Khi 66 (2) o 4sinx.cos2x + 2cos2x = 2 o sinx(2sin^x + sinx - 1) = 0
<:> sinx 6 -1;0;^
2
(3) Doi chi6u vdi di^u ki^n, ta chon dif^c sinx = - .
2
V$y phifdng trinh (2) c6 cic ho nghiem \k x = - + 2 k 7 i ; x = — + 2kn , 6 6 ( k € Z ) .
2- Phi/ong p h i p sur dung ph^p bien d^i lif^ng gi^c va thay d i l u ki$n 3. Gi5i phifdng trinh: tan^ x +1 = (2-sin^2x)sin3x
cos X Hi^dng dSn giai E>i^u ki^n: cosx ?t 0 o sinx ?t ±1. Ta c6
(3)0 sin^x + cos*x = (2 - sin^2x)sin3x o (2 - sin^2x)(l - 2sin3x) = 0
<=> sin3x = - (*) (do 2 - sin^2x > 1 vdi mpi x)
c> 3sinx - 4sin'x = - (*•) <• ;- > • 2
^hay sinx = ±1 vho (**) deu khong th6a man nen c^c nghiem cua (*) chinh nghiem cua phifdng trinh (3). - ô^ •
^ i ^ i phifdng trinh (*•) tim difpc cdc ho nghiem ciia phifdng trinh (3) 1^
, = J L ^ 2 k n 5 n ^ 2 k ^ ^^^^^
18 3 18 3 IK^oa 4'r m-i Nh$n xet:
• Trong phifdng trlnh (3), dieu k i ^ n cosx * 0 bien d o i thjlnh sinx ?t + 1 , r o i thgy sinx ^ ±1 v i o phi/cfng trlnh (••) dcu khong th6a man dan den nghi^m cug phi/dng trlnh (**) chinh Ik nghi^m cua phiTdng trlnh (3). NhiT vay, khong c^n . phai t l m nghi?m cu the, ta v i n c6 the doi chieu di/dc v d i dieu k i ^ n .
• C6 the diTa phiTcJng trlnh (•) ve sin3x = s i n ^ , giai phifdng trlnh nay cung d^n
den cdc nghi^m tren, sau k h i d o i chieu v d i dieu k i ? n x ;6 - + k n . 3. PhiTdng ph^p thuT tn/c tiep vao phi/dng trinh liT^rng gi^c
D o i v d i nhiTng phi/dng trlnh m ^ dieu kien va nghiem t l m difdc kho diTa ve cdng mOt h^m liTdng gidc, ta c6 the tlm nghi?m cu t h i , r o i thay vao dieu k i ^ n de k i e m tra l a i .
Bki 4. G i a i phifdng trlnh: cos3xtan5x = sin7x HU^ng d i n giai D i e u k i ^ n cos5x ^ 0.
K h i 66 (4) o 2sin5x.cos3x = 2sin7x.cos5x o sinSx = sinI2x.
T a a m d i w c x = y; x = f+ 7^- ' ' , V d i x = — thl cos5x = c o s ^ = c o s ^ ^ 0 ô k = 2 m , ( m e
• V d i x = — + — thl c o s 5 x = c o s 20 10
2 ' 2
. 4 ^ 2 , ^ 0 .
Vay phiTdng trlnh (4) c6 nghi$m m x = m n ; x = ^ + , (m, k e Z ) .
B a i 5. G i a i phifdng trlnh: 1 >/2 ( s i n x - c o s x )
ftanx + c o t 2 x 9 t 0
D i e u k i $ n < . ô ^ c o t x - l ? t O
tanx + cot2x c o t x - 1 Hti^dng dSn giai
cosx ?tO s i n 2 x ^ 0 s i n x ; t 0 s i n x - c o s x ^ t O
(5)
K h i d6 (5) o sin2x = - > ^ s i n x o sinx(2c6sx + >/2 ) = 0
sinx = 0 cosx = — /2
OtngtyTNHH MTV DVVIIKhang Viet 0 6 r^ng sinx = 0 kh6ng th6a man dieu k i ^ n ciia b ^ i todn.
X, = — + 2k7t 4
; (k G Z ) . i^fff. ,jyf35i^jj.i , Giii cosx = - — di/qJc
3n r,, X j = + 2k7t thay dieu k i ^ n sinx * 0, cosx ^ 0 thoa man. 4
• ^ .-A s . ô f s i n 2 x 9 t 0
Thay trifc tiep cdc nghi^m X i , X2 \ko hp , ta thay chi c6 x, s i n x - c o s x ;itO
th6a man. : d
VSy nghiem cua phiWng trlnh (5) ^ x = — + 2kjr, (k e Z ) . ' ' 4
4. Phi/dng ph^p bieu di§n tren dudng tron liT^ng gi^c r6i so s^nh dl^u klf n Ta b i ^ u dien tren diTdng tr6n nhCTng diem khong th6a man dieu k i e n (ddnh diii " x " ) vk nhffng d i e m bieu dien nghi?m t l m diTdc ( d i n h d^u " o " ) . Nhffng d i l m ddnh dau " o " m^ khong triJng vdi d i e m ddnh dau " x " chinh Ik nhffng d i l m th6a m a n dieu k i $ n .
PhiTdng phdp nay c6 hipu qui k h i so diem khong thoa man dieu kien Ik it \k d vi t r i dac bi$t, dong thdi ckc phiTdng phdp da neu to ra khong hieu qua.
