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Trng THPT chuyn L T TrngCn Th
- - - - - -
GIO TRNH
BT NG THC LNG GIC
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Trng THPT chuyn L T Trng Cn Th Bt ng thc lng gicChng 1Cc bc u cs
The Inequalities Trigonometry 3
Chng 1:
CC BC U CS
bt u mt cuc hnh trnh, ta khng thkhng chun bhnh trang ln ng.Ton hc cng vy. Mun khm ph c ci hay v ci p ca bt ng thc lnggic, ta cn c nhng vt dng chc chn v hu dng, chnh l chng 1:Ccbc u cs.
Chng ny tng qut nhng kin thc cbn cn c chng minh bt ng thclng gic. Theo kinh nghim c nhn ca mnh, tc gicho rng nhng kin thc ny ly cho mt cuc hnh trnh.
Trc ht l cc bt ng thc i scbn ( AM GM, BCS, Jensen, Chebyshev) Tip theo l cc ng thc, bt ng thc lin quan cbn trong tam gic. Cui cngl mt snh l khc l cng cc lc trong vic chng minh bt ng thc (nh lLargare, nh l vdu ca tam thc bc hai, nh l vhm tuyn tnh )
Mc lc :1.1. Cc bt ng thc i scbn 4
1.1.1. Bt ng thc AM GM............................................... 4
1.1.2. Bt ng thc BCS.. 81.1.3. Bt ng thc Jensen.... 131.1.4. Bt ng thc Chebyshev..... 16
1.2. Cc ng thc, bt ng thc trong tam gic.. 191.2.1. ng thc... 191.2.2. Bt ng thc..... 21
1.3. Mt s nh l khc. 221.3.1. nh l Largare ... 221.3.2. nh l v du ca tam thc bc hai.. 251.3.3. nh l v hm tuyn tnh.. 28
1.4. Bi tp.. 29
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Trng THPT chuyn L T Trng Cn Th Bt ng thc lng gicChng 1Cc bc u cs
The Inequalities Trigonometry 4
1.1. Cc bt ng thc i s cbn :
1.1.1. Bt ng thc AM GM :
Vi mi s thc khng m naaa ,...,, 21 ta lun c
nn
n aaan
aaa...
...21
21
+++
Bt ng thcAM GM(Arithmetic Means Geometric Means) l mt bt ng thcquen thuc v c ng dng rt rng ri.y l bt ng thc m bn c cn ghi nh rrng nht, n s l cng c hon ho cho vic chng minh cc bt ng thc. Sau y lhai cch chng minh bt ng thc ny m theo kin ch quan ca mnh, tcgi chorng l ngngn v hay nht.
Chng minh :Cch 1 : Quy np kiu Cauchy
Vi 1=n bt ng thc hin nhin ng. Khi 2=n bt ng thc tr thnh
( ) 02
2
212121
+
aaaaaa
(ng!)
Gi s bt ng thc ng n kn = tc l :
kk
k aaak
aaa...
...21
21
+++
Ta s chng minh n ng vi kn 2= . Tht vy ta c :
( ) ( ) ( )( )
( )( )
kkkk
kkkk
kk
kkkkkkkk
aaaaa
k
aaakaaak
k
aaaaaa
k
aaaaaa
22121
22121
2212122121
......
......
......
2
......
+
++
++++
=
++++++
+++++++
Tip theo ta s chng minh vi 1= kn . Khi :
( ) 1 121121
1121
1121121
1121121
...1...
...
............
=
+++
=
++++
kkk
kk
k kkk
kkk
aaakaaa
aaak
aaaaaakaaaaaa
Nhvy bt ng thc c chng minh hon ton.ng thc xy ra naaa === ...21
Cch 2 : ( ligii ca Polya )
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Trng THPT chuyn L T Trng Cn Th Bt ng thc lng gicChng 1Cc bc u cs
The Inequalities Trigonometry 5
Gin
aaaA
n+++=
...21
Khi bt ng thc cn chng minh tng ng vin
n Aaaa ...21 (*)
R rng nu Aaaa n ====
...21 th (*) c du ng thc. Gi s chng khng bngnhau. Nhvyphi c t nht mt s, gi s l Aa 2
tc l 21 aAa =+= AaAaaaAaaAaaaa
2121 '' aaaa >
nn aaaaaaaa ...''... 321321
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Trng THPT chuyn L T Trng Cn Th Bt ng thc lng gicChng 1Cc bc u cs
The Inequalities Trigonometry 6
Li gii :
Ta lun c : ( ) CBA cotcot =+
1cotcotcotcotcotcot
cotcotcot
1cotcot
=++
=+
ACCBBA
CBA
BA
Khi :
( ) ( ) ( )
( ) ( )
3cotcotcot
3cotcotcotcotcotcot3cotcotcot
0cotcotcotcotcotcot
2
222
++
=++++
++
CBA
ACCBBACBA
ACCBBA
Du bng xy ra khi v ch khi ABC u.
V d 1.1.1.3.
CMR vi mi ABC nhn v *Nn ta lun c :
2
1
3tantantan
tantantan
++
++nnnn
CBA
CBA
Li gii :
Theo AM GMta c :
( ) ( )
( ) ( ) 21
33
3 3
33
3333tantantan3tantantan
tantantan
tantantan3tantantan3tantantan
=++
++
++
++=++
nnn
nnn
nnnnn
CBACBA
CBA
CBACBACBA
pcm.
V d 1.1.1.4.
Cho a,b l hai s thc tha :0coscoscoscos ++ baba
CMR : 0coscos + ba
Li gii :
Ta c :
( )( ) 1cos1cos1
0coscoscoscos
++
++
ba
baba
Theo AM GMth :
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Trng THPT chuyn L T Trng Cn Th Bt ng thc lng gicChng 1Cc bc u cs
The Inequalities Trigonometry 7
( ) ( )( )( )
0coscos
1cos1cos12
cos1cos1
+
+++++
ba
baba
V d 1.1.1.5.
Chng minh rng vi mi ABC nhn ta c :
2
3
2sin
2sin
2sin
2sin
2sin
2sin
3
2
2cos
2cos
coscos
2cos
2cos
coscos
2cos
2cos
coscos+
++++
ACCBBA
AC
AC
CB
CB
BA
BA
Li gii :
Ta c
=
=
BABA
BA
BA
AA
A
A
cotcot4
3
2sin
2sin
2cos
2cos4
coscos4
3
2cot2sin
2cos2
cos
Theo AM GMth :
+
+
BABA
BA
BA
BABA
BA
BA
cotcot4
3
2sin
2sin
3
2
2cos
2cos
coscos
2
cotcot4
3
2sin
2sin
2cos2cos4
coscos4
32
Tng t ta c :
+
+
AC
AC
AC
AC
CBCB
CB
CB
cotcot4
3
2sin2sin3
2
2cos
2cos
coscos
cotcot4
3
2sin
2sin
3
2
2cos
2cos
coscos
Cng v theo v cc bt ng thc trn ta c :
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Trng THPT chuyn L T Trng Cn Th Bt ng thc lng gicChng 1Cc bc u cs
The Inequalities Trigonometry 8
( )ACCBBAACCBBA
AC
AC
CB
CB
BA
BA
cotcotcotcotcotcot
2
3
2
sin
2
sin
2
sin
2
sin
2
sin
2
sin
3
2
2cos
2cos
coscos
2cos
2cos
coscos
2cos
2cos
coscos
+++
++
++
2
3
2sin
2sin
2sin
2sin
2sin
2sin
3
2+
++=
ACCBBApcm.
Bc u ta mi ch c bt ng thcAM GMcng cc ng thc lnggic nnsc nh hng n cc bt ng thc cn hn ch. Khi ta kt hpAM GMcngBCS,Jensen hay Chebyshev th n thc s l mt v kh ng gm cho cc bt ng thclnggic.
1.1.2. Bt ng thc BCS :
Vi hai b s ( )naaa ,...,, 21 v ( )nbbb ,...,, 21 ta lun c :
( ) 2222
1
22
2
2
1
2
2211 ......... nnnn bbbaaabababa +++++++++
Nu nhAM GMl cnh chim u n trong vic chng minh bt ng thc thBCS (Bouniakovski Cauchy Schwartz) li l cnh tay phi ht sc c lc. Vi
AM GM ta lun phi ch iu kin cc bin l khng m, nhng i vi BCS ccbin khng b rng buc bi iu kin , ch cn l s thc cng ng. Chng minh btng thc ny cng rt ngin.
Chng minh :
Cch 1 :
Xt tam thc :
( ) ( ) ( )22222
11 ...)( nn bxabxabxaxf +++=
Sau khi khai trin ta c :
( )22
2
2
12211222
2
2
1 ......2...)( nnnn bbbxbababaxaaaxf ++++++++++= Mt khc v Rxxf 0)( nn :
( ) +++++++++ 2222
1
22
2
2
1
2
2211 .........0 nnnnf bbbaaabababa pcm.
ng thc xy ran
n
b
a
b
a
b
a=== ...
2
2
1
1 (quy c nu 0=ib th 0=ia )
Cch 2 :
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Trng THPT chuyn L T Trng Cn Th Bt ng thc lng gicChng 1Cc bc u cs
The Inequalities Trigonometry 9
S dng bt ng thc AM GM ta c :
( )( )2222
1
22
2
2
1
22
2
2
1
2
22
2
2
1
2
......
2
......nn
ii
n
i
n
i
bbbaaa
ba
bbb
b
aaa
a
++++++
+++
++++
Cho ichy t 1 n n ri cng v c n bt ng thc li ta c pcm.y cng l cch chng minh ht sc ngngn m bn c nn ghi nh!
By gi vi s tip sc caBCS,AM GMnhc tip thm ngun sc mnh, nhh mc thm cnh, nhrng mc thm vy,pht huy hiu qu tm nh hng ca mnh.Hai bt ng thc ny b p b sung h tr cho nhau trong vic chng minh bt ngthc. Chng lng long nht th, song kim hp bch cngph thnh cng nhiubi ton kh.
Trm nghe khng bng mt thy, ta hyxt cc v d thy r iu ny.
V d 1.1.2.1.
