Bồi dưỡng kiến thức học sinh giỏi lượng giác: Phần 1

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Tài liệu Bồi dưỡng học sinh giỏi lượng giác cung cấp cho người đọc một số lượng lớn các bài toán chọn lọc đủ thể loại, mỗi phần của tài liệu có thể xem như một chuyên đề riêng trình bày trọn vẹn vấn đề một cách hoàn chỉnh. Mời các bạn cùng tham khảo nội dung phần 1 tài liệu.

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Nội dung Text: Bồi dưỡng kiến thức học sinh giỏi lượng giác: Phần 1

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Cty Timn MTV DVVH Kbang VIft Lcdnoiddu O A I N C T H U ' C LirCfNC G I A C 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 § 1. CAC CONG THirC LUONQ GIAC CO BAN vien trirdng Trung hoc Quoe gia Chu Van An. Quycn sach gom 5 chiTdng: 1. Cac thuTc Irfcjfng giac cd ban - Chirring 1: Dang IhiJc liTdnggiae. sin^a + cos'a = 1 sin a - ChU'ring 2: Baft dang Ihu'e lu'ring giac. • tana = - ChU'ring 3: Ding thite lUdng giac trong lam giae. cos a cota = cos a - ChU'ring 4: Bat dang thiJc trong tarn giac. sma - Chirring 5: Vai ifng diing cua li/dng giac trong vi^c giai cdc b^i to^n sd cap. 1 + tan^a = 1 1 + cot a == 1 tana.cota Cuon sach nay sc cung ca'p cho ban doc mot so lifdng raft Idn c^c bai toan chon cos^ a 2 ^ _ sin^ a loc gom du the loai. Moi phan cua cuon sach c6 the xem nhiT mot chuyen de rieng 2. Cac c6ng thrfc CQng cung: trinh bay tron ven mot van de mot each hoan chinh (rieng phan phu'dng trinh lifdng sin(a + P) = sinacosP + sinPcosa giac se du'dc de cap trong cuon khac). sin(a - P) = sinacosP - sinPcosa Ngoai vi$c he thong va phan loai bai tap, chiing toi luon chu y ve vice phat cos(a + P) = cosacosP - sinasinP trien mot bai tap theo cac hifdng tong quat hoa, dac bi§t hoa va so sanh binh luan cos(a - P) = cosacosP + sinasinP cac phiTdng phap giai khac nhau. Trong mpl chijfng mifc nha'l dinh chung loi manh tana + tanP dan di^a ra cac giai phap ve khia canh sif pham de dung cuon sach nay. Vi le do tan(a + P) = 1 - tan a tan p cuon sach nay se la mot tai lieu tot de cac ban giao vien lham khao vao muc dich tana - tanp day cho hoc sinh mot each tifduy mot bai loan (chuf khong ddn ihuan giang day cho tan(a - p) = 1 + tan a tan P hoc sinh hicu mot bai loan cu the). 3. Cac cong thitc nhan cung: 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' sin2a = 2sinacosa cua mot lam giac vdi cac nghiem cua mot phu'dng trinh bac ba Wring iJng. Theo cos2a = cos^a - sin'a = 2cos^a - ] = 1 - 2sin'^a quan niem ciia chiing loi day la mot trong nhCfng phan dac sac cua cuon sach. ,tan2a „ = 2tana— Mac dij hel siJc c6' gang trong qua trinh bien soan cuon sach, nhi/ng vdi mot 1-tan^a dung lu'ring qua Irin can truyen tiii cho ban doc nen cuon sach khong the tranh khoi sin3a = 3sina - 4sin'a nhffng khic'm khuyet. l/t > cos3a = 4cos^a - 3cosa Chiing loi rat mong nhan diTric sif gop y cua ban doc de cuon sach hoan ihien 3tana-lan'^a tan3a = hrin nCa trong cac Ian tai ban sau. l-3tan^a Thi^ lij' gop y xin gufi ve 4. Cac cong thtfc blen tong thanh tich Phan Huy Khiii Tri/dng Trung hoc Quoc gia Chu Van An. . „ ^ . a+p a-p 10 - pho Thuy Khuc - quan Tay Ho - Ha NQI sina + sinP = 2.sin ^cos Xin chan lhanh cam dn! --^ 2 2 . „ „ a+p . a-P sina - sinP = 2 cos ^-sin
