Bồi dưỡng học sinh giỏi lượng giác Tài liệu dùng cho học sinh chuyên Toán và học sinh giỏi

- ChU'ring 4: Bat dang thiJc trong tarn giac.. Moi phan cua cuon sach c6 the xem nhiT mot chuyen de rieng trinh bay tron ven mot van de mot each hoan chinh rieng phan phu'dng trinh lifd

^ - ^ V r y T R L T O N G TRUNG HOC Q U 6 c GIA CHU V A N AN

P H A N H U Y KHAI (Chu bien)

Lcdnoiddu

Boi dUdng hoc sink gidi lM(/ng gidc la quyen sach md dau trong bo sach viet

cho hoc sinh chuyen Toan va boi diTdng hoc sinh gioi ve mon Toan cua nhom giao

vien trirdng Trung hoc Quoe gia Chu Van An

Quycn sach gom 5 chiTdng:

- Chirring 1: Dang IhiJc liTdnggiae

- ChU'ring 2: Baft dang Ihu'e lu'ring giac •

- ChU'ring 3: Ding thite lUdng giac trong lam giae

- ChU'ring 4: Bat dang thiJc trong tarn giac

- Chirring 5: Vai ifng diing cua li/dng giac trong vi^c giai cdc b^i to^n sd cap

Cuon sach nay sc cung ca'p cho ban doc mot so lifdng raft Idn c^c bai toan chon

loc gom du the loai Moi phan cua cuon sach c6 the xem nhiT mot chuyen de rieng

trinh bay tron ven mot van de mot each hoan chinh (rieng phan phu'dng trinh lifdng

giac se du'dc de cap trong cuon khac)

Ngoai vi$c he thong va phan loai bai tap, chiing toi luon chu y ve vice phat

trien mot bai tap theo cac hifdng tong quat hoa, dac bi§t hoa va so sanh binh luan

cac phiTdng phap giai khac nhau Trong mpl chijfng mifc nha'l dinh chung loi manh

dan di^a ra cac giai phap ve khia canh sif pham de dung cuon sach nay Vi le do

cuon sach nay se la mot tai lieu tot de cac ban giao vien lham khao vao muc dich

day cho hoc sinh mot each tifduy mot bai loan (chuf khong ddn ihuan giang day cho

hoc sinh hicu mot bai loan cu the)

Can nhan manh them rang trong cuon sach nay danh mot phan de trinh bay each

thic't lap cac he thuTc li^dng giac trong lam giac diTa vao moi quan he giiJa cac yeu to'

cua mot lam giac vdi cac nghiem cua mot phu'dng trinh bac ba Wring iJng Theo

quan niem ciia chiing loi day la mot trong nhCfng phan dac sac cua cuon sach

Mac dij hel siJc c6' gang trong qua trinh bien soan cuon sach, nhi/ng vdi mot

dung lu'ring qua Irin can truyen tiii cho ban doc nen cuon sach khong the tranh khoi

nhffng khic'm khuyet l/t >

Chiing loi rat mong nhan diTric sif gop y cua ban doc de cuon sach hoan ihien

hrin nCa trong cac Ian tai ban sau

Thi^ lij' gop y xin gufi ve

Phan Huy Khiii

Tri/dng Trung hoc Quoc gia Chu Van An

10 - pho Thuy Khuc - quan Tay Ho - Ha NQI

Xin chan lhanh cam dn! ^

O A I N C T H U ' C LirCfNC G I A C

§ 1 CAC CONG THirC LUONQ GIAC CO BAN

1 Cac thuTc Irfcjfng giac cd ban

sin^a + cos'a = 1

cos a cota = sma

1 + tan^a = 1 cos^ a

sin a tana =

tana + tanP tan(a + P) =

1-tan^a

sin3a = 3sina - 4sin'a cos3a = 4cos^a - 3cosa tan3a = 3tana-lan'^a l-3tan^a

4 Cac cong thtfc blen tong thanh tich

„ ^ a + p a - p sina + sinP = 2.sin ^cos

2 2

„ „ a + p a - P sina - sinP = 2 cos ^-sin „ - a + p a - P 2 2

cosa + cosP - 2cos !-cos 2 2

^SlduSng hfc sinb gioi Lupag gldc - rhan Huy Kbal

^ „ sin(a + P)

tana + tanp = - cos a cos (3

sin(a - P) tana - tanp =

cota + cotp =

cota - cotp =

cosa cosp sin(a+p) sin a sin P sin(p-a) sin a sin p

5 Cac cong thitc bi6n tich thanh ts'ng

sin(a + P) + sin(a - p) sinacosp = ^—^ —

6 Gia tri Itf^ng giac cua cac goc (cung) c6 lidn quan dac bi^t

- Hai goc doi nhau:

, 1 , , , , : , <

§ 2 D A N G THirC LUONQ QiAc K H 6 N G D I E U K I ^ N Cdc bai toan trong muc nay co dang sau day: ChiJug minh cac he thiJc ii/cJng

giac khong CO kern theo dieu kien gi - ^ u i

PhiTctng phap giai cac bai toan n^y thuin tiiy dura vao cac phep bien doi lu'cJng giac ^, ^

1 Taco: sin54" = cos.36"

3 sin 18" - 4sin^ 18" = 1 - 2sin^ 1S"

o 4sin' 18" - 2sin^ 18" - 3sin 18" + 1 = 0 (sinl8"-I)(4sin^l8" + 2sinl8"-1) = 0 (1)

Do 0" < 18" < 90" => 0 < sinl8" < 1, nen (l)o4sin'l8" + 2sinl8"-1 =0 (2) Lai tuf 0 < sinlS" < 1, nen tir (2) suy ra sinl8" = ^^^^-^^ => dpcm

4

2 Ap dung cong thuTc sin3a = 3sina - 4sin^a, va gia thiet phan chiJng sinl" 1^

so hffu ti, khi do theo tinh cha't ciia cac phep tinh vdi so' hi?u ti suy ra sin3", sin9", sin27", sinSl" la sohifuti

Do sin8l" = cos9" va sin 18" = 2sin9"cos9", nen suy ra sin 18" la so hffu ti TiT phan 1/ta c6: la so hiJu ti, tiTc la ^/izl = £ vdi p,q e N 4 4 q

= > N/5 = 4 - + 1, vay N/S la so hffu ti

4

Do la dieu vo li vi N/5 nhtf da biet la so v6 ti Vay gia thiet phan chi?ng la

sai, neri sinl" la so'v6 ti => dpcm > '

Nhdn xet:

1 Bkng each suf dung cong thiJc: cos3a = 4cos''a - 3cosa,

Vdi phep giai tu'cfng tU", ta chuTng minh diTdc cosl" la so v6 ti

2 Bay gicf xet bai toan sau:

ChiJng minh rang cos20'^ la so v6 t i Khi do earh giai hoan toan khac each

Vay cos20" la nghiem cua phu'dng trinh Sx"* - 6x - 1 = 0 (3)

Ta CO ket qua quen bie't sau trong l i thuye't da thiJc

Xet phiTdng trinh da thuTc: anx" + a n i x " " ' + + a i x ' + a „ = 0, (4)

Trong do a; la so'nguycn vdi moi i = 0, 1, 2, ,n

Goi P la tap hdp tat ca cac \idc cua ao con Q la tap hdp tat ca cac \idc cUa ap

Khi do ne'u (4) co nghiem hiJu ti, thi nghiem do phai c6 dang

x = i ^ , v d i p e P, q e Q

q

Ap dung vao (3), ta thay moi nghiem hSu ti ciia (3) ne'u c6, thi chung deu

thuoc vao tap hcfp sau:

Q = {±1/8; ± 1/4; ±1/2; ± 1}

Tuy nhien bang each thuf trifc tie'p ta thay moi phan tii" ciia Q deu khong phai

la nghiem ciia (3) Noi each khac moi nghiem ciia (3) deu la so' v6 t i V i

eos20" la mot nghiem ciia (3) nen eos20"la so v6 ti Do la dpem

Bai 2 ChiJng minh:

4cos36" + cot7"3()' + ^ + + S + S +

Giai Theobai 1, ta CO sinl8"=

= > c o s 3 6 " = l - 2 s i n ^ l 8 " = l - 2 - ^ " ^ " ' ^ '

16

•4cos36" = 4 - =S + \ (1)

Ap dung cong thiJc: Neu 0 < a < 90", Ihi c o t y = c o l a + V l + c o l ^ a , ta c6:

cot7"30' = col 15"+ Vl + col^ 15" (2)

-Bai 3 Chu'ng minh rting

tan-10" + tan-50" + tan-70" = 9

Ap dung cong thu'c 1 + tan'a =

dUctng vdi dang thu'c sau:

eos^lO" ' cos-50" ' eos^ 70

^ cos^ 50" cos' 70" + cos^ lOcos^ 70" + cos^ lOcos^ 50 _

^ cos2l0"cos-5)"cos2 70" ~ •

Goi A va B lu'dng vtng la tii' so' va mau so ciia vc' trai ciia (1), la co:

A = cos'(60" - 10")cos'(60" + lO") + eos'10"[cos'(60" + lO") + cos'(60" - lO")]

= [(cos60eoslO + sin60sinl0)(cos60eosl0- sin60sinlO)]^ ', + cos-10"[(cos60cosl0- sin60sinl0)' + (cos60cosl0 + sin60sinlO)^]

= (cos'60eos-10 - sin^60sin^l0)- + co,s^lO"(2cosYiO"eos'lO" + 2sin'60sin^l0")

BOI auOng a y e ainti glol Lupng gUc - rhmn Huy KluU

M a t khac tiTcfng tiT nhtfbien doi A, ta cung c6:

hay Stan'-lO" - 27lan''lO" + 33tan^lO" - 1 = 0 (5) '

Tif (5) suy ra tan^lO" la nghiem cua phi/dng trinh:

3 x ' - 27x^ + 33x - 1 = 0 (6) ;

Tu'dng tiT tan^'>0, tan'70 cfing la nghiem cua (6)

M a t khac de tha'y tan-lO", tanl^O", tan^70 la ba so khac nhau

(cu the ta CO tan^lO < tanl^O < tan^70), nen tan'10, lan-50, tan'70 la ba

nghiem khac nhau cua (6)

Theo dinh H Viet vdi phU'rtng trinh bac ba, ta c6:

/ 27^

tan^l()" + tanl'50 + tan^70 = = 9 => dpcm

Chu y: X i n nhSc lai vdi phu'dng trinh bac ba:

Cdch 3: Dang thuTc csln chifng minh tU'rtng dU'dng vdi dang thiJe sau:

tan-10" + tan'5()" + tan'70 = 3tan'60"

o (lan-60 - tan^ 10) + (lan'60 - tan^50) + (tan'60 - tan-70") = 0 (7)

A p dung cong thiJc: •

cos-10 cos-50" cos70"

o 4sin.'50"sin70"cos50"cos7()" + 4sinl0"sinl 10"cos-10cos^70" '

- 4sinlO'sin 13()"cos'l()cosl*i0 = 0 e> sinlOO'sinl4()"cos50cos70 + sin20sinl40"cosl0cos70

- sin20sinlOOcosl()cos5() = 0 (do sinl 10 = sin 70; sin 130 = sin50 va sin2a = 2sinacosa)

« > cosl0cos70cos-50 + coslOcos5()cos^70 - cos50cos7()cos-10 = 0 (8) (do sinlOO = coslO; s i n l 4 0 = cos5(); sin20 = cos70; s i n l 4 0 = cos50; )

Do cosl()cos70cos50 ^ 0, nen (8) o c o s 5 0 + c o s 7 0 - c o s 10 = 0

o 2cos60coslO - coslO = 0

o c o s l O - c o s 10 = 0 (9)

V i (9) diing nen (7) diing => dpcm

Cdch 4: A p dung cong IhiJc: tan'a = 1 - thi dang thiJc can chuTng minh

tan 2a

• ' f t

tiTdng diTctng vc'Ji dang thiJc sau:

tan 10 ^ tan 50 ^ tan 70 ^

<=> Ian50tan4()tan2() + tan7()tan8()tan2() - tanl()ian8()tan40 = 3tan8()tan40tan20

<=> lan20 + tan80 - tan 40 = 3tan2()lan4()tan80

(do Ian50lan40 = tan70tan2() = tanl0tan8() = 1 ) '

<=> tan2()( 1 - tan40tan8()) + tan8()(l - tan20lan40)

- t a n 4 0 ( l + t a n 2 0 t a n 8 0 ) = () (11) i

i ^ , r. I'ln oc + tan p ^

A p dung cong thiTc: tan(a + P) = , nen

1-tan a tan P tan80 + t a n 4 0 Ian4'0 + tan20 (11) CO tan20 + tan 80 tan 40(1 + tan 20 tan 80) ^ 0

tan 120 tan 60 c:> —^-— tan 4()(tan 80 - tan 20) = tan 40(1 + tan 80 tan 20)

TiT (12) suy ra (11) dung => dpcm

Binh luan: Trong 4 each giai, c6 3 each su" dung thuan tiiy bien ddi lu'dng giac,

eon 1 each ket http vciii cac kicn thiJe vc tinh chat nghiem ciia mot phu'dng

trinh dai so' (cu the sii' dung dinh l i Viet trong thi du nay)

Bai 4 Chu'ng minh rang —^— + — + — ^ — = 4

71 371 571 COS cos COS

TiT do suy ra xet phu'cfng trinh sau day:

eos3x - -cos4x hay eos3x + eos4x = 0 (1)

Ta eo: cos3x = 4eos x - 3cosx

cos4x = 2cos"2x - 1

= 2(2eos^x - 1)' - 1 = 8cos^x - 8eos'x + 1

V i the (1) CO 4COS-X - 3cosx + Scos'x - 8eos^x + 1 = 0

CO 8cos'*x + 4 e o s \ 8eos\ 3cosx + 1 = 0

CO (cosx + l)(8eos''x - 4eos'x - 4cosx + 1) = 0 (2)

2S, sin — = 2sin—cos — + 2sin —cos—- + 2sin—cos 5n

1 I n 4n lit 6% An

= sin — + sin sin — + sin sin —

Cty Train nrV D VVH lUiang Vift

Bai 5. Chufng minh rang:

cos7x = cos6xcosx - sin6xsinx

= cosx(4cos^2x - 3cos2x) - 2sinxsin3xcos3x

= cosx[4(2cos\ l)"* - 3(2cos^x - 1)] - 2sinx(3sinx - 4sin\)(4cos^x - 3cos)

= cosx|4(8cos\ 12cos''x + 6cos'x - I ) - 6cos^x + 3J

- 2sin'x(3 - 4sin^x)(4cos\ 3cosx)l

= eosx(64cos''x - 112cos''x + 56cos-x - 7) (1)

T r o n g ( l ) t h a y x - — , va do c o s — = cos—= 0, c o s — ^ 0, nen tijf (1) suy ra

14 14 2 14 • phircfng trinh: 64x' - 112x' + 56x - 7 = 0 (2)

nhan x, = c o s ' — lii nghicm Tifcfng M X j =:cos^—, x^ = c o s ^ — cung I ^

64 16

• dpcm

Bai da&ng hpc slnh giol Lupng gidc - rhan Ituy Khai

Bai 6 Chu'ng minh rang

Ro rang thoa man phu'dng trinh: sin"4x = '-in^Sx (1)

Dat t - sinx va ap dung cac cong thiJc:

Nhan xet:

1 Dang ihrfc Iren co each chu'ng minh khac sau day

De thay dang thufc dau bai tu'dng dU'dng vc'ti dang thiJc sau

COS _^ cos ^ cos ^- i tJl '•'0 fe' { H i f

Ko rang — ; — ; — thoa man phu'dng trinh: cos4x = cos3x (1)