CMR vi mi ,,ba ta c :
( )( )2
21cossincossin
++++
baba
Li gii :
Ta c :
( )( ) ( )( )
( ) ( )( ) ( )12cos12sin12
1
2
2cos12sin
22
2cos1
coscossinsincossincossin 22
++++=
++
++
=
+++=++
abbaab
abba
abbaba
Theo BCSta c :
( )2cossin 22 BAxBxA ++
p dng ( )2 ta c :
( ) ( ) ( ) ( ) ( )( ) ( )31112cos12sin 2222 ++=++++ baabbaabba Thay ( )3 vo ( )1 ta c :
( )( ) ( )( )( ) ( )41112
1cossincossin 22 ++++++ baabba
Ta s chng minh bt ng thc sau y vi mi a, b :
( )( )( ) ( )52
11112
12
22
++++++
babaab
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Trng THPT chuyn L T Trng Cn Th Bt ng thc lng gicChng 1Cc bc u cs
The Inequalities Trigonometry 10
Tht vy :
( ) ( )( )
( )( )2
211
24111
2
1
22
15
2222
2222
++++
++
+++++
baba
abbaba
ab
( )( ) ( ) ( ) ( )62
1111
2222 +++
++ba
ba
Theo AM GMth ( )6 hin nhin ng ( )5 ng.T ( )1 v ( )5 suy ra vi mi ,,ba ta c :
( )( )2
21cossincossin
++++
baba
ng thc xy ra khi xy ra ng thi du bng ( )1 v ( )6
( )
++=
=
+=
=
=+
=
Zkkab
baarctg
ba
abbatg
ba
abba
ba
2121
12cos1
2sin
22
V d 1.1.2.2.
Cho 0,, >cba v cybxa =+ cossin . CMR :
33
222 11sincos
ba
c
bab
y
a
x
+++
Li gii :Bt ng thc cn chng minh tng ng vi :
( )*cossin
11cos1sin1
33
222
33
222
ba
c
b
y
a
x
ba
c
bab
y
a
x
++
++
+
Theo BCSth :
( ) ( )( )222
1
2
2
2
1
2
2211 bbaababa +++
vi
==
==
bbbaab
bya
axa
21
21
;
cos;sin
( ) ( )23322
cossincossin
ybxabab
y
a
x++
+
do 033 >+ ba v ( )*cossin =+ cybxa ng pcm.
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Trng THPT chuyn L T Trng Cn Th Bt ng thc lng gicChng 1Cc bc u cs
The Inequalities Trigonometry 11
ha
x
yz
N
Q
P
A
B C
M
ng thc xy ra22
2
2
1
1 cossin
b
y
a
x
b
a
b
a==
+=
+=
=+
=
33
2
33
2
22
cos
sin
cossin
cossin
ba
cby
ba
cax
cybxa
b
y
a
x
V d 1.1.2.3.
CMR vi mi ABC ta c :
R
cbazyx
2
222++
++
vi zyx ,, l khong cch t im M bt k nm bn trong ABC n ba cnhABCABC ,, .
Li gii :
Ta c:
( )
++++=++
=++
=++
++=
cba
cbacba
abc
ABC
MCA
ABC
MBC
ABC
MAB
MCAMBCMABABC
h
z
h
y
h
xhhhhhh
h
x
h
y
h
z
S
S
S
S
S
S
SSSS
1
1
Theo BCSth :
( )cba
cba
cba
c
c
b
b
a
a hhhh
z
h
y
h
xhhh
h
zh
h
yh
h
xhzyx ++=
++++++=++
m BahAchCbhCabahS cbaa sin,sin,sinsin21
21 =====
( )R
ca
R
bc
R
abAcCbBahhh
cba222
sinsinsin ++=++=++
T suy ra :
++
++
++R
cba
R
cabcabzyx
22
222
pcm.
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Trng THPT chuyn L T Trng Cn Th Bt ng thc lng gicChng 1Cc bc u cs
The Inequalities Trigonometry 12
ng thc xy ra khi v ch khi ABCzyx
cba
==
==u v M l tm ni tip ABC .
V d 1.1.2.4.
Chng minh rng :
+
2;08sincos 4
xxx
Li gii :
p dng bt ng thc BCSlin tip 2 ln ta c :
( ) ( )( )( )( ) ( )( )
4
2222222
2224
8sincos
8sincos1111
sincos11sincos
+
=+++
+++
xx
xx
xxxx
ng thc xy ra khi v ch khi4
=x .
V d 1.1.2.5.
Chng minh rng vi mi s thc a vx ta c
( ) 11
cos2sin12
2
+
+
xaxax
Li gii :
Theo BCS ta c :
( )( ) ( ) ( ) ( )
( )( ) ( )
( ) 11
cos2sin1
1cos2sin1
21421
cossin21cos2sin1
2
2
2222
42242
2222222
+
+
++
++=++=
+++
xaxaa
xaxax
xxxxx
aaxxaxax
pcm.
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Trng THPT chuyn L T Trng Cn Th Bt ng thc lng gicChng 1Cc bc u cs
The Inequalities Trigonometry 13
1.1.3. Bt ng thc Jensen :
Hm s )(xfy = lin tc trn on [ ]ba, v n im nxxx ,...,, 21 ty trn on
[ ]ba, ta c :
i) 0)('' >xf trong khong ( )ba, th :
++++++
n
xxxnfxfxfxf nn
...)(...)()( 2121
ii) 0)(''
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Trng THPT chuyn L T Trng Cn Th Bt ng thc lng gicChng 1Cc bc u cs
The Inequalities Trigonometry 14
V d 1.1.3.1.
Chng minh rng vi mi ABC ta c :
2
33sinsinsin ++ CBA
Li gii :
Xt xxf sin)( = vi ( );0x
Ta c ( );00sin)('' =
2;00
cos
sin2''3
xx
xxf . T theo Jensenth :
==
++
+
+
3
6sin3
3
2223222
CBA
fC
fB
fA
f pcm.
ng thc xy ra khi v ch khi ABC u.
V d 1.1.3.3.
Chng minh rng vi mi ABC ta c :
21
222222
32
tan2
tan2
tan
+
+
CBA
Li gii :
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Trng THPT chuyn L T Trng Cn Th Bt ng thc lng gicChng 1Cc bc u cs
The Inequalities Trigonometry 15
Xt ( ) ( ) 22tanxxf = vi
2;0
x
Ta c ( ) ( )( ) ( ) ( ) 1221221222 tantan22tantan122' + +=+= xxxxxf
( ) ( )( )( ) ( )( )( ) 0tantan1122tantan112222'' 2222222 >++++= xxxxxf Theo Jensenta c :
=
=
++
+
+
2122
36
33
2223222
tg
CBA
fC
fB
fA
f pcm.
ng thc xy ra khi v ch khi ABC u.
V d 1.1.3.4.
Chng minh rng vi mi ABC ta c :3
2
3
2tan
2tan
2tan
2sin
2sin
2sin ++++++
CBACBA
Li gii :
Xt ( ) xxxf tansin += vi
2;0
x
Ta c ( ) ( )
>
=
2;00
cos
cos1sin''
4
4 x
x
xxxf
Khi theo Jensenth :
+=
+=
++
+
+
3
2
3
6tan
6sin3
3
2223222
CBA
fC
fB
fA
f pcm.
ng thc xy ra khi v ch khi ABC u.
V d 1.1.3.5.
Chng minh rng vi mi ABC nhn ta c :
( ) ( ) ( )2
33
sinsinsin
3
2sinsinsin
CBACBA
Li gii :
Ta c
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Trng THPT chuyn L T Trng Cn Th Bt ng thc lng gicChng 1Cc bc u cs
The Inequalities Trigonometry 16
++++
+=++
CBACBA
CBACBA
222
222
sinsinsinsinsinsin
coscoscos22sinsinsin
v2
33sinsinsin ++ CBA
2
33sinsinsin2 ++
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Trng THPT chuyn L T Trng Cn Th Bt ng thc lng gicChng 1Cc bc u cs
The Inequalities Trigonometry 17
Chng minh :
Bng phn tch trc tip, ta c ng thc :
( ) ( )( ) ( )( ) 0.........1,
21212211 =+++++++++ =
n
ji
jijinnnnbbaabbbaaabababan
V hai dy naaa ,...,, 21 v nbbb ,...,, 21 n iu cng chiu nn ( )( ) 0 jiji bbaa
Nu 2 dy naaa ,...,, 21 v nbbb ,...,, 21 n iu ngc chiu th bt ng thc i
chiu.
V d 1.1.4.1.
Chng minh rng vi mi ABC ta c :
3
++
++
cba
cCbBaA
Li gii :
Khng mt tnh tng qut gi s :CBAcba
Theo Chebyshev th :
33
333
=++++++
++
++
++
CBAcbacCbBaA
cCbBaACBAcba
ng thc xy ra khi v ch khi ABC u.
V d 1.1.4.2.
Cho ABC khng c gc t vA, B, C o bng radian. CMR :
( ) ( )
++++++
C
C
B
B
A
ACBACBA
sinsinsinsinsinsin3
Li gii :
Xt ( )x
xxf
sin= vi
2;0
x
Ta c ( ) ( )
=
2;00
tancos'
2
x
x
xxxxf
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Trng THPT chuyn L T Trng Cn Th Bt ng thc lng gicChng 1Cc bc u cs
The Inequalities Trigonometry 19
CBA
CBA
coscoscos
sinsinsin
Khi theo Chebyshev th :
( )CBA
CBACBA
CCBBAACBACBA
coscoscos
2sin2sin2sin
2
3sinsinsin2
3
cossincossincossin
3
coscoscos
3
sinsinsin
++
++++
++
++
++
pcm.ng thc xy ra khi v ch khi ABC u.
1.2. Cc ng thc bt ng thc trong tam gic :
Sau y l hu ht nhng ng thc, bt ng thc quen thuc trong tamgic v tronglnggic c dng trong chuyn ny hoc rt cn thit cho qu trnh hc ton cabn c. Cc bn c th dng phn ny nhmt t in nh tra cu khi cn thit.Haybn c cng c th chng minh tt c cc kt qu nhl bi tp rn luyn.Ngoi ra ticng xin nhc vi bn c rng nhng kin thc trong phn ny khi pdng vo bi tpu cn thit c chng minh li.
1.2.1. ng thc :
RC
c
B
b
A
a
2sinsinsin ===
Cabbac
Bcaacb
Abccba
cos2
cos2
cos2
222
222
222
+=
+=
+=
AbBac
CaAcb
BcCba
coscos
coscos
coscos
+=
+=
+=
( ) ( ) ( )
( )( )( )cpbpapp
rcprbprap
prCBARR
abc
CabBcaAbc
hchbhaS
cba
cba
=
===
===
===
===
sinsinsin24
sin
2
1sin
2
1sin
2
1
.2
1.
2
1.