^SlduSng hfc sinb gioi Lupag gldc - rhan Huy Kbal ^ „ s i n ( a + P) - Hai goc hdn nhau TI tana + tanp = cos a cos - (3 .sin(7i + a) = -sina sin(a - P) cos(7i + a) = -cosa tana - tanp = tan(7t + a) = tana cosa cosp sin(a+p) cot(7: + a) = cota. ' i:Rfti t>jj;if( ;>! ^ i , • ' cota + cotp = sin a sin P . , .1. ,,,,:,< § 2. D A N G THirC L U O N Q QiAc K H 6 N G D I E U K I ^ N cota - cotp = sin(p-a) sin a sin p Cdc bai toan trong muc nay co dang sau day: ChiJug minh cac he thiJc ii/cJng 5. Cac cong thitc bi6n tich thanh ts'ng 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 sinacosp = sin(a + P) ^—^+ sin(a - — p) lu'cJng giac. ^, ^ _ cos(a + P) + cos(a - P) Bai 1. cosacosP = — — 2 1. ChiJug minh r^ng sin 18" = ^^^zl. . „ cos(a - P) - cos(a + P) 4 sinasinP = — —. 2. Chtyng minh sinl" la s6' v6 ti. 2 Giai 6. Gia tri Itf^ng giac cua cac goc (cung) c6 lidn quan dac bi^t 1. Taco: sin54" = cos.36" - Hai goc doi nhau: 3 sin 18" - 4sin^ 18" = 1 - 2sin^ 1S" sin(-a) = -sina o 4sin' 18" - 2sin^ 18" - 3sin 18" + 1 = 0 cos(-a) = cosa (sinl8"-I)(4sin^l8" + 2sinl8"-1) = 0. (1) tan(-a) = -tana Do 0" < 18" < 90" => 0 < sinl8" < 1, nen cot(-a) = -cota. ( l ) o 4 s i n ' l 8 " + 2sinl8"-1 =0. (2) - Hai goc bu nhau Lai tuf 0 < sinlS" < 1, nen tir (2) suy ra .sinl8" = ^^^^-^^ => dpcm. sin(7t - a) = sina 4 cos(7t - a) = -cosa 2. Ap dung cong thuTc sin3a = 3sina - 4sin^a, va gia thiet phan chiJng sinl" 1^ tan(7i - a) a = =-tana so hffu ti, khi do theo tinh cha't ciia cac phep tinh vdi so' hi?u ti suy ra sin3", sin cosa cot(7t2- a) = -cota. sin9", sin27", sinSl" la sohifuti. Hai goc - cos 71 phu nhau Do sin8l" = cos9" va .sin 18" = 2sin9"cos9", nen suy ra sin 18" la so hffu ti. TiT = sina phan 1/ta c6: la so hiJu ti, tiTc la ^/izl = £ vdi p,q e N — a 4 4 q = > N/5 = 4 - + 1, vay N/S la so hffu ti. tan u 7C = cota 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. >. ' cot aa = tana.
Nhdn xet: cot7"30' = col 15"+ Vl + col^ 15" . (2) 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. - ^773 2. Bay gicf xet bai toan sau: Vi col 15 = . = ^ ^ " ^ ^ = 2 + \/3, nen thay vao (2) va Vl-cos30" 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 co: cos7"30' = 2 + 73 + V l + 4 + 3 + 4V3 = 2 + ^3 + 78 + 4^3 ^ 8cos'20 - 6cos20 - 1 = 0 . = 2+ V3+^(V2+ ^ ) ' Thay a = 20", ta co: ^ = 4cos^20" - 3cos20" = ^/2+73 + ^/4+76. (3) ^ 8cos^20 - 6cos20 - 1 = 0 . T i r ( l ) , (3) suy ra: Vay cos20" la nghiem cua phu'dng trinh Sx"* - 6x - 1 = 0. (3) 4cos36" + cot7"30' = V[ + V2+>/3 + 74 + V5 + 76 => dpem. Ta CO ket qua quen bie't sau trong l i thuye't da thiJc. Chu y: De