Dc thay ( 1 ) 0 2cos"2x - 1 = 4eos'x - 3cosx , •

o 2(2cos"x - 1) - 1 = 4cos'x - 3cos

« > (cosx - 1 )(Kcos\ 4cos"x - 4cosx - 1) = () ( 2 )

~'' -y'- khong thoa man phu'dng trinh cosx - 1 = 0 , nen no thoa man

phu'dng trinh: 8cos'x + 4cos"x - 4cosx - 1 = 0

V i le d o t| = 2 c o s - ^ , 1 2 = 2 c o s ^ , 1 3 = 2 c o s - ^ la ba nghiem phan b i e t c i i a phiTdng trinh: t^ + t" - 2 l - 1 = 0 ( 3 ) Theo dinh l i Viet a p dung vdi ( 3 ) , ta co:

l | + l 2 + t 1 = - ' t|l2 + t 2 t 3 + t 3 l | = - 2 (4) t|t2t3 = 1

D a t A = ^ + ^ + ^ ,

B = ^ + ^/hl7 + thi A-^ = l i + t 2 + t 3 - 3 ^ l | t 2 t 3 + 3 A B ; '*

T i l f( 4 ) ( 5 ) s u y r a A ' = - 4 + 3 A B ' (6) Lap luan tu'dng liT l a eo: B* = - 5 + 3AB (7) Nhan tijrng ve (6) (7), ta di den: ( A B ) ' = (3AB - 4)(3AB - 5)

Dat " - " " J ' l^hi do d^ng thufc can chtfng minh tiTdng difdng vdi dang thiJc

sau: tan3a + 4sin2a = A / F T (1)

<=> sin3a + 4sin2acos3a = Vfl cos3a

o (sin3a + 4sin2acos3a)' - 11 cos'3a = 0

(do ca hai ve cua (1) dcu la so diTOng) ^•

o (sin3a + 2sin5a - 2sina)" - 1 lcos'3a = 0

<=> sin'^3a + 4sin'5a + 4sin^a + 4sin5asin3a 4sin3asina

-8sin5asina - 1 lcos^3a = 0

o ^ ' - " ' ' ^ ^ + 2(1 - c o s l O a ) + 2(1 - c o s 2 a ) + 2(cos2a - c o s 8 a )

2

-2(cos2a - cos4a) - 4(cos4a - cos6a) - 11 ^'^s6a _ ^

<=> - 1 - 2(cos2a + cos4a + cos6a + cos8a + coslOa) = 0 (2)

Do sina = sin— 9^ 0, nen

11 (2) o -2sina(cos2a + cos4a + cos6a + cos8a + coslOa) = sina (3)

A p diing cong thi?c: 2sinacosb = sin(a + b) + sin(a - b), nen

(3) <=> sina - sin3a + sin3a - sinSa + sin5a - sin7a +sin7a - sin9a

+ sin9a - sin 1 l a = sina c:i>-sinlla = 0 (4)

D o a = — , n c n s i n l l a = sin7t = 0

11 Vay (4) dung dpcm

Bai 9 Chiirng minh rang

tan'2()" + tan'40" + tan-60" = 33273

Giai

<, 3 tan a - t a n "'a , SU" dung cong thiTc: tan3a = , suy ra neu thay a = 20 , ta c6:

1 - 3 t a n ^ a

^ _ 3tan20"-lan''20 ^ ^ _ tan^ 20"(3-tan^ 20")^ ,

l - 3 t a n ' 2 0 " ~ (1-3tan^ 20")^

o lan'20" - 33tan''2()" + 27tan-20" - 3 = 0 ( { a

Vay X| - tan*2()" la nghicm cua phiTdng trinh: f> f

x ' - 3 3 x ' + 2 7 x - 3 = 0 (1) Lap luan tUrtng luf, (1) con nhan \2 = tan''4{)" va x, = tan^80" la nghiem Vay

XI X:, X3 la ba nghiem khac nhau cua (1) ^, ^ ,

Ta c6: tan''2()" + lan'4()" + tan'SO" = x, + x^ + x^

Thay vao (2), vdi luti y x^ + X j + x^ = (x, + Xj + x^)^ - 2( X | X 2 + X2X3 + X3X1)

tan'20" + tan''40" + tan'SO" = 33(33' - 3.27) + 9 = 33273 => dpcm

Chii y: Hoan toan tu'dng tif, ta co d i n g thuTc sau:

lan'lO" + tan'50" + tan'7()" = 433

Bai 10 Chiang minh rhng:

3 t a n a - t a n ^ a 2 tan a

= l - 3 t a n ^ a 1 - t a n ^ a ^ t a n a tan"* a - lOtan^ a + 5

^ ( 3 t a n a - t a n 3 a ) ( 2 t a n a ) 5 tan^ a - lOtan^ a + 1 ( l - 3 t a n ^ a ) ( l - t a n ^ a )

Trong (1) thay = va chu y rang: tann = 0; t a n - j ^ 0, nen tuf (1) suy ra

t a n ^ - - 1 0 t a n 2 - + 5 - 0

5 5

Vay l a n ^ la nghicm ciia phiTdng trinh trung phiTcfng x'* - lOx^ + 5 = 0 (2)

TiTdng t y t ^ " — - ' ^ ^ " ^ nghiem khac cua (2)

r>. 471 371 471 , 7 71 9 371

Do t a n - = - t a n — ; t a n — = - l a n — , nen y, = t a n ' ' - ; y 2 = tan^ — la hai

nghiem phan biet cua phiTdng Irinh y^ - lOy + 5 = 0 (3)

A p dung dinh l i Viet v d i (3), ta c6: y i y 2 = 5 ( 4 )

1 ChiTng minh rang: tanatan(60" + a)tan(60" - a) = tan3a, vdi gia thiet cac

tang xet trong he thiJc deu c6 nghia

2 A p diing phan 1 hay chiJng minh:

71 67t 1271 71 471 271 r:

V T = tan — tan — tan = tan — tan — tan — = tan — = v 3

9 21 11 9 9 9 3

Bai 12

1 Chtfug minh rang:

tana + lan{a + 60") + tan(a + 120") = 3tan3a vdi giii thiet la cac tang c6 mat trong he ihiJc dcu c6 nghla

2 Chifng minh rang

tan5" + tan65" + tan 125" = 3tan 15"

tan9" + tan69" + tan 129" - 3tan27"

tan53" + Ian 113" + tan 173" = 3tanl59"

tan57" + tanl 17" + t a n l 7 7 " = 3 t a n l 7 l " Cong tifng ve 15 diing thtfc tren va co:

S = 3(tan3" + tan 15" + lan27" + + Ian 159" + t a n l 7 l " )

T V

= 3((tan3" + tan63" + I a n 123") + ( I a n 15" + lan75" + tan 135") +

(tan27" + lan87" + tan 147") + (tan39" + tan99" + tan 159") +

(tan5l" + t a n l l l " + tanl7l")J

Lai ap dung phan 1/ta c6: '•'''^

S = 9(tan9" + tan45" + tan8l" + tanl 17" + tan 153")

= 9(tan9" + tan8l" + tanl 17" + tan 153") + 9

2 2 2 ( s i n 5 4 " - s i n l 8 " ) sin 18" sin 54 sin 18" sin 54"

1 Chu'ng minh: tan'a + lan'(60" - a) + tan^(60 + a) = 9tan\3a + 6

2 Chu'ng minh: tanl5" + tan'lO" + tan'15" + + tan-80" + tan85" = 198^,

2 Dat ve'traicua diing thiJc can chiirng minh la S Ta c6:

S = (lan'5" + tan'55" + tan^65") + (tan^lO + lanl50 + tan^70) + (tan'15" + tan'45" + tan'75) + (tan'20 + tan^40 + tan^80) + (tan'25" + tan'35" + tan^85") + tan'3()" + tan-60 (1) Tuf (1) va ap dung phan 1/ ta c6:

S = (9tan-15" + 6) + (9lan-30" + 6) + (9tan'45" + 6) + (9tan^60" + 6) +

(9tan'75" + 6) + i +

1

= 9(tan' 15" + tan'45" + tan'75") + 33 + ^ + 9(tan'30" + tan'60") (2)

TCr (2) lai ap dung philn 1/ ta Ihu di/dc:

4

2 Ta c6:

VT = (Sin2".sin62"sin58")(sinl8"sin42"sin78")(sin22"sin38"sin82") (1)

B S i dudng h p c s i n h g i d i L u ^ n g g i i c - fhan Huy K h a l

TufCl) va apdung phan l/c6:

87t 1-cos —

Dat y = cos'x > 0, thi (7) co dang:

(2cos'2x - 1)^ = (4cos''x - 3cosx)'

<=> [2(2cos^x - 1)' - IJ- = cos^x(4cos^x - 3)^

T u ' ( 8 ) s u y r a : 64y^ - 144y' + l()4y^ - 25y + 1 = 0

B6I duaing hpc sbib gkil Lu^ng gUc - Fhmn Htiy lOuU

Do cos^ cos^ cos^ ^ deu khac nhau khdc 1, nen phirong irinh

,3 1 : 0 hay 64y' - 80y' + 24y - 1 = 0

Nhan y, = cos y ' y 2 -^'^^ ~'^y^= '-"os — la nghiem

Thay (2) ( 3 ) vao ( 1 ) vii c6 dpcm

Bai 17 Cho n la so nguycn du'dng Chiang minh

2 + \/2TV2^+^^^^^? ~ 2cos — ^

n clii u ca n

Giai

Ta chrfng minh bhng nguycn ly quy nap loan hoc:

Vdi n = 1, la c6 -Jl = 2 - ^ = 2cos— = 2cos—^

Vay dang ihiJc can chiJng minh dung vcKi n = 1

B6I daang hpc sinh gidl Lupng gidc - rhan Huy Khal

+ Gia su" dang thtfc can chrfng m i n h da dung den n = k, tiJc la ta c6:

V a y tiT (3) suy ra dang thiJc can chiJng m i n h ciing dung v d i n = k + 1

T h c o nguyen ly quy nap toan hoc suy ra d i e u phai chiJng m i n h

B a i 18 Cho n la so' nguyen du'dng Chufng m i n h x\ng

Cong tufng v c ' n - 1 dang thuTc Iren va co:

B a i 2 3 Churng m i n h r a n g v6i m o i n n g u y e n du-dng, ta c 6 :

0, ne'u X = 2kn

Sn = sinx + sin2x + + sinnx - <

X 2 n + i cos cos X

2 2 X

2 2

2 sin ^

2

dpcm

B a i 2 4 (ufng d u n g h e thuTc lu'dng g i a c t r o n g b a i t o a n d a y s o )

C h o d a y so' x a c d i n h nhu" sau:

B a i 25 (iJng d u n g h e thiJc lu-dng g i a c t r o n g b a i t o a n d a y so)

H a i d a y so { U n } , { v „ ) du'Oc x a c d i n h b a n g c o n g thiJc truy h o i nhuTsau:

cos cos r cos

2.3 2^3 2".3 (ban doc xin tiC nghicm lai)

Tu'Ong lu" nhu'cach giai trong bai 17, la c6 he thu"c sau:

2.3 2^.3 2".3 2" sin 2 ^ ' s n

2".3 2".3

Thay lai vao (1), va c6: u^ =

V 3 c o s 2" 3

Ccic bai loan trong muc nay c6 dang sau day: Gia suT cac dai lifdng trong he

IhiJc can chiyng minh thoa man mot dieu kien nao do Ta phai chi^ng minh he

thuTc da cho la dung Dieu kien cho triTdc c6 the cho difcKi dang hinh hoc, dai

so,, PhiTctng phap giai cac bai loan nay la sir van dung kheo leo ke't hdp

giCTa ciic phep bien lUOng giac cO ban va iriel de siir dung cac dieu kien da

cho trong de bai

V ^ ^ cosa + cosP+cosy sina + smB+sinv

B a i 1 Cho — — — — = = m

cos(a + p + y) sin(a + p + Y)

ChiJng minh rang cos(a + P) + cos(p + y) + cos(y + a) = m

G i a i Dat a + p + y = S;u = P + y ; v = y + a ; w = a + p. i -

K h i do de thay a = S - u ; P = S - v ; y = S - w % >

cosa c o s ( S - u ) cosScosu + sinSsinu

Ta c6:

cosS cosS cosS

= cosu + sinu.tanS, sina sin(.S-u) s i n S c o s u - s i n u c o s S sinS sinS

= (cosu + cosv + cosw) + (sinu + sinv + sinw)tanS (4)

sinS = (cosu + cosv + cosw) - (sinu + sinv + sinw)cotS (5) Trtr tiTng ve (4) (5) va tif giii thiel ta c6:

(sinu + sinv + sinw)(tanS + cotS) = 0

V i |tanS + cot S| = |tan S| + jcol S| > 2,

nen noi rieng tanS + cotS ^ 0, vay tu" (6) suy ra ,

sinu + sinv + sinw = 0

; TO(7), ( 4 ) v a g i a t h i e t s u y r a cosu + cosv + cosw = m o cos(P + y) + cos(y + a) + cos(a + p) = m

cos3a + cos3p + cos3y = 4cos'a + 4cos'p + 4cos''y - 3(cosa + cosp + cosy)

= 4(cos-^a + cos^'p + cosV) • - - (1)

B a i 3 Cho

(do cosa + cosp + cosy = 0)

A p dung hang dang thiJc, la c6:

cos^a + cos'p + cosV = (cosa + cosP + cosy)^ - 3cosacosp(cosa + cosP)

- 3cosPcosy(cosP + cosy) - 3cosycosa(cosy + cosa) - 6cosacosPcosy (2)

V I cosa + cosP + cosy - 0, nen ?| -t » l i l t

cosa + cosP - -cosy; cosp + cosy = - cosa; cosycosa = -cosp (3)

Thay (3) vao (2) di den cos'a + cos^P + cos y = 3cosacosPcosy (4)

Lai thay (4) vao (1), va co: cosacospcosy = ^ ( c o s 3 a + cos3p + cos3y)

Tir gia Ihiet: acos(a + cp) + bcos(p + cp) = 0, ta co:

acosacoscp - asinasincp + bcosPcoscp - bsinPsincp = 0

=> coscp(acosa + bcosP) - sin(p(asina + bsinP) = 0 (1)

Tif gia Ihiel acosa + bcosP = 0, va sincp ^ 0 (do 9 k n , k e Z), nen tijf (1) suy

ra: asina + bsinP = 0 (2)

VcHi moi x, la co:

(•(x) = acosxcosa - asinxsina + bcosxcosp - bsinxsinP

= (acosa + bcosP)cosx - (asina + bsinP)sinx (3)

Do acosa + bcosP = 0 (gt) va ihco (1) ihi asina + bsinP = 0, tijf do tir (3) suy

= 0

Do a = b = 0, nen ro rang Vx e R, ta c6 f(x) = 0 => dpcm

B a i 5 Cho sin(a + 2p) = 2sina Chiang minh rang tan(a + p) = 3tanp, "

vdi gia Ihiet tan(a + P) va tanp co nghia

G i a i

V i e l lai gia thiel du-citi dang sau: sinf(a + P) + p] = 2 s i n [ ( a + P) - p]

T i r ( l ) , la co:

sin(a + P)cosp + sinPcos(a + P) = 2sin(a + P)cosP - 2sinPcos(a + p)

=> 3sinPcos(a + p) = sin(a + P)cosp

Do lan(a + P) va tanP co nghia nen cos(a + P) ;^ 0; cosp ^ 0

Chia ca hai ve cho cos(a + p)cosP va co: tan(a + P) 3tanP => dpcm

(1)