2
1
2
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Trng THPT chuyn L T Trng Cn Th Bt ng thc lng gicChng 1Cc bc u cs
The Inequalities Trigonometry 20
4
22
4
22
4
22
2222
2222
2222
cbam
bac
m
acbm
c
b
a
+=
+=
+=
ba
Cab
l
ac
Bca
l
cb
Abc
l
c
b
a
+=
+=
+=
2cos2
2cos2
2cos2
( )
( )
( )
2sin
2sin
2sin4
2tan
2tan
2tan
CBAR
Ccp
Bbp
Aapr
=
=
=
=
+
=+
+
=+
+
=+
2tan
2tan
2tan
2tan
2tan
2tan
AC
AC
ac
ac
CB
CB
cb
cb
BA
BA
ba
ba
S
cbaCBA
S
cbaC
SbacB
S
acbA
4cotcotcot
4cot
4cot
4cot
222
222
222
222
++=++
+=
+=
+=
( )( )
( )( )
( )( )ab
bpapC
ca
apcpB
bc
cpbpA
=
=
=
2sin
2sin
2sin
( )
( )
( )ab
cppC
ca
bppB
bc
appA
=
=
=
2cos
2cos
2cos
( )( )( )
( )( )
( )
( )( )
( )cppbpapC
bpp
apcpB
appcpbpA
=
=
=
2tan
2tan
2tan
( )
CBACBA
R
rCBACBA
CBACBA
CBACBA
R
pCBACBA
coscoscos21coscoscos
12
sin2
sin2
sin41coscoscos
coscoscos12sinsinsin
sinsinsin42sin2sin2sin
2cos
2cos
2cos4sinsinsin
222
222
=++
+=+=++
+=++
=++
==++
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The Inequalities Trigonometry 21
1cotcotcotcotcotcot
12
tan2
tan2
tan2
tan2
tan2
tan
2cot
2cot
2cot
2cot
2cot
2cot
tantantantantantan
=++
=++
=++
=++
ACCBBA
ACCBBA
CBACBA
CBACBA
( ) ( ) ( ) ( ) ( ) ( ) ( )
( )
( ) ( ) ( ) ( ) ( ) ( ) ( )
( )
( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( )
( )
( ) kCkBkAkCkBkA
kCkBkAkCkBkA
Ck
Bk
Ak
Ck
Bk
Ak
Ak
Ck
Ck
Bk
Bk
Ak
kAkCkCkBkBkA
kCkBkAkCkBkA
kCkBkAkCkBkA
Ck
Bk
AkCkBkAk
kCkBkAkCkBkA
Ck
Bk
AkCkBkAk
k
k
k
k
k
k
coscoscos212sinsinsin
coscoscos211coscoscos
212cot
212cot
212cot
212cot
212cot
212cot
12
12tan2
12tan2
12tan2
12tan2
12tan2
12tan
1cotcotcotcotcotcot
tantantantantantan
coscoscos4112cos2cos2cos
212sin
212sin
212sin41112cos12cos12cos
sinsinsin412sin2sin2sin
212cos
212cos
212cos4112sin12sin12sin
1222
222
1
+
+
+=++
+=++
+++=+++++
=++++++++
=++
=++
+=++
++++=+++++
=++
+++=+++++
1.2.2. Bt ng thc :
acbac
cbacb
bacba
+
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The Inequalities Trigonometry 22
1cotcotcot
9tantantan
4
9sinsinsin
4
3coscoscos
222
222
222
222
++
++
++
++
CBA
CBA
CBA
CBA
2cot
2cot
2cot
12tan2tan2tan
2sin
2sin
2sin
2cos
2cos
2cos
222
222
222
222
CBA
CBA
CBA
CBA
++
++
++
++
33
1cotcotcot
33tantantan
8
33sinsinsin
8
1coscoscos
CBA
CBA
CBA
CBA
332
cot2
cot2
cot
33
1
2
tan
2
tan
2
tan
8
1
2sin
2sin
2sin
8
33
2cos
2cos
2cos
AAA
AAA
CBA
CBA
1.3. Mt s nh l khc :
1.3.1. nh l Lagrange :
Nu hm s ( )xfy = lin tc trn on [ ]ba ; v c o hm trn khong ( )ba ; th tn ti 1 im ( )bac ; sao cho :
( ) ( ) ( )( )abcfafbf = '
Ni chung vi kin thc THPT, ta ch c cng nhn nh l ny m khng chng minh.V chng minh ca n cn n mt s kin thc ca ton cao cp. Ta ch cn hiu cchdng n cng nhng iu kin i km trong cc trng hp chng minh.
V d 1.3.1.1.
Chng minh rng baRba
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The Inequalities Trigonometry 23
Xt ( ) ( ) xxfxxf cos'sin == Khi theo nh l Lagrangeta c
( ) ( ) ( ) ( )
abcabab
cabafbfbac
=
cossinsin
cos:;:
pcm.
V d1.3.1.2.
Vi ba
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The Inequalities Trigonometry 24
CMR nu 0>x thxx
xx
+>
++
+
11
1
11
1
Li gii :
Xt ( ) ( )( ) 0ln1ln1
1ln >+=
+= xxxx
xxxf
Ta c ( ) ( )1
1ln1ln'
++=
xxxxf
Xt ( ) ttg ln= lin tc trn [ ]1; +xx khvi trn ( )1; +xx nn theo Lagrange th :
( ) ( )
( ) ( )
( ) ( ) 01
1ln1ln'
1
1'
1
ln1ln:1;
>+
+=
+>=
+
++
xxxxf
xcg
xx
xxxxc
vi > 0x ( )xf tng trn ( )+;0
( ) ( )
xx
xx
xx
xxxfxf
+>
++
+>
++>+
+
+
11
1
11
11ln
1
11ln1
1
1
pcm.
V d 1.3.1.5.
Chng minh rng + Zn ta c :
1
1
1
1arctan
22
1222+
++
++ nnnnn
Li gii :
Xt ( ) xxf arctan= lin tc trn [ ]1; +nn
( )21
1'
xxf
+= trn ( ) ++ Znnn 1;
Theo nh l Lagrange ta c :( ) ( )
( ) ( )( )
( )( )
++=
+
++
+=+=
+
+
+=+
1
1arctan
1
1
11
1arctanarctan1arctan
1
1
1
1':1;
22
2
nnc
nn
nnnn
c
nn
nfnfcfnnc
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The Inequalities Trigonometry 25
( ) 111; +
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The Inequalities Trigonometry 26
ng thc xy ra khi v ch khi :
cbaCBAzyxBzCyx
BzCy::sin:sin:sin::
coscos
sinsin==
+=
=
tc zyx ,, l ba cnh ca tam gic tng ng vi ABC .
V d 1.3.2.2.
CMR Rx v ABC bt k ta c :
( )CBxAx coscoscos2
11 2 +++
Li gii :
Bt ng thc cn chng minh tng ng vi :
( )( ) ( )
02
sin2
sin4
12
cos2
sin4
2sin4
2cos
2cos2
cos12coscos'
0cos22coscos2
22
22
2
2
2
2
=
=
+=
+=
++
CBA
CBA
ACBCB
ACB
ACBxx
Vy bt ng thc trn ng.
ng thcxy ra khi v ch khi :
==
=
+=
=
CBx
CB
CBx cos2cos2coscos
0
V d 1.3.2.4.
CMR trong mi ABC ta u c :2
222
2sinsinsin
++++
cbaCcaBbcAab
Li gii :
Bt ng thc cn chng minh tng ng vi :( )
( ) ( )BbccbCcAb
BbccbCcAbaa
2cos22cos2cos'
02cos22cos2cos2
222
222
+++=
+++++
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The Inequalities Trigonometry 27
( ) 02sin2sin 2 += CcAb Vy bt ng thc c chng minh xong.
V d 1.3.2.4.
Cho ABC bt k. CMR :
2
3coscoscos ++ CBA
Li gii :
t ( )BACBCB
CBAk ++
=++= cos2
cos2
cos2coscoscos
01
2
cos
2
cos2
2
cos2 2 =++
+
kBABABA
Do 2
cosBA +
l nghim ca phng trnh :
012
cos22 2 =+
kxBA
x
Xt ( )122
cos' 2 +
= kBA
. tn ti nghim th :
( )
2
3coscoscos
2
31
2cos120' 2
++
CBA
kBA
k
pcm.
V d 1.3.2.5.
CMR Ryx , ta c :
( )2
3cossinsin +++ yxyx
Li gii :
t ( )2
sin212
cos2
sin2cossinsin 2yxyxyx
yxyxk +
++
=+++=
Khi 2
sinyx +
l nghim ca phng trnh :
012
cos22 2 =+
kxyx
x
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The Inequalities Trigonometry 28
( )
2
3
0121'
=
k
k
pcm.
1.3.3. nh l v hm tuyn tnh :
Xt hm ( ) baxxf += xc nh trn on [ ];
Nu( )
( ) ( )Rk
kf
kf
th ( ) [ ]; xkxf .
y l mt nh l kh hay. Trong mt s trng hp, khi m AM GM b tay,BCS u hng v iu kin th nh l v hm tuyn tnh mipht huy ht sc mnhca mnh. Mtpht biu ht sc ngin nhng li l li ra cho nhiu bi bt ngthc kh.
V d 1.3.3.1.
Cho cba ,, l nhng s thc khng m tha :
4222 =++ cba
CMR : 82
1+++ abccba
Li gii :
Ta vit li bt ng thc cn chng minh di dng :
082
11 ++
cbabc
Xt ( ) 82
11 ++
= cbabcaf vi [ ]2;0a .
Khi :
( ) ( )
( ) 08882822
0888280 22
=
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The Inequalities Trigonometry 29
V d 1.3.3.2.
CMR cba ,, khng m ta c :
( )( ) ( )3297 cbaabccbacabcab +++++++
Li gii :
tcba
cz
cba
by
cba
ax
++=
++=
++= ;; . Khi bi ton tr thnh :
Chng minh ( ) 297 +++ xyzzxyzxy vi 1=++ zyx
Khng mt tnh tng qut gi s { }zyxx ,,max= .
Xt ( ) ( ) 27977 ++= yzxyzzyxf vi
1;
3
1x
Ta c :
( )
( )
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The Inequalities Trigonometry 30
1.4.5.CBA
CBAsinsinsin8
9cotcotcot ++
1.4.6. CBAACCBBA
sinsinsin82
cos2
cos2
cos
1.4.7. CBACBA sinsinsincoscoscos1 +
1.4.8.Sbacacbcba 2
33111 4
++
++
+
1.4.9. 32++cba m
c
m
b
m
a
1.4.10.2
33++
c
m
b
m
a
m cba
1.4.11. 2plmlmlm ccbbaa ++
1.4.12.abcmcmbma cba
3111222
>++
1.4.13. ( )( )( )8
abccpbpap
1.4.14. rhhh cba 9++
1.4.15.
+
+
+
4
3sin
4
3sin
4
3sinsinsinsin
ACCBBACBA
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Trng THPT chuyn L T Trng Cn Th Bt ng thc lng gicChng 2 Cc phngphp chng minh
The Inequalities Trigonometry 31
Chng 2 :
Cc phng php chng minh
Chng minh bt ng thc i hi k nng v kinh nghim. Khng th khi khi m tam u vo chng minh khi gp mtbi bt ng thc. Ta s xem xt n thuc dngbino, nndng phngphp no chng minh. Lc vic chng minh bt ng thcmi thnh cng c.