thay: Xet phiTdng trinh da thuTc: anx" + a n . i x " " ' + ... + a i x ' + a „ = 0, (4) , ..2 « 2 cos' Trong do a; la so'nguycn vdi moi i = 0, 1, 2,...,n. r 'i cos cos a 1 1 + cos a 2 cola + V1 + cot a = —— - + Goi P la tap hdp tat ca cac \idc cua ao con Q la tap hdp tat ca cac \idc cUa ap. sma sin a sma a a 2 h->.,y' sm cos - Khi do ne'u (4) co nghiem hiJu ti, thi nghiem do phai c6 dang Bai 3. Chu'ng minh rting x = i ^ , v d i p e P, q e Q. tan-10" + tan-50" + tan-70" = 9 q Giai Ap dung vao (3), ta thay moi nghiem hSu ti ciia (3) ne'u c6, thi chung deu thuoc vao tap hcfp sau: Ap dung cong thu'c 1 + tan'a = thi dang thu'c can chu'ng minh tu'dng Q = {±1/8; ± 1/4; ±1/2; ± 1}. cos a dUctng vdi dang thu'c sau: 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. V i eos^lO" ' cos-50" ' eos^ 70 eos20" la mot nghiem ciia (3) nen eos20"la so v6 ti. Do la dpem. ^ cos^ 50" cos' 70" + cos^ lOcos^ 70" + cos^ lOcos^ 50 _ Bai 2. ChiJng minh: (1) 4cos36" + cot7"3()' +^ + +S +S + ^ 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: Giai A = cos'(60" - 10")cos'(60" + lO") + eos'10"[cos'(60" + lO") + cos'(60" - lO")] Theobai 1, ta CO sinl8"= = [(cos60eoslO + sin60sinl0)(cos60eosl0- sin60sinlO)]^ ', + cos-10"[(cos60cosl0- sin60sinl0)' + (cos60cosl0 + sin60sinlO)^] =>cos36"=l-2sin^l8"=l- 2-^"^"'^' = (cos'60eos-10 - sin^60sin^l0)- + co,s^lO"(2cosYiO"eos'lO" + 2sin'60sin^l0") 16 ( 1 T -cos-10--sin^lO •co.sMo" - c o s - l ( ) + - s i n ^ l O n^-' •4cos36" = 4 - =S + \. (1) 4 4 Ap dung cong thiJc: Neu 0 < a < 90", Ihi c o t y = cola + V l + c o l ^ a , ta c6: cos^lO-- + cos' lO" - - c o s ^ l O " _9_ (2) 2 l6
BOI auOng a y e ainti glol Lupng gUc - rhmn Huy KluU M a t khac tiTcfng tiT nhtfbien doi A, ta cung c6: ta CO dinh l i V i c t sau: B = c o s ^ l O ' W ( 6 0 " - 10)cos^(60+ 10) N c u X|, X2, X i la ba nghiem ciia no, thl ta c6: = cos^l0"(cos^60cos^l0- sin^60sin^l0)^ b X| + X^ + Xi = — 2 /- a ( 3^ = COSMIC COSMIC — = c o s ' ' 1 0 - - c o s 10 = —(4cos''lO"-3coslO")^ C A : V 16^ / I 4^ X1X2 + X2X, + X,X| =. — a . 1 = —(cos30*')^ (do cos3a = 4cos'a - 3cosa) d 16 X|X2X,= . a 1 3 (3) Cdch 3: Dang thuTc csln chifng minh tU'rtng dU'dng vdi dang thiJe sau: . 16 4 64 tan-10" + tan'5()" + tan'70 = 3tan'60" 9 o (lan-60 - tan^ 10) + (lan'60 - tan^50) + (tan'60 - tan-70") = 0 (7) T C f ( l ) ( 2 ) ( 3 ) suy ra: V T ( 1 ) = 12 A p dung cong thiJc: • 64 2 , 2n s i n ( a - P ) s i n ( a + P) tan a - tan P = j - ——, Vay (1) dung => dpcm cos a c o s ' P Nh^n xet: va chu y rang cos-6()" = —, nen dc thay X e t cac each giai khac t h i du tren sau day: 4 ^'u^ k A .w. , -> 3tana-lanV ^ , 4sin50"sin70" 4sin lO" sin 1 lO" 4 s i n ( - 1 0 " ) s i n 130" ^ Lack 2: A p dung cong thi/c: tan3a = , ta co: (7) o + — n + ^ -7, = 0 l-3tan''a cos-10 cos-50" cos70" 9 tan^ a - 6 tan"* a + tan^ a tan" 3 a = (4) o 4sin.'50"sin70"cos50"cos7()" + 4 s i n l 0 " s i n l 10"cos-10cos^70" ' l - 6 t a n ' a + 9tan^a - 4sinlO'sin 13()"cos'l()cosl*i0 = 0 V i tanl30 = tan^l50 = tan^210 = ^, nen ti!f (4) suy ra k h i thay a = 10*', ta c6: e> sinlOO'sinl4()"cos50cos70 + sin20sinl40"cosl0cos70 - sin20sinlOOcosl()cos5() = 0 1 _ 9 t a n ^ l 0 " - 6 t a n ' ' l 0 " + tan'^10" (do sinl 10 = sin 70; sin 130 = sin50 va sin2a = 2sinacosa) 3~ l - 6 t a n ^ l 0 + 9tan^l0" « > cosl0cos70cos-50 + coslOcos5()cos^70 - cos50cos7()cos-10 = 0 (8) hay Stan'-lO" - 27lan''lO" + 33tan^lO" - 1 = 0 . (5) ' (do sinlOO = coslO; s i n l 4 0 = cos5(); sin20 = cos70; s i n l 4 0 = cos50;...) Tif (5) suy ra tan^lO" la nghiem cua phi/dng trinh: Do cosl()cos70cos50 ^ 0, nen 3 x ' - 27x^ + 33x - 1 = 0 . (6) ; (8) o c o s 5 0 + c o s 7 0 - c o s 10 = 0 • 'ft. Tu'dng tiT tan^'>0, tan'70 cfing la nghiem cua (6) o 2cos60coslO - coslO = 0 M a t khac de tha'y tan-lO", tanl^O", tan^70 la ba so khac nhau o c o s l O - c o s 10 = 0. (9) (cu the ta CO tan^lO < tanl^O < tan^70), nen tan'10, lan-50, tan'70 la ba V i (9) diing nen (7) diing => dpcm. nghiem khac nhau cua (6). Cdch 4: A p dung cong IhiJc: t a n ' a = 1 - thi dang thiJc can chuTng minh tan 2 a Theo dinh H Viet vdi phU'rtng trinh bac ba, ta c6: tiTdng diTctng vc'Ji dang thiJc sau: / 27^ tan^l()" + tanl'50 + tan^70 = = 9 => dpcm. tan 10 ^ tan 50 ^ tan 70 ^ Ian20^unl00^tanl40~~ Chu y: X i n nhSc lai v d i phu'dng trinh bac ba: ^ t"n50 tan70 tan 10 , a x ' +bx^ + cx + d = 0 (a ;^ 0) Rd rang (10) + = 3
Ian50tan4()tan2() + tan7()tan8()tan2() - t a n l ( ) i a n 8 ( ) t a n 4 0 = 3tan8()tan40tan20 371 571 Pos— COS — , c o s — la ba nghiem phan biet ciia phu'cfng trinh: lan20 + tan80 - tan 40 = 3tan2()lan4()tan80 7 7 7 (do Ian50lan40 = tan70tan2() = tanl0tan8() = 1 ) ' 8 x ' - 4 x ' - 4x + 1 = 0 tan2()( 1 - tan40tan8()) + tan8()(l - tan20lan40) n Dat X | = cos-y; X2 = e o s ^ ; X3 - c o s - ^ , ta c6 theo dinh l i Viet: • * - t a n 4 0 ( l + t a n 2 0 t a n 8 0 ) = (). (11) i 7 i . ^ ,. ,s. - r. I'ln oc + tan p ^ / 4^ 1 A p dung cong thiTc: tan(a + P) = , nen X| +X2 +X3 = 2 1-tan a tan P tan80 + t a n 4 0 Ian4'0 + tan20 -4 _ (4) (11) CO tan20 + tan 80 tan 40(1 + tan 20 tan 80) ^ 0 X | X 2 + X 2 X 3 + X3X1 = " 8~ ~ ~ 2 tan 120 tan 60 -1 c:> —^-— tan 4()(tan 80 - tan 20) = tan 40(1 + tan 80 tan 20) X1X2X3 - tan 60 8 1 1 1 1 < = > — 5 — ( i a n 8 ( ) - t a n 2 ( ) ) = 1 + tan80tan20 Ta eo: —^-— + — ^ tan 60 71 37: 57: — X| +— X2 +— X3 tan 8 0 - t a n 20 cos - cos cos = tan 60 1 1 1 1 + t a n 80 tan 20 (5) X|X2 +X2X3 +X3X1 CO tan60 = lan60. (12) XI X 2X3 _ 1 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, Thay (4) vao ( 5 ) va c6: = —— = 4 = 0 dpcm. + 1 3n 57: eon 1 each ket http vciii cac k i c n thiJe vc tinh chat nghiem ciia mot phu'dng cos cos COS 7 7 7 8 trinh dai so' (cu the sii' dung dinh l i V i e t trong thi du nay). Nhdn xet: X e t each g i a i khac bang each thuan luy bien ddi lu'dng giac nhu'sau: B a i 4. Chu'ng minh rang —^— + — + — ^ — = 4. 