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

516.24076 B452D 5NG TRUNG HOC QUdC GIA CHU VA» PHAN HUY KHAI (Chu bien) CHU" XUAN DUNG - HOANG VAN PHU - CU PHUONG ANH HOCSmUGWl c NHA XUAT BAN DAI HOC QUOC GIA HA NOI I Ha NQll " • ^-^VryTRLTONG TRUNG HOC QU6c GIA CHU VAN AN PHAN HUY KHAI (Chu bien) ChCr XUAN DUNG - HOANG VAN PHU - CU PHUONG ANH -a Boitftmig Hocsimcwi llJdNG GlAC Tdi lieu dCing cho hoc sinh chuyfin Todn hoc sinh gidi. 'HI/ VI!:N 1 NHA XUi^T BAN BAI HOC QUOC GIA HA Nfil 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! ^ Cty Timn MTV DVVH Kbang VIft OAINC THU'C LirCfNC GIAC § 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 = cos a tana.cota = 1 2^ _ 1 + cot a = sin^ a 2. Cac c6ng thrfc CQng cung: sin(a + P) = sinacosP + sinPcosa sin(a - P) = sinacosP - sinPcosa cos(a + P) = cosacosP - sinasinP cos(a - P) = cosacosP + sinasinP tana + tanP tan(a + P) = tan(a - p) = 1 - tan a tan p tana - tanp 1 + tan a tan P 3. Cac cong thitc nhan cung: sin2a = 2sinacosa cos2a = cos^a - sin'a = 2cos^a - ] = 1 - 2sin'^a , „ 2tana tan2a = — 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 2 2 „ - a+p a-P 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 = ^—^ — _ cos(a + P) + cos(a - P) cosacosP = — — 2 . „ cos(a - P) - cos(a + P) sinasinP = — —. 2 6. Gia tri Itf^ng giac cua cac goc (cung) c6 lidn quan dac bi^t - Hai goc doi nhau: sin(-a) = -sina cos(-a) = cosa tan(-a) = -tana cot(-a) = -cota. - Hai goc bu nhau sin(7t - a) = sina cos(7t - a) = -cosa tan(7i - a) = -tana cot(7t - a) = -cota. - Hai goc phu nhau sin cos a 2 tan 71 — a u a 7C cot a = cosa = sina = cota = tana. - Hai goc hdn nhau TI .sin(7i + a) = -sina cos(7i + a) = -cosa tan(7t + a) = tana cot(7: + a) = cota. ' i:Rfti t>jj;if( ;>! ^i,•' . , .1. ,,,,:,< § 2. DANG THirC LUONQ QiAc KH6NG DIEU KI^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. -^ui PhiTctng phap giai cac bai toan n^y thuin tiiy dura vao cac phep bien doi lu'cJng giac. ^, ^ Bai 1. 1. ChiJug minh r^ng sin 18" = ^^^zl. 4 2. Chtyng minh sinl" la s6' v6 ti. Giai 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 ti. Khi do earh giai hoan toan khac each giai tren. ' Ap dung cong thufc: cos3a = 4cos^a - 3cosa ^ 8cos'20 - 6cos20 -1=0. Thay a = 20", ta co: ^ = 4cos^20" - 3cos20" ^ 8cos^20 - 6cos20 -1=0. Vay cos20" la nghiem cua phu'dng trinh Sx"* - 6x - 1 = 0. (3) Ta CO ket qua quen bie't sau trong li thuye't da thiJc. Xet phiTdng trinh da thuTc: anx" + an.ix""'+ + aix'+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^, vdip 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 ti. Vi 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"= =>cos36"=l-2sin^l8"=l- 2-^"^"'^' 16 •4cos36" = 4- =S + \. (1) Ap dung cong thiJc: Neu 0 < a < 90", Ihi coty = cola + Vl + col^a, ta c6: cot7"30' = col 15"+ Vl + col^ 15" . (2) Vi col 15 = . Vl-cos30" - ^773 = ^^"^^ = 2 + \/3, nen thay vao (2) va co: cos7"30' = 2 + 73 + Vl+4 + 3 + 4V3 = 2 + ^3 + 78 + 4^3 = 2+ V3+^(V2+ ^)' = ^/2+73 + ^/4+76. (3) Tir(l), (3) suy ra: 4cos36" + cot7"30' = V[ + V2+>/3 + 74 + V5 + 76 => dpem. Chu y: De thay: , 2 « r 'i cos cola + V1 + cot a = —— cos a 1 - + sma sin a 1 + cos a 2 cos' sma a a 2 sm cos - 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: Giai 2 h->.,y' cos a thi dang thu'c can chu'ng minh tu'dng (1) 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") n^-' ( 1 T -cos-10 sin^lO 4 4 •co.sMo" -cos-l() + -sin^lO cos^lO + cos' lO" cos^lO" 2 _9_ l6 (2) BOI auOng aye ainti glol Lupng gUc - rhmn Huy KluU Mat khac tiTcfng tiT nhtfbien doi A, ta cung c6: B = cos^lO'W(60"- 10)cos^(60+ 10) = cos^l0"(cos^60cos^l0- sin^60sin^l0)^ ( 3^ 2 /- = COSMIC COSMIC — = I 4^ V cos''10 cos 10 = —(4cos''lO"-3coslO")^ 16^ / 1 = —(cos30*')^ (do cos3a = 4cos'a - 3cosa) 16 1 3 16 4 64 (3) 9 TCf(l)(2)(3) suy ra: VT(1)= 64 12 Vay (1) dung => dpcm Nh^n xet: Xet cac each giai khac thi du tren sau day: ^'u^ k A .w. , -> 3tana-lanV ^ , Lack 2: Ap dung cong thi/c: tan3a = , ta co: l-3tan''a tan" 3a = 9 tan^ a - 6 tan"* a + tan^ a l-6tan'a + 9tan^a (4) Vi tanl30 = tan^l50 = tan^210 = ^, nen ti!f (4) suy ra khi thay a = 10*', ta c6: 1 _9tan^l0"-6tan''l0" + tan'^10" 3~ l-6tan^l0 + 9tan^l0" hay Stan'-lO" - 27lan''lO" + 33tan^lO" -1=0. (5) ' Tif (5) suy ra tan^lO" la nghiem cua phi/dng trinh: 3x'- 27x^ + 33x - 1=0. (6) ; Tu'dng tiT tan^'>0, tan'70 cfing la nghiem cua (6) Mat 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: Xin nhSc lai vdi phu'dng trinh bac ba: ax' +bx^ + cx + d = 0 (a ;^ 0) ta CO dinh li Vict sau: Ncu X|, X2, Xi la ba nghiem ciia no, thl ta c6: b X| + X^ + Xi = — a C A : X1X2 + X2X, + X,X| =. — a . d X|X2X,= . a 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) Ap dung cong thiJc: • 2 , 2n sin(a-P)sin(a + P) tan a - tan P = j- ——, cos a cos'P va chu y rang cos-6()" = —, nen dc thay 4 4sin50"sin70" 4sin lO" sin 1 lO" 4sin(-10")sin 130" ^ (7) o + —n + ^ -7, = 0 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; sinl40 = cos5(); sin20 = cos70; sinl40 = cos50; ) Do cosl()cos70cos50 ^ 0, nen (8) ocos50 + cos70-cos 10 = 0 o 2cos60coslO - coslO = 0 ocoslO-cos 10 = 0. (9) Vi (9) diing nen (7) diing => dpcm. Cdch 4: Ap dung cong IhiJc: tan'a = 1 - thi dang thiJc can chuTng minh tan 2a • 'ft. tiTdng diTctng vc'Ji dang thiJc sau: tan 10 ^ tan 50 ^ tan 70 ^ Ian20^unl00^tanl40~~ ^ t"n50 tan70 tan 10 , Rd rang (10) <=> + = 3 tan 80 tan 40 tan 20 <=> 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) - tan40(l+tan20tan80) = (). (11) i i . ^ , . r. I'ln oc + tan p ^ Ap dung cong thiTc: tan(a + P) = , nen 1-tan a tan P tan80 +tan40 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) tan 60 <=> —5—(ian8()-tan2()) = 1 + tan80tan20 tan 60 tan 80-tan 20 <=> = tan 60 1 +tan 80 tan 20 CO tan60 = lan60. (12) 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 li Viet trong thi du nay). Bai 4. Chu'ng minh rang —^— + —+ —^— = 4. 