B a i 6 Cho cos(2a + P) = 1 ChlTng minh he thiTc tan(a + p) tana = 2 t a n

-2 vdi gia thiet cac tang co mat trong he thiJc deu co nghia ;,

1 + tanp

B a i 7, Cho ^'"^ = i ChUng minh = Lz^^R^^ = ^ ^''

-sin(2a + P) m m + n m - n j H

B61 duSng bpc ainb gfol Lupng g U c - Fhan Huy Khal

Giai

DiTa d i n g thtfc can chiJng minh ve dang tifcJng diTdng sau:

J sin a sin (3 tana + tan 3 cos a cos (3 sin(a + 9) cos(ft + p)

(m + n ) t a n a m - n (m + n)tanacosacos3 ( m - n ) c o s t t c o s P

<=> (m - n)taji(a + P) = (m + n)tana (1)

Tijrgia thiet ta c6:

msinp = nsin(2a + P) => msin[(a + p) - a ] = nsin[(a + P) + a ]

=:> msin(a + P)cosa - msinacos(a + P) = nsin(a + P)cosa + nsinacos(a + p)

=> (m - n)sin(a + p)cosa = (m + n)sinacos(a + P) (2)

Chia ca hai vc cua (2) cho cosacos(a + P) ta c6:

(m - n)tan(a + p) = (m + n)tana

Vay (1) dung => dpcm

Chu y: Ta hicu r i n g khi de bai bat chu'ng minh mot he thu'c nao do, thi mac

nhien da giai thiet la cac bleu thu'c c6 mat trong he thuTc do triTdtc het phai c6

Bai 9. Cho tan(a + P+ = 3tana

Chtirng minh: sin(2a + 2P) + sin2a = 2sin2p

2sm23= f ' + f • (3) ^ '

Q t V l O t ^ + l , ^ 2tan(a + 3) 2 t a n a •

Mat khac: sm[2(a + P)] + sm2a = — + r — (4)

l + tan^(a + 3) l + tan^a Thay tana = t, tan(a + P) = 3t vao (4) roi rut gpn ta cung c6:

sin(2a + 2p) + sin 2a = (5)

9 t V l 0 t ^ + 1 Tur (3) (5) suy ra dpcm

, aa, + b b , _ bb, _ s i n ( x - 3 ) co.s(x-3)

ab|+a|b ^_^'^±_ s m ( x - a ) ^ c o s ( x - a )

b b, s i n ( x - 3 ) c o s ( x - 3 )

_ sin(x — a)cos(x — a) + sin(x — 3)cos(x — 3)

sin(x - a)cos(x - 3) + sin(x - 3)cos(x - a) sin(2x - 2a) + sin(2x - 23)

2 s i n [ 2 a - ( a + 3)]

2 sin[2x - (a + 3)] cos(a - 9)

2 s i n [ 2 x - ( a + 3)]

Bai 11. Cho cos(cp - a ) = a; sin((p - P) = b

Chu'ng minh r i n g : a^ - 2absin(a - P) + b^ = cos^(a - P)

= cos(a - P) => dpcm

B6I duoing hpc abib glol Lupng gidc - Phan Huy Kbal

Giai

Tir gia thiet ta c6:

VT = cos^((p - a) - 2sin((p - P)cos((p - a)sin(a - P) + sin^((p - p)

1 + cos(2ip - 2a) l - c o s ( 2 i p - 2 3 )

= — H 2sin(u3 - (3)cos(u3 - a ) s i n ( a - 3)

2 2

= 1 - sin[2cp - (a + P)]sin(a - P) - 2sin((p - P)cos((p - a)sin(a - P)

= 1 + sin(a - P) {sin[2(p - (a + P)] - 2sin(q) - P)cos((p - a)}

= 1 + sin(a - p) {sin[2(p - (a + p)] - sin[2(p - (a + p)] - sin(a - p)}

Ta c6: cos(a + 2P) = co.sacos2P- sinasin2p (1)

Tif 3sin^a + 2sin'P = 1 => 3sin^a = 1 - 2sin'p = cos2p (2)

Tir 3sin2a - 2sin2p = 0 => sin2p = | s i n 2 a = 3sinacosa (3)

Thay (2), (3) vao (1), suy ra:

cos(a + 2P) = 3sin^acosa - 3sin^acosa = 0 (4)

sin'a + sin'P = sin(a + P)

Chufng minh rang: (x-V^ = ^

Cty Trmn nTV.DVVH Khaag Vlft

Giai Tiif gia thict la c6: 1 - cos2a + 1 - cos2p = 2sin(a + P)

<=> 2 - 2cos(a + P)cos(a - P) = 2sin(a + P) c=> cos(a + P)cos(a - p) + sin(a + P) = 1

D o ( ) < a < ^ ; 0 < P < ^ = > ( ) < a + P<7t

(1)

(2)

Gia thict philn chufng:

Khi do tiT (2) suy ra chi c6 hai kha nang sau day xay ra:

a + p ^ —

2

1 0 < a + p < J Khi do: ^ > a + p > | a - 3| > 0 (3)

Tur (3) suy ra: cos(a - P) > cos(a + P) > 0

=> cos(a - P)cos(a + P) > cos'(a + P) = 1 - sin'(a + P)

=> cos(a - P)cos(a + P) + sin'(a + P) > 1 (4)

Vi (3), ncn c6 sin(a + P) > sin'(a + P) (5) Tir (4) (5) di den: cos(a - p)cos(a + p) + sin(a + P) > 1 (6)

Tir (6) va ( I ) suy ra mau thuan

2 Ncu — < a + p < 71

2

D o ( ) < a < - , 0 < P < - z : : > ( \ - 3 < - = > cos(a - P) > 0

2 2 2 ^

Mat khac vi — < a + p<7i=:> cos(a + P) < 0

Tir do ta c6: cos(a - p)cos(a + P) < 0

=> cos(a - P)xos(a + P) + sin(a + P) < 1 (do sin(a + P) < 1) Tir (7) va (1) cung suy ra mau thuan

Vay gia thict phan chufng la sai => a + p = ^ => dpcm

Bai 14 Cho a, P deu thuoc khoang (0; ^ ) va a ^ p Gia su"

cos X —cos (V sin ^ a cos 3

cosxcosasin'P - cosxsin'acosP = sin^Pcos'a - sin^acos^P

Luting fjliic — Fhan tluy Khal

cosx(cosasin^P - cosPsin^a) = c o s ' a ( l - cos^p) - cos'P(I - cos^a)

= cos^a - cos^p (1)

M a t khac lai c6:

cosasin'p - cosPsin^a = c o s a ( l - cos^p) - cosPd - cos'a)

= (cosa - cosP)( 1 + cosacosP) (2) Tir (1) (2) va do gia thiet a, P e (0; ^ ) va a ; t P nen ta c6:

cosa - cosP ^ 0, tijf do d i den:

Ta c6: sinP(sinP + siny) = 2sinPsin ^ 2

Thay (2) (3) v a o ( l ) suy ra:

sinP(sinP + siny) = 2sinPcosPsina = sin2Psina = sin^a

(3)

.(4)

(2)

B 6 I duSng h p c alnb g l o l Lugng glAc - fban Huy Khal

cosx cos2x cos3x

1 Cho ^ , , • , V •> X 2aT ~ a i — 'AT

ChiTng minh rSng: sin' — = — ' ^

^ ^, sinx sin3x sin5x ^, , ,

2 Cho = = — ChiTng minh rang: ' ^

= = = sin — =>apcm

4cos2x 2 2

2 Phan 2/ chrfng minh tifdng tiT nhif phan 1/ Ban doc hay tif nghicm lai dieu ay

Bai 19 Cho a ?t 0, b 0 va ihoa man he sau:

TiJ" he thu-c: (a + b)sin(x - a) = (a - b)sin(x + a)

=> alsin(x - a) - sin(x + a)l = -b|sin(x - a) + sin(x + a)]

c b tv + tan

Bai 20 Cho eosa = tanP; cosp - lany; cosy = tana Va a, p, y e

Chiirng minh rSng: sina = sinp - siny =

Giai

0;