71 371 571 Ta co: COS cos COS 7: 37: 37: 57: . 57: 71 7 7 7 COS COS +COS COS +COS -cos- 1 1 1 Giai _ 7 _ _ 7 _ _ _ _ 7 _ _ _ 7 L _ _ _ ^ - _ _ (6) 7t 371 571 ^ , , , , 7: + ^ +- 371 57: 7t 37: 57: cos cos COS cos COS cos V i —; — ; — nam trong so cac nghiem cua phu'cfng trinh: 7. 7 7 7 7 7 71 37: 55 3x + 4x = (2k + 1)71, vdi k G Z Dat Si = cos— + c o s — + cos TiT do suy ra xet phu'cfng trinh sau day: 7 7 1 71 37: 37: 57: 57: 7: eos3x - -cos4x hay eos3x + eos4x = 0. (1) Ta eo: cos3x = 4eos x - 3cosx S2 - COS—COS — + C O S — c o s — + cos — c o s — 71 37: 5TC cos4x = 2cos"2x - 1 S I = cos —COS—COS—, = 2(2eos^x - 1)' - 1 = 8cos^x - 8eos'x + 1. 1 1 1 V i the (1) CO 4COS-X - 3cosx + Scos'x - 8eos^x + 1 = 0 (7) CO 8cos'*x + 4 e o s \ 8eos\ 3cosx + 1 = 0 thi (6) CO 1 n + 1 37: r - + I 57: S , S, COS COS COS CO (cosx + l)(8eos''x - 4eos'x - 4cosx + 1) = 0 (2) 7 7 7 Do sin — ;^ 0, nen ta c6: . f 71 37: 57: 1 , , , ,^ ^ 7 K h i X 6
Cty Train nrV D VVH lUiang Vift 5n B a i 5. Chufng m i n h rang: 2S, sin — = 2sin—cos — + 2sin —cos—- + 2sin—cos 1 4 7t 4 37t 4 571 . I n . 4n .lit .6% .An cos + COS + COS = sin — + sin sin — + sin sin — 14 14 14 ^ 3 . 2 n 2 37t 2 571 1 1 1 1 1 4 C O S - COS -cos . 6TZ . n 14 14 14 - s i n — = : s i n —. 7 7 Giai Taco: -,f Do sin — 5 ^ 0 , =>S| = — (8) 7 2 cos7x = cos6xcosx - sin6xsinx Ta c6: = cosx(4cos^2x - 3cos2x) - 2sinxsin3xcos3x 2 / = cosx[4(2cos\ l)"* - 3(2cos^x - 1)] - 2sinx(3sinx - 4sin\)(4cos^x - 3cos) 7t 371 571^ 2 " 2 37C 2 COS — + COS + COS — COS — + COS + COS — = cosx|4(8cos\ 12cos''x + 6cos'x - I ) - 6cos^x + 3J 7 7 7 7 7 - 2sin'x(3 - 4sin^x)(4cos\ 3cosx)l 27t 671 IOTC = eosx(64cos''x - 112cos''x + 56cos-x - 7). (1) 1 + COS 1+COS— 1 + cos -~ 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 2 14 14 2 14 • phircfng trinh: 6 4 x ' - 112x' + 56x - 7 = 0 (2) 27t 671 l()7:~ 3+ C O S — + COS — + cos (9) 2 ' 4 7 7 7 nhan x, = c o s ' — l i i nghicm. Tifcfng M X j =:cos^—, x^ = c o s ^ — cung I ^ 14 • 14 • 14 rr • 27t 57t 67t 7t IOTI 37t . . n g h i c m cua (1). R6 rang X|, X:, X 3 la ba nghicm khac nhau cua (2), nen theo Ta CO c o s — = - c o s — ; c o s — = -cos—; c o s ^ ^ = - c o s — , n e n tif (9), ta co: djnh l i V i e t , ta c6: 112 S2-^Sf-^l3-S,|. (10) XI + X2 + x, = 2 4 64 56 Thay(8)vao(l())vac6: 8 2 = - - - +- =- - . ( I I ) X1X2 4- X 2 X 3 + X3X1 — 64 ^ 8 4 8 2 Ta l a i c6: 7 X1X2X3 = 64 371 57t 1 871 27t Si = cos —cos—cos^^— = —cos — cos + COS — 4 7t 4 371 4 571 7 7 7 7 7 COS + COS + COS 2 2 2 Taco: 14 14 14^Xi+X2+X3 1 7C 1 37r 7t ir,1 + cos 27r 1 — + — cos 37t 7t + COS — . 4cOS 2 ^ - COS 2 371 COS 2 571 4XIX',XT ^'^l'^2'^3 = COS" — + — cos + COS — 14 14 14 2 7 4 4 7 7 7 7 - _1 1 COS 57t + COS 371 7t + COS — (12) _ ( X | + X 2 + X 3 ) ^ - 2 ( X | X 2 + X 2 X 3 +X3X1 4X1X2X3 ~ 4 4 7 7 7 4 4 ' T h a y ( l l ) ( 1 2 ) vao (7) va c6: 112 2 56 56 1 en-.' 64 '64_64t 64 , 1 1 (Ipcm. 71 +- 37t 571 1 64 16 COS - COS COS 7 7 7 8 • dpcm.