71 371 571 COS cos COS 7 7 7 Giai 7t 371 571 ^ , , , , Vi —; —; — nam trong so cac nghiem cua phu'cfng trinh: 3x + 4x = (2k + 1)71, vdi k G Z 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. Vi the (1) CO 4COS-X - 3cosx + Scos'x - 8eos^x +1=0 CO 8cos'*x + 4eos\ 8eos\ 3cosx +1=0 CO (cosx + l)(8eos''x - 4eos'x - 4cosx + 1) = 0 (2) . f 71 37: 57: 1 ,, , ,^ ^ Khi X 6 <^ —; —; — \i cosx + 1 0, nen suy ra l 7 7 7 I ),n« 371 571 7 8x' - 4x' - 4x + 1 = 0 n 7 X| +X2 +X3 = X|X2 + X2X3 + X3X1 = Pos— COS — , cos— la ba nghiem phan biet ciia phu'cfng trinh: 7 7 7 Dat X| = cos-y; X2 = eos^; X3 - cos-^, ta c6 theo dinh li Viet: • * / 4^ 1 2 -4 _ "~~~2 ,s. - 8 (4) X1X2X3 - -1 8 Ta eo: —^-— + —^ 1 71 37: 57: cos - cos cos 111 X|X2 +X2X3 +X3X1 XI X 2 X 3 Thay (4) vao (5) va c6: 1 1 1 — + — + — X| X2 X3 (5) _ 1 + = —— = 4 =0 dpcm. 3n 57: 1 cos cos COS 7 7 7 8 Nhdn xet: Xet each giai khac bang each thuan luy bien ddi lu'dng giac nhu'sau: Ta co: 1 1 1 + ^ + - 7: 371 57: cos cos COS 7 7 7 7: 37: 37: 57: . 57: 71 COS COS +COS COS +COS -cos- _7__7____7___7L___^-__ 7t 37: 57: cos COS cos 7. 7 7 (6) 71 37: Dat Si = cos— + cos— + cos 7 7 55 1 71 37: 37: 57: 57: 7: S2 - COS—COS —+ COS—cos—+ cos —cos — 71 37: 5TC SI = cos —COS—COS—, 111 11 IS, thi (6) CO + r- + n 37: 57: S, COS COS COS 7 7 7 (7) Do sin — ;^ 0, nen ta c6: 7 2S, sin — = 2sin—cos — + 2sin —cos—- + 2sin—cos 5n 1 .In . 4n .lit .6% .An = sin — + sin sin — + sin sin — 11111 . 6TZ . n - sin—=:sin —. 7 7 Do sin —5^0, =>S| = — 7 2 (8) Ta c6: 7t 371 571^ COS — + COS + COS — 7 7 7 2 / 2 " COS — + COS 7 7 2 37C 2 + COS — 1 + COS 27t 671 IOTC 1 + COS— 1 + cos -~ 2 ' 4 3 + 2 l()7:~ 27t 671 COS— + COS — + cos 7 7 7 (9) rr • 27t 57t 67t 7t IOTI 37t . . Ta CO cos— = -cos—; cos— = -cos—; cos^^ = -cos—, nen tif (9), ta co: S2-^Sf-^l3-S,|. (10) 2 4 Thay(8)vao(l())vac6: 82= + - = (II) ^ 8 4 8 2 Ta lai c6: 371 57t 1 Si = cos —cos—cos^^— = —cos — 7 7 7 1 7C 1 = COS" —+ — 2 7 4 37r 7t cos + COS — 7 7 871 27t cos + COS — 7 7 ir, 27r 1 + cos — 1 + — 4 37t 7t cos + COS — 7 7 - _1 1 ~ 4 4 57t 371 7t COS + COS + COS — 7 7 7 4 4 ' (12) Thay(ll)(12) vao (7) va c6: 1 en ' 1 1 + - 71 37t COS - COS COS 7 7 7 571 1 (Ipcm. 8 Cty Train nrV D VVH lUiang Vift Bai 5. Chufng minh rang: 4 7t 4 37t 4 571 cos + COS + COS 14 14 14 ^3 . 2 n 2 37t 2 571 4COS - COS -cos 14 14 14 Giai Taco: -,f 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) Trong(l)thay x-—, va do cos—= cos—= 0, cos—^ 0, nen tijf (1) suy ra 14 14 2 14 • phircfng trinh: 64x' - 112x' + 56x - 7 = 0 (2) nhan x, =cos'— lii nghicm. Tifcfng M Xj =:cos^—, x^ =cos^— cung I^ 14 • 14 • 14 nghicm cua (1). R6 rang X|, X:, X3 la ba nghicm khac nhau cua (2), nen theo djnh li Viet, ta c6: 112 XI + X2 + x, = 64 56 X1X2 4- X2X3 + X3X1 — X1X2X3 = 64 7 64 4 7t 4 371 4 571 COS + COS + COS 2 2 2 Taco: 14 14 14^Xi+X2+X3 . 2 ^ 2 371 2 571 4XIX',XT 4cOS - COS COS ^'^l'^2'^3 14 14 14 _ (X| +X2 +X3)^ -2(X|X2 +X2X3 +X3X1 4X1X2X3 112 64 64 2 56 56 '64_64t , 64 16 • dpcm. [...]