Dat X = sin'a; y = sin'P; / = sin'y

Tijr gia thiet suy ra:

cos^ acos^ [3 = sin^ 3

cos' (\ tan^ i3 cos^ = tan" ~) <^

cos = tan a TirCl) suy ra: x = Ttr (2) suy ra: z =

cos^ i.3cos^ ~t = sin^ -j o cos^ -^eos^ a = sin^ «

( l - x ) ( l - y ) = y ( l - y ) ( l - / ) = z ( l - z ) ( l - x ) = x

l - 2 y 1-y 1-y

2 - y Thay (4) (5) vao (3) va eo:

=> y- - 3y + 1 = 0

(4) (5)

(1)

(2) (3)

/J • <

1 - l - y 2 - y j l - 2 y

(6) 1-y i - 2 y 1-y

B6I dudnfj h^c sinh gUii I irpng glAc - fhan Huy Kbal

=> a^cos''a + b^sin'*a + ab(cos''a - cos^a) + ab(sin''a - sin^a) = 0

=> a^cosV + b^sin'*a - abcos^a(l - cos^a) - absin^a(l - sin^a) = 0

=> a^cos'*a + b^sin'*a - 2abcos'asin^a = 0

=> (acos^a - bsin^a)^ = 0 => acos^a - bsin^a = 0

=> a(l - sin^a) - bsin^a = 0

=> sin^ a = ^ • (1)

a + b

* Lap luan tifdng tif, ta c6: cos^ (v = (2)

sin « + sin 3 + sin = 0

Chu'ng minh rang: y - P = P - a = —

Tfirgia Ihict ta c6:

G i a i

cosrt + COS (3 — — cos~<

sincv + sin3 = —sin^i cos^a + cos^i3 + 2cosacos^3 = cos^ (1)

sin^ ft+ sin^ ^3 + 2sinasinf3 = sin (2)

Cong lirng ve (1) (2), di den 2 + 2cos(a - p) = 1

Hoan loan tifdng tif cung c6: cos(P - y) = cos(y - a) = - ^ (4)

Do 0 < a < p < Y < 271, ncn suy ra:P - a; y - P; Y - « deu thupc [0; 2n]

Vi vay tu" (3) (4) suy ra P - a; y - P; Y - ct chi nhan mot trong hai gia tn

Bai 24 Cho a + k i r , k e Z Y6\i n nguycn dtfdng, dat

coscv cos 2ft cosna

cos ft cos (V cos ft

cos ft cos ft

Cong tiTng v c n dang thiJc dang (1) voti k = l , n , di den

v%sin(k + l)ft sinkft • ^ c o s k w

/ 1;: - / TTi t-sin> r — • (2)

'-•OS ft ^ cos ft ^ COS ft

TijC (2) sau khi u'dc lu"dng so hang giong nhau d ca hai ve suy ra

-^H?iili-^ = sin(v + sin(HS„ ~ l ) = S„.sin<v (3)

Tijf (3) de dang thay rang neu a k7i (va dT nhien ^ + theo gia thiet

kiT^ , „ sin(n + l)o; *' 'o.f;j

= ^ a ^ — ) , taco: S„ = ^ ^ •

Neu k chan thi coskn - 1 => (cosa)'' = 1 j,- I Uii

Neu k le ihl cosk;: = - 1 ^ (cosa)'' = - 1

GAums2. BAT OANC THlTC Ll/dNC C l A C

§ 1. B A T D A N G THirC L U O I N G G I A C K H O P i G C 6 D I E U K I E I N

PhiTdng phap giai cac bai loan thuoc muc nay cung giong nhU' phtfdng phap

giai cac bai loan trong chuyen muc "ChiJng minh cac dang thuTc lifting giac

khong dieu k i c n " nghia la can ket hdp mpt each nhuan nhuyen giffa viec sijf

dung cac k i e n thtfc va k l nang chrfng minh ba'l dang thtfc vt'Ji viec ap dung

linh hoal cac cong thuTc bien doi Itfdng giac can thiel

B&i 1, Khong dung bang so hang hay may tinh ca nhan, chiJug minh rSng

(VI a, 11" dcu G (0; 90") va y = tanx la ham dong bien trong khoang (0; 90"))

2 2tan(\ 5

BSI duong hpc sinh gidi Lufing giic — Fban Huy Khal

Tu" do diTa vao tinh dong bien cua ham so y = 3x - Ax" suy ra:

-<sin20" < — y 3 20 <y(sin20")<y f 7

Ap dung cong thu"c sin3a = 3sina - 4sin"^a, la c6:

y(sin20") = 3sin20" - 4sin^20 = sin60", nen

TT T T 2TT 2TT 3TT 3TT T T

C O S —7 7 7 7 7 7 7 = cos—COS hcos — cos hcos—cos— (3)

Cty Train nrv DVVH Kbang V/ft

VI X > y > z > 0, nen (5) hien nhien dung => dpcm

Bai 5 ChiJug minh rang cos38" > tan38" (khong diing bang so' hay may tinh)

: cosa > tana neu 0 < a < X(, < 90" (2) ^

Ta CO nhan xet sau: cos36" > tan36" (3)

That vay (3) cosl36" > tan'36" (do cos36" va tan36" deu la cac so diTcJng)

B 6 I duSng hpc sinh QHH I ir<,tiig gUc — nan Huy KItdl

=> 4cos72" = > 1 (5)

cos36"

Tilf (5) suy ra (4) d u n g v a y n h a n xet (3) di/tfc c h i J n g m i n h

Bay gicf tif (2), (3) s u y r a x „ > 36" (6)

2 4 200 Tir (8) (9) suy ra ,sinx, > — > — (10)

Tir (2) v^ (11) suy ra cos38" > tan38" =^ dpcm

Bai 6. Chtfug minh rang vdi moi a, ta c6:

4sin3a + 5 > 4cos2a + 5sina

Giai

Ta c6: 4sin3a + 5 > 4cos2a + 5sina t O J f +

<=> 4(3sina - 4 s i n V ) + 5 > 4(1 - 2sin^a) + 5sina

<=> 16sin'a - 8.sin^a - 7sina - 1 < 0

Chufng minh: m < ' ^ L ^ M , d day

cosa, +cosa2 + + cosa„

M = max tanttj; m = min tanttj

m(cosai +.,.+ cosa„) < sinai + + sina„ < M(cosai + +cosan)

Vay ta se chtfug minh (3) la diing khi n > 2

Vay (6) dung ^ (5) dung khi n = 2

* Gia %\i (5) da dung khi n = k, ttfc la ta c6: ksina > sin(ka)

Theo gia thiet quy nap ta c6:

(k + l)sina = ksina + sina > sin(ka) + sina , ; ,

NhiT the (5) cung dung khi n = k + 1

Thco nguyen ly quy nap thi (5) dung Vn > 2

Do 0 < X < — , nen theo bat d i n g thiJc ccf ban ta c6: tanx > x

Vay f(x) la ham dong bien tren sina + tana - 2a > 0

=> sina + tana > 2a (2)

Thay (2) vao (1) va c6 dpcm

Chu y: Ta luon suT dung bat dang thiJc cd ban sau:

Neu 0 < a < — => sina < a < tana (3)

2

Ba't d i n g thdrc nay ngoai viec cMng minh b^ng c^ch suT dung dao ham, ta

con C O each giai khac sau day

Tren diTcfng tron ddn vi dat B O A = a

Khi do ta c6: SOAT = S q u a i O A B > SAOAB

O - O A A T > — > - O A ^ sin a

2 271 2

o tana > a > sina

(do 0 A = 1; A T = tana)

Vay (3) diTcJc chtfug minh

Bai 10 Chtfng minh 4tan5"tan9" < 3tan6"tanlO"