Bai da&ng hpc slnh giol Lupng gidc - rhan Ituy Khai Cty rnilN l*rrv own Khang Vl^t B a i 6. Chu'ng m i n h rang 2. Ta CO b a i l o a n tU'Ong t u " s a u : 1 1 1 •+ = 8. Chu'ng m i n h rang: — ^ — + !—— + - = 415. . 2 2:1 ' . 2 371 . 2 4 71 4 2Tt 4 371 sin - sin sin COS _^ cos ^ cos ^- i tJl ' • ' 0 fe' { Hif 7 7 7 Giai V a n tat Idi giai nhu" sau: 'f".*' 71 271 37t , , ^ , , , T , iri Ro rang thoa man phu'dng trinh: sin"4x = '-in^Sx. (1) Do —; — ; — thoa man phu'dng trinh: cos'4x = cos"3x, "r '-M 7 7 7 f 2 ^ -'271 2 371 , , Dat t - sinx va ap dung cac cong thiJc: nen suy r a t| =eos y , t , = cos" — , 13 = cos — la b a nghiem phan b i e l cua sin3x = 3sinx - 4 s i n \ 3t - 4r^ phu'dng trinh: 64t - 801' + 24t - 1 = 0, tiT d o suy ra dpcm. sin^4x - 16sin'xc()s'xcos'2x = 1 6 r ( l - r ) ( l - l^f, khi do (1) cr> dang sau: B a i 7. Chu'ng minh rang: 6 4 t ' - 1121^ + 5 6 1 - 7 = 0. (2) Theo nhan xet tren (2) eo 3 nghicm phan biet: ,1 2^ ^ 4^ ,1 8^ 3/5-3^ . 2 271 • 2 37: , . 2 67t cos— + cos— + COS— = •:\ . t t| = sin — ; t 2 = sin — U ^ ^ s i n — . fi1 V 7 V 7 V 7 V 2 1 1 ^ t2t3+t|t3+t|t2 Giai Ta eo: , 271 471 87: , ., , , . 2 271 + 371 671 t, u Ko rang — ; — ; — thoa man phu'dng trinh: cos4x = cos3x (1) sin sin sin Dc thay ( 1 ) 0 2cos"2x - 1 = 4eos'x - 3cosx , • 56 o 2(2cos"x - 1) - 1 = 4cos'x - 3cos I | l 2 + t 2 l 3 + t 3 t | =• 64 « > (cosx - 1 )(Kcos\ 4cos"x - 4cosx - 1) = () (2) A p dung dinh i i Viet v d i (2), thi (4) t|t2t3= ~'' -y'- khong thoa man phu'dng trinh cosx - 1 = 0 , nen no thoa man 64 Thay (4) vao (3) di den VP (3) = 8 dpcm. phu'dng trinh: 8cos'x + 4cos"x - 4cosx - 1 = 0 . Nhan xet: 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 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 phiTdng trinh: t^ + t" - 2 l - 1 = 0. (3) 2 27t , . 3;: , 7 67t Theo dinh l i Viet a p dung vdi ( 3 ) , t a co: cot"^ 1 col" 1 col 1 = 2 (5) 7 7 l|+l2+t.1=-' t|l2 + t 2 t 3 + t 3 l | = - 2 (4) sur dung cong ihiJc 1 + c o r a = sin^ a t|t2t3 =1. L a i ap dung cong thiJc cot"a - 1 = 2cot2acota, thi DatA = ^ + ^ + ^ , 471 27: 67: 371 127T 671 , (5) o cot — c o t — - + c o t — c o t — + cot col — = 1 B = ^ + ^/hl7 + thi 7 7 7 7 7 7 A-^ = l i + t 2 + t 3 - 3 ^ l | t 2 t 3 + 3 A B . ; '* 471 271 71 471 271 7t , ^.