Nhvy, c th ng u vi cc bt ng thc lng gic,bn c cn nm vngcc phngphp chng minh. s lkim ch nam cho ccbi bt ng thc. Nhngphngphp cng rt phongph v a dng : tng hp, phn tch, quy c ng, clng non gi, i bin, chn phn t cc tr Nhng theo kin ch quan ca mnh,nhng phng php tht s cn thit v thng dng s c tc gi gii thiu trongchng 2 : Cc phng php chng minh.
Mc lc :2.1. Bin i lng gic tng ng ... 322.2. S dng cc bc u cs ... 382.3. a v vector v tch v hng .. 462.4. Kt hp cc bt ng thc c in .. 482.5. Tn dng tnh n diuca hm s 572.6. Bi tp . 64
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The Inequalities Trigonometry 32
2.1. Bin i lng gic tng ng :
C th ni phngphp ny l mt phngphp xa nhTrit.N s dng cccng thc lng gic v s bin i qua li gia cc bt ng thc. c th s dng
tt phngphp ny bn c cn trang b cho mnh nhng kin thc cn thit v bin ilng gic (bn c c th tham kho thm phn 1.2. Cc ng thc,bt ng thctrong tamgic).
Thng thng th vi phng php ny, ta s a bt ng thc cn chng minh vdng bt ng thc ng hay quen thuc. Ngoi ra, ta cng c th s dng hai kt ququen thuc 1cos;1sin xx .
V d 2.1.1.
CMR :7
cos3
14sin2
14sin1
>
Li gii :
Ta c :
( )17
3cos
7
2cos
7cos
14sin2
14sin1
7
3cos
7
2cos
7
cos
14
sin2
14
5sin
14
7sin
14
3sin
14
5sin
14sin
14
3sin
14sin1
++=
++=
++=
Mt khc ta c :
( )27
cos7
3cos
7
3cos
7
2cos
7
2cos
7cos
7
2cos
7
4cos
7cos
7
5cos
7
3cos
7cos
2
1
7cos
++=
+++++=
t7
3cos;
7
2cos;
7cos
=== zyx
Khi t ( ) ( )2,1 ta c bt ng thc cn chng minh tng ng vi :
( ) ( )33 zxyzxyzyx ++>++
m 0,, >zyx nn :
( ) ( ) ( ) ( ) ( )403 222 >++ xzzyyx
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The Inequalities Trigonometry 34
V d 2.1.4.
Cho ( )Zkk +
2
,, l bagc tha 1sinsinsin 222 =++ . CMR :
222
2
tantantan213
tantantantantantan
++
Li gii :
Ta c :
222222222
222
222
222
tantantan21tantantantantantan
2tan1
1
tan1
1
tan1
1
2coscoscos
1sinsinsin
=++
=+
++
++
=++
=++
Khi bt ng thc cn chng minh tng ng vi :
( ) ( ) ( ) 0tantantantantantantantantantantantan
tantantantantantan3
tantantantantantan
222
222222
2
++
++
++
pcm.
ng thc xy ra
tantantan
tantantantan
tantantantan
tantantantan
==
=
=
=
V d 2.1.5.
CMR trong ABC bt k ta c :
++++
2tan
2tan
2tan3
2cot
2cot
2cot
CBACBA
Li gii :
Ta c :
2cot
2cot
2cot
2cot
2cot
2cot
CBACBA=++
t2
cot;2
cot;2
cotC
zB
yA
x === th
=++
>
xyzzyx
zyx 0,,
Khi bt ng thc cn chng minh tng ng vi :
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The Inequalities Trigonometry 35
( ) ( )
( ) ( )( ) ( ) ( ) 0
3
3
1113
222
2
++
++++
++++
++++
xzzyyx
zxyzxyzyx
xyz
zxyzxyzyx
zyxzyx
pcm.ng thc xy ra CBA cotcotcot ==
CBA ==
ABC u.
V d 2.1.6.
CMR :xxx cos2
2sin31
sin31
+
+
+
Li gii :
V 1sin1 x v 1cos x nn :0sin3;0sin3 >>+ xx v 0cos2 >+
Khi bt ng thc cn chng minh tng ng vi :( ) ( )
( )
( )( ) 02cos1cos
04cos6cos2
cos1218cos612
sin92cos26
2
2
2
+
+
+
xx
xx
xx
xx
do 1cos x nn bt ng thc cui cng lun ng pcm.
V d 2.1.7.
CMR2
;3
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The Inequalities Trigonometry 37
V d 2.1.9.
Cho ABC khng vung. CMR :
ACCBBACBACBA 222222222222 tantantantantantan9tantantan5tantantan3 +++++
Li gii :
Bt ng thc cn chng minh tng ng vi :( ) ( )( )( )
( )
( ) ( )
( )
( )( ) ( ) 0sincoscos2
01coscos4cos4
01cos4coscos2
01cos42cos2cos2
4
3
cos2
2cos1
2
2cos1
4
3coscoscos
coscoscos
1
coscos
1
coscos
1
coscos
1
coscoscos
4
coscoscos
183
cos
1
cos
1
cos
141
cos
11
cos
11
cos
14
tan1tan1tan18tantantan4tantantan4
22
2
2
2
2
222
222222222222
222222222
222222222
+
+
+++
+++
+
+
+
+
++
++
++
+++++
BABAC
BACC
CBABA
CBA
C
BA
CBA
CBAACCBBACBA
CBACBACBA
CBACBACBA
pcm.
V d sau y, theo kin ch quan ca tcgi, th ligii ca nxng ng l bcthy v bin i lng gic. Nhng bin i tht s lt lo kt hp cng bt ng thcmt cch hp l ng ch mang n cho chng ta mt bi ton tht s c sc !!!
V d 2.1.10.
Cho na ng trn bn knh R , C l mt im ty trn na ng trn. Trong haihnh qut ni tip hai ng trn, gi M v N l hai tip im ca hai ng trn vi
ng knh ca na ng trn cho. CMR : 122 RMN
Li gii :
Gi 21 ,OO l tm ca hai ng trn. t 2=CON (nhvy 20
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The Inequalities Trigonometry 38
NM O
O1O2
C
Vy :
cottancot2
cot 2121 RRRRONMOMN +=+
=+=
Trong vung MOO1 c :
( )
( )
cos1
coscoscos1
cos2sin
11
111
+==+
=
=
RRRR
RROOR
Tng t :
( )
sin1
sinsinsin 2222
+===
RRRROOR
Do :
( )( )
1cossin
2
2cos
2sin
2cos
1
2cos2.
2cos
2sin
2cos
2sin
2cos2
cos1sin1
1cossin
sin1cos
cos1sin
sin
cos
sin1
sin
cos
sin
cos1
cos
2
2
++=
+
=
+
+
=
++
++=
++
+=
+
++
=
R
R
R
R
RR
RRMN
m ( )=+
+ 122
12
22
42cossin R
RMN
pcm.
ng thc xy ra MNOC=4
.
2.2. S dng cc bc u cs :
Cc bc u cs m tcgi mun nhc n y lphn 1.2. Cc ng thc, btng thc trong tam gic. Tas a cc bt ng thc cn chng minh v cc bt ngthc cbn bng cch bin i v s dng cc ng thc cbn.Ngoi ra, khi tham giacck thi, tcgi khuyn bn c nn chng minh cc ng thc, bt ng thc cbns dng nhmt b cho bi ton.
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The Inequalities Trigonometry 39
C1
C
B1
B
A1
A
V d 2.2.1.
Cho ABC .ng phngic trong ccgc CBA ,, ct ng trn ngoi tip ABC
ln lt ti 111 ,, CBA . CMR :
111 CBAABCSS
Li gii :
Gi R l bn knh ng trn ngoi tip ABC th n cng l bn knh ng trnngoi tip 111 CBA .
Bt ng thc cn chng minh tng ng vi :( )1sinsinsin2sinsinsin2 111
22 CBARCBAR
Do 2;2;2111
BA
C
AC
B
CB
A
+
=
+
=
+
= nn :
( )
( )22
cos2
cos2
cos2
cos2
cos2
cos2
sin2
sin2
sin8
2sin
2sin
2sinsinsinsin1
CBACBACBA
BAACCBCBA
+++
V 02
cos2
cos2
cos >CBA
nn :
( ) 8
1
2sin
2sin
2sin2
CBApcm.
ng thc xy ra ABC
u.
V d 2.2.2.
CMR trong mi tamgic ta u c :
2sin
2sin
2sin4
4
7sinsinsinsinsinsin
CBAACCBBA +++
Li gii :
Ta c :2
sin2
sin2
sin41coscoscos CBACBA +=++
Bt ng thc cho tng ng vi :
( )1coscoscos4
3sinsinsinsinsinsin CBAACCBBA +++++
m :
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The Inequalities Trigonometry 40
BABAC
ACACB
CBCBA
coscossinsincos
coscossinsincos
coscossinsincos
=
=
=
nn :
( ) ( )243
coscoscoscoscoscos1 ++ ACCBBA
Tht vy hin nhin ta c :
( ) ( )3coscoscos3
1coscoscoscoscoscos
2CBAACCBBA ++++
Mt khc ta c :2
3coscoscos ++ CBA
( )3 ng ( )2 ng pcm.ng thc xy ra khi v ch khi ABC u.
V d 2.2.3.
Cho ABC bt k. CMR :
1coscos4cos21
1
coscos4cos21
1
coscos4cos21
1
+++
+++
++ ACCCBBBAA
Li gii :
t v tri bt ng thc cn chng minh l T.Theo AM GMta c :
( ) ( )[ ] ( )19coscoscoscoscoscos4coscoscos23 ++++++
ACCBBACBAT m :
2
3coscoscos ++ CBA
v hin nhin :( )
4
3
3
coscoscoscoscoscoscoscoscos
2
++
++CBA
ACCBBA
( ) ( ) ( )29coscoscoscoscoscos4coscoscos23 ++++++ ACCBBACBA
T ( ) ( )2,1 suy ra 1T pcm.
V d 2.2.4.
CMR vi mi ABC bt k, ta c :
( ) ( ) ( )222222 34 accbbaScba +++++
Li gii :
Bt ng thc cn chng minh tng ng vi :
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The Inequalities Trigonometry 42
+
2tansin
2tansin
2
1
2sin
2sin2
AB
BA
BA
Tng t ta c :
+
+
2tansin
2tansin
2
1
2sin
2sin2
2tansin
2tansin
2
1
2sin
2sin2
CA
AC
AC
BC
CB
CB
T suy ra :
( ) ( ) ( )
+++++
++
BAC
ACB
CBA
ACCBBA
sinsin2
tansinsin2
tansinsin2
tan2
1
2sin
2sin
2sin
2sin
2sin
2sin2
++++
2
sin
2
sin
2
sin
2
sin
2
sin
2
sin2coscoscosACCBBA
CBA
Khi :
( )
( ) ( ) ( )4
1coscoscos
4
11coscoscos
4
1coscoscos
2
1
1coscoscos4
1
2sin
2sin
2sin
2sin
2sin
2sin
=++=++++
++++
CBACBACBA
CBAACCBBA
m2
3coscoscos ++ CBA
( )8
51coscoscos
4
1
2sin
2sin
2sin
2sin
2sin
2sin ++++ CBA
ACCBBA
( )2 ngpcm.