(1) ( d o x > 0 , x + 2 y > 0 ) sin(x + 2y) sinx

x + 2y e t h a m s o f ( t ) = — v d i 0 < t < —

V i 0 < X < X + 2y < 571 l ( x + 2y) < r(x) vay (1) dung => dpcm

1 G i a i nhu" bai 9 bhng each x c t ham so" r(x) = 3x - x^ vdi 0 < x < — )

2 G i a i nhiT bai 9 bang each x c t ham so f(x) = tanx - sinx - — v d i 0 < x < — )

Giai tifdng W nhif bai 13

max f(x) = /max f^(x); min f(x) = /min f^(x)

Dap so: max t(x> = 2 ( ^ 2 + l ) ;

Dat a = 1 + cos2a; b = 1 + cos4a; c = 1 - cos6a

Do a > 0; b > 0, c > 0 va do ve trai co nghla nen a > 0, b > 0, c > 0

T a c 6 : a + b + c = 3 + (cos4a + cos2a - cos6a)

<=> sin2a = - - s i n 4 a 2 sin^ 4a = 1

B6i ducing hpc sinh glolLapngglAc -rii.m liuy Khal

X ' + y" + z^ > - 2 ( x y + yz + zx) = 2 TCr (3) (4) suy ra (2) dung => (1) dung dpcm

Dau bang xay ra<::>x + y + z = ()

^ cos(a + P) ^ cos(P + y) ^ cos(y + a ) _ ^

s i n ( a - p ) s i n ( P - y ) s i n ( y - a )

B a i 19 Cho a va P la hai so thifc va k la so tiT nhien

ChiJng minh bat d^ng thiJc sau (neu ve trai cua no c6 nghla)

(1) ro rang dung khi n = 1

Gia sur (1) da diing den n = p, tiJc la

< pisinxl + Isinxl = (p + l)|sinx| (theo (2))

Vay (1) cung dijng khi n = p + 1

Theo nguyen ly quy nap suy ra (1) dung Vn nguyen dUdng, Vx e R

A p dung (1) v(3i cac so' tu" nhien r, s ta c6 Vx k n

cosx + sinx cos px

<

Cty TimnMTVDVVHHhang Vl^t

sinrx < r - sinsy sinx — i , siny <s (3)

Tiif (3) suy ra V X , y ^ k n , ta c6:

sin rx sin sy sinx siny sin ry sin sx

< r s

< r s

s m x s m y Tir (4) (5) suy ra:

sin rx sin sy + sin ry sin sx

(4) (5)

sinrx sinsy sin X sin y

sin ry sinrx

s m x s m y

Vay Vr, s nguyen du'dng, Vx, y ^ kn, ta c6:

sin ry sin sx + sin rx sin sy

Tu-dng tu- cQng c6: sin rx sinsy = ^ [ c o s ( k p + a ) - cos(ka + P)]

-ai CO 2sinysinx = cos(y - x) - cos(y + x) = cosp - cosa (7)

iTiir tren ta c6: sinrysinsx + sinrxsinsy = cosacoskp - coskacosp

T i , ^ „ / - 7 x / g x - //rx V c o s k p c o s a - c o s k a c o s p , 2

1 hay (7) (8) vao (6) va co: t l < - 1

cos p - cos a

Do la dpcm

Bai 20, Cho ham so r(x) = cos2x + acos(x + a)

Chu-ng minh rllng: {min +1 max > 2

Tir (1) (2) suy ra max f(x) > 1

7t

V 1) < - l /min f(x)'i^

\R /

Bay gifJ lif (3), (5) ccS dpcm

Bai 21 Tim gia tn Idn nhal va be nhal cua ham so

Phu'dng trinh asinx + bcosx = c c6 nghiem <=> + b^ > c^)

NhiT vay ton lai x,, G R sao cho sin2X() + cos2xo - 1 = 0 ==> f(X()) = 0

Nhir vay la di den min f(x) = 0 - ,

xeR

Gpi m la mot gia tri luy y cua ham so y = sin2x + cos2x-l

Khi do phifdng trinh sau day (an x)

2 + sin2x sin2x + cos2x-l

2 + sin2x

De lhay do 2 + sin2x > 0 Vx, nen

(1) o sin2x + cos2x - 1 = m{2 + sin2x)

1 Tim a, b, c de ham so'

f ( X ) = Vacos^ X + bsin^ x + c + Vasin^ x + bcos^ x + c x^c dinh Vx G R

2 Vdi a, b, c tim di/dc d cau 1 hay tim gia Iri Idn nha'l cua ham so

Vi (2) cung chinh la dieu kicn dc Vasin' x + bcos' x + c xac dinh V x G R,

nen (2) la dicu kicn can va dii dc f(x) xac djnh Vx e R

2 Vdi dicu kicn (2) xct ham so

F(x) = Vacos^ X + bsin" x + c + Vasin^ x + bcos^ x + c + —sin2x, '

2 vdi X G R

Vi f(x) > 0 V X G R, nen ta xct ham so:

Do vay Irong Iru'dng hdp nay, ta c6:

max F ( x ) = — + max r ( x ) = J2(ii + b + 2c) +

V i the Irong lru"i<ng hc.tp nay ta c6:

max F ( x ) = + max f ( x ) = + 72(a + b + 2c)

1 Kethcfp l a i , ta di de'n k e l q u a sau: max F(x) = - | m | + 72(a + b + 2 c )

B a i 23 T i m gia tri Idn nha't va nho nhat ciia bieu thuTc

P = sin-(x + y)eos(x - y) + sin-(x - y)cos(x + y) khi x, y e R

G i a l

Ta c6:

2P = cos(x - y)11 -cos(2x + 2 y ) l + cos(x + y ) | 1 - cos(2x - 2y)]

= cos(x - y) - cos(x - y)|2cos-(x + y) - 1 ] +

cos(x + y) - cos(x + y)[2cos'(x - y) - 1]

- 4cosxcosy - 4cosxcosy(cos'xcos^y - sin"xsin"y)

V a y P = 2cosxcosy[2 - (cos'x + cos'y)|

=> |p| = 2|cosx||cosy|(2-(cos-x + cos^y)) (1)

(do 2 - (cos^ X + cos- y) > 0 V X, y e R )

Cty TPamnfvDVVnKHang Vlft

Tuf (1) va theo ba't d^ng thrfc Co si, ta c6:

|P| <(cos^ X + cos^ y) 2-(cos^ x + cos'^y)

L a i theo bat d i n g thtfc Co si, ta co:

(cos^ X + cos^ y ) [ 2 - (cos^ x + cos^ y )

(cos x + cos y ) + [ 2 - (cos x + cos^ y )

D c thay 1 - sinxsiny > 0 v d i gia thiet da cho

H i e n nhien ta c6: (sin'x - l)(sin-y - 1) > 0

=> s i n ' x s i n ' y + 1 > sin'x + sin'y

=> (1 - s i n x s i n y ) ' > (sinx - siny)"

* - 1- 1- • sinx - s i n y , , „ Tu^ do de dang suy ra - 1 < — < 1 hay - 1 < P < 1

1 - s i n x s i n y

L a i CO k h i X = 0; y = - thi P = - 1 ; con k h i x = - , y = 0 thi P = 1

2 2 ^

Tijr do di den ke't quit sau: max P = 1; m i n P = - 1

B a i 25 T i m gia tn Idn nhat va nho nha't ciia ham so

r(x) - V l + 2cos- X + \ / l + 2sin-x khi x e R

G i a i

G o i m la gia t n tiJy y ciia ham so, khi do phu'tJng trinh sau day (an x )

V l + 2cos" X + V l + 2 s i n ' x = m (1) cd nghiem

Dat u==\/l + 2 c o s ^ x ; \ yfl + lsin^ \

K h i do (1) cd nghiem k h i va chi khi he sau day:

.•At-."'