... cac dai lifdng trong he IhiJc can chiyng minh thoa man mot dieu k i e n 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 TO(7), (7) (4)vagiathietsuyra cosu + cosv + cosw = m o cos(P + y) + cos(y + a) + cos(a + p) = m Do la dpcm B a i 2 Cho cosa + cosP + co.sy = 0 Chiirng minh rang cosacospcosy = (cos3a + cos3p + cos3y) so,, PhiTctng phap giai... 6 Cho cos(2a + P) = 1 ChlTng minh he thiTc tan(a + p) - tana = 2 t a n 2 (•(x) = acosxcosa - asinxsina + bcosxcosp - bsinxsinP B a i 4 Cho Do lan(a + P) va tanP co nghia nen cos(a + P) ;^ 0; cosp ^ 0 (2) VcHi m o i x, la co: e (1) T i r ( l ) , la co: Giai f(x) = 0, Vx TiTdng tir b = B a i 5 Cho sin(a + 2p) = 2sina Chiang minh rang tan(a + p) = 3tanp, " Do la dpcm ra (do D ^ 0) D smx (3) B a i 3 Cho. .. Bai 14 Cho a, P deu thuoc khoang (0; ^ ) va a ^ p Gia su" cos X —cos (V Bai 13 Cho 0 < a < — , O < 0 < — va thoa man dieu kien: 2 2 sin'a + sin'P = sin(a + P) Chufng minh rang: (x-V^ = ^ sin ^ a cos 3 cos X - c o s 3 => dpcm sin^3cos(.v Lhu^ng mmh rang: ^ tan" — = tan —tan — 22 2 2 Giai Tir giii thict ta c6: cosxcosasin'P - cosxsin'acosP = sin^Pcos'a - sin^acos^P (7) Khaag Vlft BAl duding hgc sinh Luting... sin(x - 3)cos(x - a) (2) sin(2x - 2a) + sin(2x - 23) 2 s i n [ 2 a - ( a + 3)] 2 sin[2x - (a + 3)] cos(a - 9) T i r ( l ) (2) suy ra dpcm Bai 9 Cho tan(a + P+ = 3tana Chtirng minh: sin(2a + 2P) + sin2a = 2sin2p 2 s i n [ 2 x - ( a + 3)] = cos(a - P) => dpcm 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) B6I duoing hpc abib glol Lupng gidc - Phan... Chia ca hai vc cua (2) cho cosacos(a + P) ta c6: / ^ 2tan(a + 3) — 2tana r— + l + tan^(a + 3) Chu y: Ta h i c u r i n g khi de bai bat chu'ng minh mot he thu'c nao do, thi mac ' • (4) l + tan^a nhien da giai thiet la cac bleu thu'c c6 mat trong he thuTc do triTdtc het phai c6 Thay tana = t, tan(a + P) = 3t vao (4) r o i rut gpn ta cung c6: nghla sin(2a + 2p) + sin 2 a = Bai 8 Cho t a n - = 4 t a n —... (r-l)-(r-i)cosa (4) (2) B 6 I duSng h p c alnb g l o l Lugng Tir(4) (5) suy ra glAc - fban r-1 => Do la dpcm Bai 18 1 Cho cosx cos2x 2 Huy Khal + cty 2 Ian —lan — = r + 1 2 2 r-1 2« 2 2 2 1^ 2aT ~ a i — 'AT ChiTng minh rSng: sin' — = — ' ^ ^ ^, sinx sin3x sin5x ^, , ' , a-, 2 a, - a 4a2 2 Cho = = — ChiTng minh rang: ' ^ a, cos3x 33 aj ^ , , • , V • X > a3 Giai , , a^cosx cos 2x cos2x ; 'd.y = aTCOs3x... - >9 Va b cy (6) T i r ( 5 ) ( 6 ) di den - + - + - >2 a b c i 1 1 i + cos2a l + cos4a i-cos6a >2 =>dpcm B^i 18 Cho a, p, y doi mot khac nhau va khac - + kn, k e Z • uu«' cos^(a + p) C O S ^ ( P + Y) cos^(Y + a ) ChiJng mmh bat dang thtfc: — —+ — ~ - — — + — -^2 ou 2 \R / Giai < sinpx| + |sinx| < pisinxl + Isinxl = (p + l)|sinx| "it ta c6- = ^ [ c o s ( a - kp) - cos(ka - p) Giai sinpx| . 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). 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