V d 2.2.6.
Cho ABC bt k. CMR :
2tan
2tan
2tan
cotcotcot
2223
222
CBA
cba
CBA
cba
++
++
Li gii :
Ta c :
SCBA
cba4
cotcotcot
222
=++
++
nn bt ng thc cho tng ng vi :
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The Inequalities Trigonometry 43
( )1
2tan
2tan
2tan
64222
3
CBA
cbaS
Mt khc ta cng c :
2sin4
cos22cos2
22
2222
Abca
AbcbcaAbccba
+=
SAbcA
Abc
A
a4sin2
2tan
2sin4
2tan
22
==
Tng t ta cng c :
SC
cS
B
b4
2
tan
;4
2
tan
22
( )1 ng pcm.
V d 2.2.7.
CMR trong mi tamgic ta c :( ) ( ) ( ) 3cos1cos1cos1 ++++++++ CabbaBcaacAbccb
Li gii :
Ta c v tri ca bt ng thc cn chng minh bng :( ) ( ) ( ) ( )[ ] ( )BcaAbcCabCbaBacAcbCBA coscoscoscoscoscoscoscoscos ++++++++++
t :
( ) ( ) ( )
BcaAbcCabR
CbaBacAcbQ
CBAP
coscoscos
coscoscos
coscoscos
++=
+++++=
++=
D thy2
3P
Mt khc ta c :( ) ( ) aARCBRBCCBRBcCb ==+=+=+ sin2sin2cossincossin2coscos
Tng t :
cbaQ
cAbBa
bCaAc
++=
=+
=+
coscos
coscos
V ta li c :
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The Inequalities Trigonometry 45
22
3
8
23
8
++
++
+
R
S
ac
caca
cb
bcbc
ba
abab
r
S
Li gii :
Theo AM GM ta c :
2
cabcab
ac
caca
cb
bcbc
ba
abab ++
++
++
+
Do( )
623
822
cba
r
SprS
++=
=
Li c :
( )62
2cbacabcab ++
++
++
++
+
ac
caca
cb
bcbc
ba
abab
r
S2
23
8
v tri c chng minh xong.Ta c :
( )
33
2
33sinsinsin
sinsinsin2
Rcba
CBA
CBARcba
++
++
++=++
Theo AM GM ta c :
( )( ) ( )( ) ( )( )8
2 abcpapcpcpbpbpappS =
( ) ( ) ( )accbba
abc
cba
abc
cba
abcp
R
S
+++++=
++=
++
9
2
9
33
8
3
8
3
82
2
Mt ln na theo AM GM ta c :
( ) ( ) ( ) ( )( )( ) accaca
cb
bcbc
ba
abab
accbba
abc
accbba
abc
++
++
+
+++
+++++ 3.3
99
v phi chng minh xongBt ng thc c chng minh hon ton.
V d 2.2.10.
Cho ABC bt k. CMR :4
2
8
2
8
2
8
3
6
2cos
2cos
2cos
++
R
abc
C
c
B
b
A
a
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The Inequalities Trigonometry 46
Li gii :
p dng BCS ta c :
( )
2cos2cos2cos2cos2cos2cos
222
2444
2
8
2
8
2
8
CBA
cba
C
c
B
b
A
a
++
++++
m :
( )224
222
16
4
9
2cos
2cos
2cos
SR
abc
CBA
=
++
V th ta ch cn chng minh : 2444 16Scba ++ Trc ht ra c : ( ) ( )1444 cbaabccba ++++
Tht vy : ( ) ( ) ( ) ( ) 01 222222 ++ abcccabbbcaa
( ) ( ) ( ) ( ) ( ) ( ) 0222222222 ++++++++ babacacacbcbcba (ng!)Mt khc ta cng c :
( )( )( ) ( )( )( )( ) ( )21616 2 bacacbcbacbacpbpappS +++++==
T ( ) ( )2,1 th suy ra taphi chng minh : ( )( )( ) ( )3bacacbcbaabc +++ t :
bacz
acby
cbax
+=
+=
+=
v cba ,, l ba cnh ca mt tam gic nn 0,, >zyx Khi theo AM GMth :
( )( )( ) ( )( )( )( )( )( )bacacbcbaxyz
zxyzxyxzzyyxabc +++==
+++=
8
222
8
( )3 ng pcm.
2.3 a v vector v tch v hng :
Phng php ny lun a ra cho bn c nhng li gii bt ng v th v. N ctrng cho s kt hp hon gia i s v hnh hc. Nhng tnh cht ca vector li mang
n ligii thtsngsa v p mt. Nhng s lng cc bi ton ca phngphp nykhng nhiu.
V d 2.3.1.
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The Inequalities Trigonometry 47
A
BC
e
e
e
1
2
3
O
A
B C
CMR trong mi tamgic ta c :
2
3coscoscos ++ CBA
Li gii :
Ly cc vector n v 321 ,, eee ln lt trn cc cnh CABCAB ,, .
Hin nhin ta c :
( )( ) ( ) ( )
( )
2
3coscoscos
0coscoscos23
0,cos2,cos2,cos23
0
133221
2
321
++
++
+++
++
CBA
CBA
eeeeee
eee
pcm.
V d 2.3.2.
Cho ABC nhn. CMR :
2
32cos2cos2cos ++ CBA
Li gii :
Gi O, G ln lt l tm ng trn ngoi tip v trng tm ABC .
Ta c : OGOCOBOA 3=++ Hin nhin :
( )( ) ( ) ( )[ ]
( )
2
32cos2cos2cos
02cos2cos2cos23
0,cos,cos,cos23
0
22
22
2
++
+++
+++
++
CBA
BACRR
OAOCOCOBOBOARR
OCOBOA
pcm.
ng thc xy ra ABCGOOGOCOBOA ==++ 00 u.
V d 2.3.3.
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The Inequalities Trigonometry 48
O
A
B C
Cho ABC nhn. CMR Rzyx ,, ta c :
( )2222
12cos2cos2cos zyxCxyBzxAyz ++++
Li gii :
Gi O l tm ng trn ngoi tip ABC .Ta c :
( )
( )222
222
222
2
2
12cos2cos2cos
02cos22cos22cos2
0.2.2.2
0
zyxCxyBzxAyz
BzxAyzCxyzyx
OAOCzxOCOByzOBOAxyzyx
OCzOByOAx
++++
+++++
+++++
++
pcm.
2.4. Kt hp cc bt ng thc c in :
V ni dung cng nhcch thc s dng cc bt ng thc chng ta bn chng1:Cc bc u cs. V th phn ny, tas khng nhc li m xt thm mt s vdphc tp hn, th v hn.
V d 2.4.1.
CMR ABC ta c :
2
39
2cot
2cot
2cot
2sin
2sin
2sin
++
++
CBACBA
Li gii :
Theo AM GM ta c :
3
2
sin
2
sin
2
sin
3
2sin
2sin
2sin
CBA
CBA
++
Mt khc :
2sin
2sin
2sin
2cos
2cos
2cos
2cot
2cot
2cot
2cot
2cot
2cot
CBA
CBA
CBACBA==++
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The Inequalities Trigonometry 50
( )
CBA
CCBBAA
CBA
CCBBAA
CBA
CBA
coscoscos2
cossincossincossin
2
3
coscoscos2
cossincossincossin
coscoscos
2sin2sin2sin4
1
3
++=
++
=
Suy ra :
( )( )
( )1tantantan2
9
coscoscos
cossincossincossincoscoscos
2
9tantantancoscoscos
3
3
CBA
CBA
CCBBAACBACBACBA
=
++++
Mt khc : 33tantantan CBA
( )22
3933
2
9tantantan
2
9 33= CBA
T ( )1 v ( )2 suy ra :
( )( )2
39tantantancoscoscos ++++ CBACBA
pcm.
V d 2.4.3.
Cho ABC ty. CMR :
34
2tan
1
2tan
2tan
1
2tan
2tan
1
2tan
++
++
+C
C
B
B
A
A
Li gii :
Xt ( )
=
2;0tan
xxxf
Khi : ( ) =xf ''
Theo Jensen th : ( )132
tan2
tan2
tan ++CBA
Xt ( )
= 2;0cot
xxxg
V ( ) ( )
>+=
2;00cotcot12'' 2
xxxxg
Theo Jensen th : ( )2332
cot2
cot2
cot ++CBA
Vy ( ) ( )+ 21 pcm.
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The Inequalities Trigonometry 51
V d 2.4.4.
CMR trong mi tamgic ta c :3
3
21
sin
11
sin
11
sin
11
+
+
+
+
CBA
Li gii :
Ta s dng b sau :B : Cho 0,, >zyx v Szyx ++ th :
( )12
11
11
11
1
3
+
+
+
+
Szyx
Chng minh b :Ta c :
( ) ( )2111111111 xyzzxyzxyzyxVT +
+++
+++=
Theo AM GM ta c :
( )399111
Szyxzyx
++++
Du bng xy ra trong ( )3
3S
zyx ===
Tip tc theo AM GM th :33 xyzzyxS ++
( )4271
27 3
3
Sxyzxyz
S
Du bng trong ( )4 xy ra3
Szyx ===
Vn theo AM GM ta li c :
( )51
3111
3
2
++
xyzzxyzxy
Du bng trong ( )5 xy ra3
Szyx ===
T ( ) ( )54 suy ra :
( )627111
2Szxyzxy++
Du bng trong ( )6 xy ra ng thi c du bng trong ( ) ( )3
54S
zyx ===
T ( )( )( )( )6432 ta c :
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The Inequalities Trigonometry 52
( )3
32
31
2727911
+=+++
SSSSVT
B c chng minh. Du bng xy ra ng thi c du bng trong ( )( )( )643
3
Szyx ===
p dng vi 0sin,0sin,0sin >=>=>= CzByAx
m ta c2
33sinsinsin ++ CBA vy y
2
33=S
Theo b suy ra ngay :3
3
21
sin
11
sin
11
sin
11
+
+
+
+
CBA
Du bng xy ra2
3sinsinsin === CBA
ABC
u.
V d 2.4.5.