B S I auang A p e ainh gkil LUfing gi'.ic - fhan Huy lUial

De thay diTdng thang

no'i M , P CO phiTdng trinh

u + V = \/3 + 1

Con tiep tuyen vcfi difdng tron (3) va song song vdi M P co phifcTng trinh

u + V = 2^/2

Tir do suy ra diTc^Jng thang u + v = m phai n3m giiya hai drfdng noi tren

Nhir vay 7 3 + l < m < 2 V 2 la dieu k i c n can va du de he (2) (3) (4) co

nghiem (ttfc la (1) CO nghiem)

Do do max f ( x ) = 2 ^ 2 ; min f ( x ) = 1 + N/3

B a l 26 ChiJng minh rang V a , V p la co:

a^(l + sin^p) + 2a(sinP + cosp) + 1 + cos^P > 0

Giai

X e t tarn thtfc bac hai P(a) vdi

P(a) = a^(l + sin^P) + 2a(sinp + cosp) + 1 + cos^p,

Bat 27 Cho ai (i = l,n) la cac goc tCly y do bkng rad, va m la so nguyen diTcfng

Ic Chifng minh:

n ^ ^ t t i + 2 ] ^ c o s a , + 2n - 2 ^ s i n ^ a ; i=l V i=l '=• / i = l

i=l

(1)

Tilf (1) suy ra Sn > 0 Vn la so' nguyen dufdng

Dau b^ng trong (1) xay ra khi va chi khi V i = 1,n ta co:

<=>

costti = 0

n cosaj + ^ a j = 0

i=l

tt: = —+ k:7t, k: e Z ' 2 ' '

Do n la so nguyen le, ncn (3) khong the xay ra, do do trong bat d i n g thuTc

S n > 0 khong the xay ra dau bang, tiJc la S „ > 0 dpcm! *

B a i 28 Cho a, b, c, d > 0 T i m gia tri l(5h va nho nhat ciia bieu thiJc: ^

^ asin"* x + bcos"* y acos"* x + bsin"* y „ ^ ^- •> u — ' u.v^

P = + tren m i e n xdc dinh cua bieu thiTc csin^ X + dcos^ y ccos^ x + dsin^ y

umiiy uyi miui

0 < cosa + cosP < — va 0 < cosacosP < —

4 4 Dat A = cosa + cosp va B - cosacosP khi do 0 < A < 1 va 0 < B < —

4 Luc nay ba't dang thuTc can chuTng minh c6 dang:

2 - A 1 - A + B

A V B ( 2 - A ) - ^ 1 - A + B -7 _ A

Da'u bang xay ra o A = l

Ta CO nhan xet sau:

Ne'u a, P la hai g()c tuy y ma |a - P = 1 (rad) thi ta c6

max||sina|,|sinp > - 1

Nhan x c l chi^ng minh nhifsau:

Ve vong iron dcin v j Hai dtfiJng thang y = ± ^ cat diffJng iron tai A , B (giao diem ben phiii true lung)

Dal A O B = 2(v, khi do la c6:

eSl duBri^hpc sMh ^Sflu^ng giAc - nan nuy taiai

Nhan xet diTcJc chtfng minh

D a t ve t r a i ciia bat d i n g thtfc can chrfng m i n h la S, k h i do ta c6:

S > —^(Isinnl + sin(n +1)1) + - ^ ( | s i n ( n + 2)| + |sin(n + 3)|) + +

n + r n + 3 '

— ! — ( | s i n ( 3 n - 2)| + |sin(3n - 1 ) | ) 3n — 1

Theo nhan x e t tren vdi mpi k = n, n + 2 3n - 2 ta c6:

Bai 31 Cho 0 < p < a < ^ ChuTng minh rang v d i m o i so nguyen diTdng n, ta c6

ba't dang thtfc sau: sin" a - sin" P >

Dau bang xay ra <=> a = p

Bai 32 Cho 0 < a < - Chtfng minh bat dang thtfc ' '

(tan a tan P + tan Plan y + tan y tana)^

3

Do sin^a + sin^p + sin^y = 1

=> cos^a + cos'P + cos^y = 2

BSI duOHIg hgc sTnh gidJ Jirpng^giac - man nuy

nimi-=> tan'alan-^p + tan-(ilan'Y + lan-ylan'a + 2tan'atan^pian'y = 1 ( l )

Dat a = tanalanP; b = lanptany; c = tanylana, thi Itf (1) c6:

<=> tana •~ Jl (do 0 < a < ) => tanp = N/3 o •

n

a = —

6

B a i 3 Cho a, p, y > 0, a + P + y < TT ChiJug minh r^ng:

sina + sinp + siny + sin(a + p + y) < sin(a + P) + sin(p + y) + sin(y + a)

G i a i

X c l hicu sau:

' S = sin(a + P) + sin{p + y) + sin(y + a) - |sina + sinp + siny + sin(a + p + y) 1

= |sin(a + P) - sin(a + p + y)| + |sin(P + y) - sinpi + |sin(y + a) - sina] - siny

= - 2 cos

= 2 s i n ^

" + P + 7

2v cos p + 1

Dau being xay ra <=> cos j — - = 0 <=> a + p + y = TT va a > 0, p > 0, y > 0

B a i 4 Cho n la so' tu' nhicn va 0 < (n + 1 )a < —

2 ChiJng minh: (1 - cos"a)(l + cos"a) < tan(na)tana

o cosnacosa - cos"" * 'acosna < sinnasina >

o cos(n + I )a < cos'"* 'acosna (2)

Ta sc chiJng minh (2) bang nguycn ly quy nap t>{ I

* V d i n = i , t h i ( 2 ) c 6 d a n g : cos2a < cos^'acosa (3)

2

75

Do gia thiet quy nap va do cosa > 0, nen tCf (5) suy ra

cos(k + l ) a c o s a < cos'''^^acoska

=> cos(k + 1 )acosa - sin(k + 1 )asina < cos^'' * ^acoska - sin(k + 1 )asina

=> cos(k + 2 ) a < cos^'' ^ ^a[co.s(k + 1 )acosa + sin(k + 1 )asina] - sin(k + 1 )asina

=> cos(k + 2 ) a < cos^'' * 'acos(k + 1 ) a + sin(k + 1 )a.sina(cos^'' * - 1) (6)

V i sin(k + l ) a > 0, sina > 0, cos^'^'^'a - 1 < 0, nen lir (6) c6:

cos(k + 2 ) a < cos''' * ^acos(k + 1 )a

Vay (2) cung diing khi n = k + 1

Theo nguyen ly quy nap suy ra dpcm

Cdch 2:

B\Xd bat d i n g thtfc da cho ve dang tU'dng di/dng sau:

tan(na)tana + cos^"a > 1 (7)

Dat f(n) = tan(na)tana + cos-"a

TriTck- het ta chiJng minh rkng V k = 0, 1, , n - 1 ta c6:

f ( k + l ) > f ( k ) (8)

That vay

(8) o tan(k + l ) a t a n a + cos'^'^'^'^a > tan(ka)tana + cos^''a

<::> tana(lan(k + 1 ) a - tan(ka)] > cos^''a( 1 - cos^a)

va thoa man he

10 cos 2a + C O S 2p + cos 2y + cos 25 > —

Chiang minh rang a, p, y, 5 0 ; ^

6

C i a i

Ta c6: cos2a + cos2p + cos2y + cos25> —

0 4 - 2(sin^a + sin^P + sin^y + s i n ^ 5 ) >Y

Do vai tro binh dang, c6 the gia siif x > y > z > t

Ro rang t > 0 That vay neu I < 0 thi tij" (1) c6: x + y + z > 1 Lai C O x^ + y ' + z* > ^ ( x + y + z)^ > ^

Dicu nay mau thuan (2) Vay t > 0 => x, y, z, t deu > 0

D a l X , = 1 1 1 1 Khi do tuf (1) c6: x, + y, + z, + t, = I

L a i C O x^ +yj^ +z^ +^'\ X +

(3)

1 ^^ ^ 1

- - t ,2

= 1 - (x + y + z + t) + X ' + y^ + z' + t^

1

= x ' + y^ + z^ + t ' Vay t i r ( 2 ) c 6 x f + y f + z ^ 4 - l f < - (4)

Tir (3) (4) ta thu lai du-dc he nhu-d) (2)