Nh§n x6t: M o i cung (hoSc gdc) li/dng gidc di/dc bieu dien bdi mpt diem tren dirdng tr6n Itfcfng gidc ddn v i (goi t i t Ik dirdng tr6n lifdng gidc).
i) x = a + 2k7r diTdc bieu dien tren diTdng tr6n li/dng giac bdi mot diem.
ii) x = a + k n diTdc bieu dien tren diTdng tr6n lifdng gidc bdi hai d i e m d6'i xrfng nhau qua gdc O.
iii) x = a + ^ ^ bieu dien trdn dirdng trbn liTdng gidc b d i ba d i e m cdch deu nhau. tao th^nh ba dinh mot tarn gidc deu noi tiep diTdng tr6n d6,
T6ng qu^t: x = a + (n > 3) bieu dien tren dUfJng tron lifdng gidc bdi n diem n
cich d6u nhau, tao thknh n dinh ciia mpt da gidc deu npi tiep diTdng tr6n d6.
<^->- u ^ L sinx + sin2x + sin3x /r ^
ằ4i 6. G i a i phiTdng trlnh: = J 3 (f>) cosx + cos2x + cos3x
Hiidng d i n giai E>i^u k i p n cosx 4- cos2x + cos3x ^
x ? t —+ k 2 7 i 4
x ^ ± — + 2k7c
. ( k e Z )
K h i d6 phiTdng trlnh (6) tiTdng difdng v d i
2x y .
4 /N , It
77t \ / ^
6
4 _ . 4
3 3
mnhS.l
Luyfn giii di trudc kp thi DH 3 mifn Mt. Trung. Nam Todn /^^.. \guy?n Van ThOng ~CShg ty TNHH MTVDWH KfiangVi?t
tan2x = V3 '^^~'^'^'Y"
Tren dtfdng tr6n Itf^ng gidc: bi^u dien x = ^ + ^ bdi 4 diem (ddnh d5, 6 2
"o"); bi^u dign x = - + — x = ± — + 2kn bdi 6 dilm (ddnh dau
4 2 3 I
(h.31). Ta thSfy c6 3 diem ddnh dau "o" khong tring vdi dau "x".
V$y phifdng trinh (6) c6 nghi$m 1^: x = - ^ + nn; x = - ^ + 2nn, (n e Z ) .
g, Giai ph^dng trinh: log. sin sinx 2
+ log, sin—+ cos2x
2 = 0 (8)
Hi^dng dSn giai
B&i 7. GiSi phtfdng trinh: cosx + sinx = 0 (7) HifOng dSn giai Khi cosx 10 thi (7) o cosx + sin3x = 0 o
X = — + kn 4 3n kn x = — + —
8 2 Bieu dien (•) trdn diTdng tr6n lir(?ng gidc ta diTdc
6 dilm dinh dS'u "o", trong d6 chi c6 3 diem n- n^m ben phdi true Oy (cosx ^ 0) (h.32) tfng vdi
x, =-—+ 2kJi ; X2 = - — + 2k7t; X3 = — + 2k7i , 4 8 8 8' ( k e Z ) .
• Khi cosx < 0 thi (7) o -cosx + sin3x = 0, gidi
(•)
. X . . X
sin —-sinx = sm—+ cos2x (•) sin sinx >0
2
( • ) o 2sin^x - sinx - 1 = 0 o
sinx = l sinx = —
2
ra dtf(?c:
X = — + kn 4
n kn x = —+ —
8 2
. ( k e Z ) (**)
Giai ra di/dc
Nhan thay
X| = — + 2kTc
X2=—- + 2kn, ( k e Z ) 6
j | ằ 0 = d ij':./
77t + 2k7t
Bi^u diSn (*•) tren di/&ng tr6n liTdng giic ta dtfdc 6 diem trong 66 chi c6 3 dilm nim ben t r i i true Oy (cosx < 0) (h.33) tfng vdi
X4 = — + 2k7i ; X . = — + 2kn ; x* = — + 2k7i ,
* 8 ^ 8 4 ( k e Z )
Vay phiTdng trinh (7) c6 cdc ho nghi^m IJl X|, X2, X3, X4, Xs, X6 nhif tren.
Khi tirng phifdng phdp rieng khd thifc hi^n, ta c6 thi ph6i hdp cdc phi/i<' phdp lai 6i doi chie'u dieu ki$n vdi nghi^m tlm difdc l^m cho b^i todn ' phiJc t^p hdn.
Hlnh 3.3
• s i n s i nX , <0 (khong thoa man (••)).
• sin—^-sinx, =sin
2 ^ I 12 + k7t + —. 1
2
Bilu dien a = - - ^ + k7r tren dufdng tr6n Itfdng Hinh 3.4 12
giic ta difdc 2 diem deu c6 sina > (h.34).
Dod6x2th6aman(**).
s i n - i - s i n x j =sin 77t
U 2 + k7t + - . Bieu dien 2
P = - ^ + k7r tren difdng tr6n liTdng gidc ta diTdc In
•lai dilm (h.3.5).
^aidiem j ^ + kn khong th6a man s i n P > - ^ .
Vay X3 khong thoa man dieu ki^n khi k 16. T6m lai, phifdng trinh (9) c6 cdc
•'Pnghiemia: x = ~ + 2kn; x = — + 4 n n , ( k , n € Z ) . > ^- • c . ^ r. h r • 6 6
Luyen guUdStruOc ky tin DUJ rrvu'n .'>VJ, irjin-. Nam ToOn hpc - Nguyen Van ThOng
cAc B A I T O A N V 6 DUdNG T H A N G , D U 6 N G T R 6 N
I . T 6 M T A T L i T H U Y E T