CMR trong mi tamgic ta c :
3plll cba ++
Li gii :
Ta c : ( ) ( ) ( )1222cos2 appcb
bc
bc
app
cb
bc
cb
A
bcla
+=
+=
+=
Theo AM GM ta c 12
+ cb
bcnn t ( )1 suy ra :
( ) ( )2appla
Du bng trong ( )2 xy ra cb = Hon ton tng t ta c :
( ) ( )
( ) ( )4
3
cppl
bppl
c
b
Du bng trong ( ) ( )43 tng ng xy ra cba ==
T ( )( ) ( )432 suy ra :
( ) ( )5cpbpapplll cba ++++ Du bng trong ( )5 xy ra ng thi c du bng trong ( )( ) ( ) cba ==432p dng BCS ta c :
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The Inequalities Trigonometry 53
( ) ( )( )63
332
pcpbpap
cbapcpbpap
++
++
Du bng trong ( )6 xy ra cba ==
T ( ) ( )65 ta c : ( )73plll cba ++
ng thc trong ( )7 xy ra ng thi c du bng trong ( ) ( ) cba ==65ABC u.
V d 2.4.6.
Cho ABC bt k. CMR :
R
r
abc
cba 24
333
++
Li gii :
Ta c : ( )( )( )cpbpappprR
abcS ===
4
( )( )( ) ( )( )( )
( )( )( )abc
abccbacaacbccbabba
abc
cbabacacb
abc
cpbpap
pabc
cpbpapp
pabc
S
R
r
2
222222882
333222222
2
+++++=
+++=
=
==
abccba
ca
ac
bc
cb
ab
ba
abccba
Rr
333333
624 ++
++++++
++=
pcm.
V d 2.4.7.
Cho ABC nhn. CMR :
abcbA
a
C
ca
C
c
B
bc
B
b
A
a27
coscoscoscoscoscos
+
+
+
Li gii :
Bt ng thc cn chng minh tng ng vi :
CBABA
A
C
CA
C
C
B
BC
B
B
A
Asinsinsin27sin
cos
sin
cos
sinsin
cos
sin
cos
sinsin
cos
sin
cos
sin
+
+
+
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The Inequalities Trigonometry 54
27coscos
coscos1
coscos
coscos1
coscos
coscos1
sinsinsin27sincoscos
sinsin
coscos
sinsin
coscos
sin
AC
AC
CB
CB
BA
BA
CBABAC
BA
CB
AC
BA
C
t
+
=
+
=
+
=
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The Inequalities Trigonometry 55
Li gii :
Bt ng thc cn chng minh tng dng vi :
( )
( ) ( )cba
abccbacba
cba
abccbacba
+++++++
+++
++++
72935
2
435
36
2222
2
222
Theo BCS th : ( ) ( )2222 3 cbacba ++++ ( ) ( ) ( )1279 2222 cbacba ++++
Li c :
++
++
3 222222
3
3
3
cbacba
abccba
( )( )( )( )
( ) ( )2728
7289
222
222
222
cba
abccba
abccbacbaabccbacba
++++
++++
++++
Ly ( )1 cng ( )2 ta c :
( ) ( ) ( )
( ) ( )cba
abccbacba
cba
abccbacbacba
+++++++
++++++++++
72935
729827
2222
2222222
pcm.
V d 2.4.9.
CMR trong ABC ta c :
6
2sin
2cos
2sin
2cos
2sin
2cos
+
+
C
BA
B
AC
A
CB
Li gii :
Theo AM GM ta c :
( )1
2sin
2cos
2sin
2cos
2sin
2cos
3
2sin
2cos
2sin
2cos
2sin
2cos
3C
BA
B
AC
A
CB
C
BA
B
AC
A
CB
+
+
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The Inequalities Trigonometry 56
m :
( )( )( )CBA
BAACCB
CC
BABA
BB
ACAC
AA
CBCB
C
BA
B
AC
A
CB
sinsinsinsinsinsinsinsinsin
2sin
2cos2
2cos
2sin2
2sin
2cos2
2cos
2sin2
2sin
2cos2
2cos
2sin2
2sin
2cos
2sin
2cos
2sin
2cos
+++=
+
+
+
=
Li theo AM GM ta c :
+
+
+
ACAC
CBCB
BABA
sinsin2sinsin
sinsin2sinsin
sinsin2sinsin
( )( )( )
( )( )( )( )28
sinsinsin
sinsinsinsinsinsin
sinsinsin8sinsinsinsinsinsin
+++
+++
CBA
BAACCB
CBABAACCB
T ( )( )21 suy ra :
683
2sin
2cos
2sin
2cos
2sin
2cos
3=
+
+
C
BA
B
AC
A
CB
pcm.
V d 2.4.10.
CMR trong mi ABC ta c : 29sinsinsinsinsinsin
++
R
rACCBBA
Li gii :
Bt ng thc cn chng minh tng ng vi :
2
2
2
36
9222222
9sinsinsinsinsinsin
rcabcab
raccbba
rACRCBRBAR
++
++
++
Theo cng thc hnh chiu :
+=
+=
+=
a
BArc
a
ACrb
a
CBra cot
2cot;cot
2cot;cot
2cot
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The Inequalities Trigonometry 57
+
++
+
+
++
+
+=++
2cot
2cot
2cot
2cot
2cot
2cot
2cot
2cot
2cot
2cot
2cot
2cot
2
22
CBBAr
BAACr
ACCBrcabcab
Theo AM GM ta c :
( )1cotcotcot42
cot2
cot22
cot2
cot22
cot2
cot2
cot2
cot 2 BACACCBACCB
=
+
+
Tng t :
( )
( )3cotcotcot42
cot2
cot2
cot2
cot
2cotcotcot42
cot2
cot2
cot2
cot
2
2
ACBCBBA
CBABAAC
+
+
+
+
T ( )( )( )321 suy ra :
( )42
cot2
cot2
cot122
cot2
cot2
cot2
cot
2cot
2cot
2cot
2cot
2cot
2cot
2cot
2cot
3222 CBABAAC
BAACBAAC
+
++
+
+
++
+
+
Mt khc ta c : ( )5272
cot2
cot2
cot332
cot2
cot2
cot 222 CBACBA
T ( ) ( )54 suy ra : ( )6363.122
cot2
cot2
cot123 222 =CBA
T ( ) ( )64 suy ra pcm.
2.5. Tn dng tnh n iu ca hm s :
Chng ny khi c th bn c cn c kin thc cbn v o hm, khost hm sca chng trnh 12 THPT. Phngphp ny thc s c hiu qu trong cc bi bt ngthc lnggic. c th s dng tt phngphp ny th bn c cn n nhng kinhnghimgii ton cc phngphp nu cc phn trc.
V d 2.5.1.
CMR :
xx
2sin > vi
2;0
x
Li gii :
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Trng THPT chuyn L T Trng Cn Th Bt ng thc lng gicChng 2 Cc phngphp chng minh
The Inequalities Trigonometry 58
Xt ( )
2sin=
x
xxf vi
2;0
x
( )2
sincos'
x
xxxxf
=
Xt ( ) xxxxg sincos = vi
2;0
x
( ) ( )xgxxxxg
vi
2;0
Li gii :
Bt ng thc cn chng minh tng ng vi :
( )
( ) 0cossin
cossin
3
1
3
1
>
>
xx
x
x
Xt ( ) ( ) xxxxf =
3
1
cossin vi
2;0 x
Ta c : ( ) ( ) ( ) 1cossin3
1cos' 3
42
3
2
=
xxxxf
( ) ( ) ( ) ( )
>+=
2;00cossin
9
4sin1cos
3
2'' 4
73
3
1 xxxxxxf
( )xf ' ng bin trong khong ( ) ( ) 00'' => fxf
( )xf cng ng bin trong khong ( ) ( ) => 00fxf pcm.
V d 2.5.3.
CMR nu a l gc nhn hay 0=a th ta c :1tansin 222 ++ aaa
Li gii :
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The Inequalities Trigonometry 59
p dng AM GM cho hai s dng asin2 v atan2 ta c :aaaaaa tansintansintansin
2222222 +
=+
Nhvy ta ch cn chng minh : aaa 2tansin >+ vi2
0
+=
+=+=
2;00
cos
cos1cos1cos1
cos
1cos2cos2
cos
1cos'
22
23
2
x
x
xxx
x
xx
xxxf
( )xf ng bin trn khong ( ) ( )0faf > vi aaaa 2tansin2
;0 >+
12tansin22222
++= aaaa
1tansin 222 ++ aaa (khi 0=a ta c du ng thc xy ra).
V d 2.5.4.
CMR trong mi tamgic ta u c :
( ) CBACBABABABA coscoscoscoscoscos12
13coscoscoscoscoscos1 ++++++
Li gii :
Bt ng thc cn chng minh tng ng vi :( ) ( )CBABABABACBA coscoscos
6
131coscoscoscoscoscos2coscoscos21 ++++++
( ) ( CBABABABACBA coscoscos6
131coscoscoscoscoscos2coscoscos 222 ++++++++
( ) ( )CBACBA coscoscos6
131coscoscos
2+++++
6
13
coscoscos
1coscoscos
+++++
CBACBA
t
2
31coscoscos =
2
3;10
11'
2ng bin trn khong .
( ) =
6
13
2
3fxf pcm.
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The Inequalities Trigonometry 60
V d 2.5.5.
Cho ABC c chu vi bng 3. CMR :
( )2
222
4
13sinsinsin8sinsinsin3
RCBARCBA +++
Li gii :
Bt ng thc cn chng minh tng ng vi :( )( )( ) 13sin2sin2sin24sin4.3sin4.3sin4.3 222222 +++ CRBRARCRBRAR
134333222
+++ abccba
Do vai tr ca cba ,, l nhnhau nn ta c thgis cba
Theo githit :2
3133 >+=++ ccccbacba
Ta bin i :
( )( )[ ]( )
( ) ( )
( ) ( )cabcc
cabcc
ababccc
abccabba
abccba
abccbaT
232333
322333
64333
4323
433
4333
22
22
22
22
222
222
+=
++=
++=
+++=
+++=
+++=
v 0230322
3>
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Trng THPT chuyn L T Trng Cn Th Bt ng thc lng gicChng 2 Cc phngphp chng minh
The Inequalities Trigonometry 61
Li gii :
Ta c :
( )( )
( )
( )( )
( )
( )( )
( )
p
cp
p
bp
p
apCBA
cpp
bpapC
bpp
apcpB
app
cpbpA
=
=
=
=
2tan
2tan
2tan
2tan
2tan
2tan
v( )( )( )
p
cp
p
bp
p
ap
p
cpbpapp
p
S
S
r
=
==
22
2
Do :2
tan2
tan2
tan2 CBA
S
r=
Mt khc :
( ) ( )
( )
( )
2cot
2cot
2cot
2cos
2sin
2sin
2sin
2cos
2cos
2cos
2cos
2tansinsinsin2
sinsinsin2
2tan
2tan2
CBA
A
A
CBA
CBA
AACBR
CBAR
Aacb
cba
Aap
cba
r
p
==
+
++=
+
++=
++=
Khi bt ng thc cn chng minh tng ng vi :
33
28
2cot
2cot
2cot
2cot
2cot
2cot
1
33
28
2cot
2cot
2cot
2tan
2tan
2tan
+
+
CBA
CBA
CBACBA
t 332
cot2
cot2
cot = tCBA
t
Xt ( )t
ttf1
+= vi 33t
( ) 3301
1'2
>= tt
tf
( ) ( ) =+==33
28
33
13333min ftf pcm.
V d2.5.7.
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The Inequalities Trigonometry 62
CMR vi mi ABC ta c :
( )( )( ) 233
38222 eRcRbRaR
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The Inequalities Trigonometry 64
V d2.5.10.
CMR :20
720sin
3
1 0
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The Inequalities Trigonometry 65
2.6.1. ( )2
5cos2cos2cos3 + BCA
2.6.2. 42cos322cos22cos3 ++ CBA
2.6.3. ( ) 542cos532cos2cos15 ++++ CBA
2.6.4. 342
tan2
tan2
tan ++ CBA vi ABC c mt gc 32
2.6.5.2222 4
1111
rcba++
2.6.6.cba r
c
r
b
r
a
r
abc 333++
2.6.7.( )( )( )
23
k Gii: Trc ht ta chng minh :B 1: 0, > yx v 01 > k th :
( ) ( ) ( )Hyxyx kkkk ++ 12
Chng minh:( ) ( ) ( ) ( ) 0121121 11 ++=
+
+
kkk
k
kk
k
aaafy
x
y
xH vi 0>= a
y
x
V ( ) ( ) ( ) 021' 11 =+= kk aakaf 1= a hoc 1=k . Vi 1=k th ( )H l ng thcng.Do 0>a v 01 >> k th ta c :
( ) 00 > aaf v 01 >> k
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lin quan n bt ng thc v lnggic
The Inequalities Trigonometry 79
( )H c chng minh.Tr libi ton 2 :T h ( )1 ta c :
+
+
k
b
k
ck
k
bck
a a
cd
a
bd
a
cd
a
bdR 12
( p dng b ( )H via
cdy
a
bdx bc == ; )
Tng t :
+
+
k
a
k
bkk
c
k
a
k
ckk
b
c
bd
c
adR
b
cd
b
adR
1
1
2
2
( )kck
b
k
a
k
kkk
c
kkk
b
kkk
a
kk
c
k
b
k
a
ddd
a
b
b
ad
a
c
c
ad
b
c
c
bdRRR
++
+
+
+
+
+
++
2
2 1
pcm.ng thc xy ra khi ABC u v M l tm tam gic. p dng ( )E ta chng minhcbi ton sau :Bi ton 3: Chng minh rng :
( )3111
2111
++++
cbacba RRRddd
Gii: Thc hinphp nghch o tm M, phng tch n v ta c :
=
=
=
c
b
a
RMC
RMB
RMA
1*
1*
1*
v
=
=
=
c
b
a
dMC
dMB
dMA
1''
1''
1''
p dng ( )E trong '''''' CBA :
( )
++++
++++
cbacba RRRddd
MCMBMAMCMBMA111
2111
***2''''''
pcm.M rng kt qu ny ta c bi ton sau :Bi ton 4: Chng minh rng :
( )42 kck
b
k
a
k
c
k
b
k
a
kRRRddd ++++
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lin quan n bt ng thc v lnggic
The Inequalities Trigonometry 80
vi 10 > k Hng dn cch gii : Ta thy ( )4 d dng c chng minh nh p dng ( )2 trongphp bin hnh nghch o tm M, phng tch n v. ng thc xy ra khi ABC uv M l tm tam gic.By gi vi 1>k th t h ( )1 ta thu c ngay :Bi ton 5: Chng minh rng :
( )( )52 222222 cbacba dddRRR ++>++ Xutpht t bi ton ny, ta thu c nhng kt qu tng qut sau :Bi ton 6: Chng minh rng :
( )( )62 kck
b
k
a
k
c
k
b
k
a dddRRR ++>++
vi 1>k Gii: Chng ta cng chng minh mt b :B 2: 0, > yx v 1>k th :
( ) ( )Gyxyx kkk ++
Chng minh:
( ) ( ) ( ) 01111 >+=+>
+ k
k
k
kk
aaagy
x
y
xG (t 0>= a
y
x)
V ( ) ( ) 1;001' 11 >>>+= kaaakag kk
( ) 1;00 >>> kaag
( )G c chng minh xong.
S dng b ( )G vobi ton ( )6 :T h ( )1 :
k
b
k
c
k
bck
aa
cd
a
bd
a
cd
a
bd
R
+
>
+ (t a
cd
ya
bd
xbc
== ; )
Tng t:
k
a
k
bk
c
k
a
k
ck
b
c
bd
c
adR
b
cd
b
adR
+
>
+
>
( )k
c
k
b
k
a
kk
k
c
kk
k
b
kk
k
a
k
c
k
b
k
a
ddd
a
b
b
ad
a
c
c
ad
b
c
c
bdRRR
++
+
+
+
+
+
>++
2
pcm.Bi ton 7: Chng minh rng :
( ) ( )72 kak
a
k
a
k
a
k
a
k
a RRRddd ++>++
vi 1
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lin quan n bt ng thc v lnggic
The Inequalities Trigonometry 81
Hng dn cch gii : Ta thy ( )7 cng c chng minh d dng nh p dng ( )6 trongphp bin hnh nghch o tm M, phng tch n v. ng thc khng th xy ratrong ( )6 v ( )7 .Xt v quan h gia ( )cba RRR ,, vi ( )cba ddd ,, ngoi bt ng thc ( )E v nhng m
rng ca n, chng ta cn gp mt s bt ng thc rt hay sau y. Vic chng minhchng xin dnh chobn c :
( )( )( )
( )( )( )ccbbccaabbaacba
cbcabacba
c
ba
b
ca
a
cb
cbacba
dRdRdRdRdRdRRRR
ddddddRRR
R
dd
R
dd
R
dd
dddRRR
+++
+++
+
++
++
222)4
)3
3)2
8)1
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lin quan n bt ng thc v lnggic
The Inequalities Trigonometry 82
ng dng ca i s vo vic pht hin v chngminh bt ng thc trong tam gic
L Ngc Anh
(HS chuyn ton kha 2005 2008Trng THPT chuyn L T Trng, Cn Th)
1/ Chng ta i tbi ton i ssau:Vi
x
0, 0, 0, 0,2222
ta lun c:
x x 2x< tg < < sinx < x
2 2 .
Chng minh:Ta chng minh 2 bt ng thc:2
sinx
x
> v2
2
x xtg
< .
t1
( ) sinf x xx
= l hm sxc nh v lin tc trong 0,2
.
Ta c:2
os x- sin x'( )
xcf x
x= . t ( ) os x- sin xg x xc= trong 0,
2
khi
( ) ( )' sin 0g x x x g x= nghch bin trong on 0,2
nn ( ) ( )0g x g< =0 vi
0,2
x
. Do ( )' 0f x < vi 0,
2x
suy ra ( )2
2f x f
> =
hay
2sin
xx
>
vi 0,2
x
.
t ( ) 1h x tgxx
= xc nh v lin tc trn 0,2
.
Ta c ( )2 2
sin' 0
2 os2
x xh x
xx c
= > 0,
2x
nn hm s ( )h x ng bin, do
( )2 2
xh x h
< =
hay
2
2
x xtg
< vi 0,
2x
.
Cn 2 bt ng thc2 2
x xtg > v sinx x< dnh cho bn c tchng minh.
By gimi l phn ng ch :Xt ABC: BC = a , BC = b , AC = b . GiA, B, Cl ln cc gc bng radian;
r, R, p, Sln lt l bn knh ng trn ni tip, bn knh ng trn ngoi tip, nachu vi v din tch tam gic; la, ha, ma, ra,tng ng l di ng phn gic, ngcao, ng trung tuyn v bn knh ng trn bng tip ng vi nhA...
Bi ton 1:Chng minh rng trong tam gic ABC nhn ta lun c:
2 2 2os os os4
p pAc x Bc B Cc C
R R
< + + <
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lin quan n bt ng thc v lnggic
The Inequalities Trigonometry 83
Nhn xt:
Tnh l hm s sin quen thuc trong tam gic ta c: sin sin sinp
A B BR
+ + = v
bi ton i sta ddng a ra bin i sau 2 2 24
os 2 os sin os2
AAc A tg c A A Ac A
< = < , t
a n li gii nhsau.Li gii:
Ta c: 2 2 24
os 2 os sin os2
AAc A tg c A A Ac A
< = < 2os sin
pAc A A
R< =
v 2 24
os sin os4
p pAc A A Ac A
R R
> = > . Ty suy ra pcm.
Trong mt tam gic ta c nhn xt sau: 12 2 2 2 2 2
A B B C C Atg tg tg tg tg tg+ + = kt hp
vi2
2
x xtg
< nn ta c
2 2 2 2 2 21
2 2 2 2 2 2
A B B C C A A B B C C Atg tg tg tg tg tg
+ + > + + =
2
. . .4
A B B C C A
+ + > (1). Mt khc2 2
x xtg > nn ta cng d dng c
12 2 2 2 2 2 2 2 2 2 2 2
A B B C C A A B B C C Atg tg tg tg tg tg+ + < + + = t y ta li c
. . . 4A B B C C A+ + < (2). T(1) v (2) ta c bi ton mi.Bi ton 2:Chng minh rng trong tam gic ABC nhn ta lun c:
2
. . . 44
A B B C C A
< + + <
Lu :Khi dng cch ny sng to bi ton mi th ton l ABC phi l nhn
v trong bi ton i sth 0, 2x
. Li gii bi ton tng tnhnhn xt trn.
Mt khc, p dng bt ng thc( )
2
3
a b cab bc ca
+ ++ + th ta c ngay
( )2 2
. . .3 3
A B CA B B C C A
+ ++ + = . Ty ta c bi ton cht hn v p hn:
2 2
. . .4 3
A B B C C A
+ +
By gita thi tcng thc la, ha, ma, ra tm ra cc cng thc mi.
Trong ABC ta lun c: 2 sin sin sin2 2a aA A
S bc A cl bl= = +
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The Inequalities Trigonometry 84
1 1 1 1
A 2 22 os2
a
b c b c
l bc b cbcc
+ + = > = +
1 1 1 1 1 1 1 1 1 1
2 sin sin sina b cl l l a b c R A B C
+ + > + + > + +
1 1 1 1 1 1 1
2a b c
l l l R A B C
+ + > + +
.
Nhvy chng ta c Bi ton 3.Bi ton 3: Chng minh rng trong tam gic ABC nhn ta lun c:
1 1 1 1 1 1 1
2a b c
l l l R A B C
+ + > + +
Mt khc, ta li c( )2 sin sin
A2 os 2sin
22 2
a
R B Cbc b c
Alc
++= =
. p dng bi ton i s ta
c:
( )( )2
2 2a
B CRR B C bc
AA l
+
+> >
( ) ( )
( )
4
a
R B C R B Cbc
B C l B C
+ +> >
+ +
4
a
bc RR
l
> > .
Hon ton tng tta c:4
c
ab RR
l
> > v
4
b
ca RR
l
> > . Ty, cng 3 chui bt
ng thc ta c:Bi ton 4: Chng minh rngtrong tam gic ABC nhn ta lun c:
12 3c a b
R ab bc ca Rl l l
< + + <
Trong tam gic ta c kt qu sin b ch h
Ac b
= = , sin c ah h
Ba c
= = v sin a bh h
Cb a
= = ,
m t kt qu ca bi ton i s ta d dng c 2 sin sin sinA B C < + + < , m
( )1 1
2 sin sin sin aA B C hb c
+ + = +
1 1 1 1b ch h
c a a b
+ + + +
, t y ta c c Bi
ton 5.Bi ton 5: Chng minh rngtrong tam gic ABC nhn ta lun c:
1 1 1 1 1 14 2a b ch h h
b c c a a b
< + + + + + + + +
( )4 2R r aA bB cC > + +
Kt hp 2 iu trn ta c iu phi chng minh.Sau y l cc bi ton c hnh thnh tcc cng thc quen thuc cc bn luyn
tp:Bi ton: Chng minh rng trong tam gic ABC nhn ta lun c:a/ ( ) ( )2 8 2 2p R r aA bB cC p R r + < + + < + .
b/ ( ) ( ) ( )( ) ( ) ( ) 22
Sp a p b p b p c p c p a S
< + + < .
c/ ( ) ( ) ( )2 2 22
abc a p a b p b c p c abc
< + + < .
d/1 1 1 1 1 1
4 2a b cl l lb c c a a b
< + + + + + = ( )' 0f x > nn hm ( )f x ng bin .
Ch 3 bt ng thc i s:1.Bt ng thc AM-GM:
Cho nsthc dng 1 2, , ..., na a a , ta lun c:1 2
1 2
......n n
n
a a aa a a
n
+ + +
Du = xy ra 1 2 ... na a a = = = .
2.Bt ng thc Cauchy-Schwarz:
Cho 2 bns ( )1 2, ,..., na a a v ( )1 2, ,..., nb b b trong 0, 1,ib i n> = . Ta lun c:
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lin quan n bt ng thc v lnggic
The Inequalities Trigonometry 87
( )222 2
1 21 2
1 2 1 2
......
...nn
n n
a a aaa a
b b b b b b
+ + ++ + +
+ + +
Du = xy ra 1 2
1 2
... n
n
aa a
b b b = = = .
3.Bt ng thc Chebyshev:Cho 2 dy ( )1 2, ,..., na a a v ( )1 2, ,..., nb b b cng tng hoc cng gim, tc l:
1 2
1 2
...
...n
n
a a a
b b b
hoc 1 2
1 2
...
...n
n
a a a
b b b
, th ta c:
1 1 2 2 1 2 1 2... ... ....n n n na b a b a b a a a b b b
n n n
+ + + + + + + + +
Du = xy ra 1 2
1 2
...
...n
n
a a a
b b b
= = =
= = =.
Nu 2 dy n iu ngc chiu th i chiu du bt ng thc.Xt trong tam gic BC c A B (A,B s o hai gc A,B ca tam gic theo
radian).
A B sin sin
A B
A B ( theo chng minh trn th hm ( )
xf x =
sinx)
2 2
A B
a b
R R
A a
B b , m A B a b . Nhvy ta suy ra nu a b th
A a
B b
(i).
Hon ton tng t : a b c
A B C
a b c v nh vy ta c
( )A
0B
a ba b
, ( ) 0
B Cb c
b c
v ( ) 0
C Ac a
c a
.Cng 3
bt ng thc ta c ( ) 0cyc
A Ba b
a b
( ) ( )2
cyc
AA B C b c
a+ + + (1).
-Cng A B C+ + vo 2 v ca (1) ta thu c:
( ) ( )3A B C
A B C a b ca b c
+ + + + + +
(2)
-TrA B C+ + vo 2 vca (1) ta thu c: ( ) ( )2cyc
AA B C p a
a
+ + (3).
Ch rng A B C + + = v 2a b c p+ + = nn (2) 3 2cyc
Ap
a
3
2cyc
A
a p
(ii), v (3) ( )
2cyc
Ap a
a
(iii).
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Trung THPT chuyn L T Trng Cn Th Bt ng thc lng gicChng 4Mt s chuyn bi vit hay,th v
lin quan n bt ng thc v lnggic
The Inequalities Trigonometry 88
Mt khc ta c thp dng bt ng thc Chebyshev cho 2 bs
, ,A B C
a b c
v ( ), , .p a p b p c Ta c: a b c A B C
a b c
p a p b p c
( )( )
3 3 3cyc
A A B Cp ap a p b p ca a b c
+ + + +
( )3
cyc
cyc
ApaA
p aa
. M
3
2cyc
A
a p
ta suy ra: ( )
32
3 3cyc
cyc
Ap p
aA pp a
a
hay ( )
3 2cyc
cyc
Ap
aAp a
a
(iv).
Ta ch n hai bt ng thc (ii) v (iii):
-p dng bt ng thc AM-GM cho 3 s , ,A B C
a b cta c:
13. .
3. .cyc
A A B C
a a b c
kt
hp vi bt ng thc (ii) ta suy ra13. . 3
3. . 2
A B C
a b c p
3. . 2
. .
a b c p
A B C
(v). Mt
khc, ta li c
1
3. .3
. .cyc
a a b c
A A B C
, m theo (v) ta ddng suy ra
1
3. . 2
. .
abc p
ABC
, t ta
c bt ng thc6
cyc
a p
A (vi).
-p dng bt ng thc Cauchy-Schwarz , ta c :
( )22 2
cyc cyc
A B CA A
a aA Aa Bb Cc Aa Bb Cc
+ += =
+ + + + (vii), m ta tm c
( ) ( )2 8 2 2p R r Aa Bb Cc p R r + < + + < + (bi tp a/ phn trc) nn
( )
2
2cyc
A
a p R r
>
(viii) (chng vi tam gic nhn).
-p dng bt ng thc AM-GM cho 3 s ( ) ( ) ( ), ,A B C
p a p b p ca b c
ta c:
( ) ( ) ( ) ( ) ( ) ( )2
3 3 3. . . . . . .
3 3 3. . 4 . 4 .
A B C ABC S ABC S ABCp a p b p c p a p b p c
a b c abc p S R p R + + = =
( )2
3. .
34 .cyc
A S A B Cp a
a p S R (4)m ( )
3 2cyc
cyc
Ap aA
p aa
(theo iv) nn t (4)
32
43
. . 729 . . .3
4 . 3 2 4cyc
cyc
Ap
aS A B C S A B C Ap
p S R R a
3
4729 . . . 3
4 2
S A B C p
R p
354 . . . . .S A B C p R (ix).
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Trung THPT chuyn L T Trng Cn Th Bt ng thc lng gicChng 4Mt s chuyn bi vit hay,th v
lin quan n bt ng thc v lnggic
The Inequalities Trigonometry 89
Xt tng
2 22y yx z x z
Tb By a Ax a Ax c Cz c Cz b By
= + + + + +
.
Ta c: 0T
2 2 2
1 1 1 1 1 1. . . 2 0y z z x x y
x a A y b B z c C ab AB bc BC ca CA
+ + + + + + +
.
. . . 2 0y z bc z x ca x y ab c a b
x aA y bB z cC AB BC CA
+ + + + + + +
. . . 2y z bc z x ca x y ab a b c
x aA y bB z cC BC CA AB
+ + + + + + +
(5).
p dng bt ng thc AM-GM ta c:
13 6
3a b c abc p
ABCBC CA AB
+ +
(6).
T(5) v (6) ta c: 6. . .y z bc z x ca x y ab px aA y bB z cC + + ++ + (7).
Thay (x, y, z) trong (7) bng (p-a, p-b, p-c) ta c:
( ) ( ) ( )
12bc ca ab p
A p a B p b C p c + +
(x)
Thay (x, y, z) trong (7) bng (bc, ca, ab)ta c:12b c c a a b p
A B C
+ + ++ + (xi).
3/ Chng ta xt bt ng thc sau:2x
sinx
vi
x
0, 0, 0, 0,2222
(phn chng minh bt
ng thc ny dnh cho bn c).
Theo nh l hm ssin ta c sin2
aA
R= v kt hp vi bt ng thc trn ta c
2 4
2
a A a R
R A , t ta ddng suy ra
12
cyc
a R
A > .
4/ Bt ng thc:2 2
2 2
sin x - x
x + x vi ( ]x 0, 0, 0, 0, (bt ng thc ny xem nhbi
tp dnh cho bn c).
Bt ng thc trn tng ng2
2 2
sin 21
x x
x x
+
3
2 2
2sin
xx x
x
+(1).
Trong tam gic ta c: 3 3sin sin sin2
A B C+ + (2) (bn c tchng minh).T(1)
v (2) ta thu c3 3 3
2 2 2 2 2 2
3 3sin 2
2 cyc
A B CA A B C
A B C
> + + + +
+ + +
3 3 3
2 2 2 2 2 2
3 32
2
A B C
A B C
> + +
+ + +
3 3 3
2 2 2 2 2 2
3 3
2 4
A B C
A B C
+ + >
+ + +.
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Trung THPT chuyn L T Trng Cn Th Bt ng thc lng gicChng 4Mt s chuyn bi vit hay,th v
lin quan n bt ng thc v lnggic
The Inequalities Trigonometry 90
Mt khc, p dng bt ng thc cho 3 gc A, B, C ta thu c2 2
2 2
sinA A
A A
>
+,
2 2
2 2
sinB B
B B
>
+ v
2 2
2 2
sin C C
C C
>
+, cng cc bt ng thc ta c:
2 2 2 2 2 22 2 2 2 2 2
sin sin sinA B C A B CA B C A B C
+ + > + ++ + +
, t y p dng nh l hm s sin
sin2
aA
R= ta c
2 2 2 2 2 2
2 2 2 2 2 22 2 2
a b c
A B CR R R
A B C A B C
+ + > + +
+ + +hay
2 2
2 22
cyc
a AR
A A
>
+ .
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90/106
Trung THPT chuyn L T Trng Cn Th Bt ng thc lng gicChng 4Mt s chuyn bi vit hay,th v
lin quan n bt ng thc v lnggic
The Inequalities Trigonometry 91