^ c o t — c o t — - + c o t — c o t — + c o t — c o t — = 1. (6) 7 7 7 7 7 7 Tilf(4)(5)suyra A ' = - 4 + 3 A B . ' (6) Ta bie't rSng neu x + y + z = 71, thi: cotxcoty + cotycotz + cotzcotx = 1. Lap luan tu'dng liT l a eo: B* = - 5 + 3 A B . (7) Va y ( 6 ) dung va do la dpcm. Nhan tijrng ve (6) (7), ta di den: ( A B ) ' = (3AB - 4 ) ( 3 A B - 5)
(3) sina - sin3a + sin3a - sinSa + sin5a - sin7a +sin7a - sin9a => ( A B - 3)' + 7 - 0 + sin9a - sin 1 l a = sina ^ AB = 3- ^ . (8) c:i>-sinlla = 0. (4) Tit (6) va (8) ta c6: A ' = 5 - 3 ^ hay A = ^ 5 - 3 ^ . (9) D o a = — , n c n s i n l l a = sin7t = 0. Ta c6: 11 J 2^ J 4^ 8^ 1J 2^ £ LL 4^ , 3/cos— + C O S — + :Mcos— = —f= Nlcos— + 2cos— + ^ 2 cos— Vay (4) dung dpcm. Bai 9. Chiirng minh rang V 7 V 7 V 7 ^ [ V 7 V 7 V tan'2()" + tan'40" + tan-60" = 33273. Giai
3tana-tan^a 2 tan a b. Ta CO do tan59" 7t 0 = l-3tan^a 1 - t a n ^ a ^ t a n a tan"* a - lOtan^ a + 5 t a n l " t a n 6 l " = tan3"tan3l" ^ (3tana-tan3a)(2tana) 5 tan^ a - lOtan^ a + 1 t a n l " t a n 6 l " t a n 5 9 " = tan3"tan3l"tan59" (1) (l-3tan^a)(l-tan^a) Theo phan 1/ thi V T (1) = lan3" Trong (1) thay = va chu y rang: tann = 0; t a n - j ^ 0, nen tuf (1) suy ra V P (1) = t a n 3 " t a n 3 l " c o t 3 l " = tan3". V a y (1) dung => dpcm. tan^--10tan2- + 5-0. Chu y : Hoan loan tufdng tii" ta co dang thufc sau: 5 5 71 27t 471 571 77t 87t IOTI UTI 137t Z7l HTl .TTl in 071 iWTl 1171 1 J7t /r lan- 27-tan — 27 tan — 27 tan — 27 tan — 27 tan — 27 tan 27 tan 27 tan 27 = v 3 . V a y l a n ^ la n g h i c m ciia phiTdng trinh trung phiTcfng x'* - lOx^ + 5 = 0 (2) That vay ap dung phan 1/ va l a m nhu" tren ta c6: 71 67t 1271 71 471 271 r: TiTdng t y t ^ " — - ' ^^"^ nghiem khac cua (2) V T = tan — tan — tan = tan — tan — tan — = tan — = v 3 . r>. 471 371 471 , 7 71 9 371 9 21 11 9 9 9 3 Do t a n - = - t a n — ; t a n — = -lan — , nen y, = t a n ' ' - ; y 2 = tan^ — la hai Bai 12. nghiem phan biet cua phiTdng Irinh y^ - lOy + 5 = 0 (3) 1. Chtfug minh rang: A p dung dinh l i V i e t v d i (3), ta c6: y i y 2 = 5 (4) tana + l a n { a + 60") + tan(a + 120") = 3tan3a . . , n 2n 3n 4n f . 2 ^ , 2 371^ v d i giii thiet la cac tang c6 mat trong he ihiJc dcu c6 nghla. L a i CO tan — t a n — t a n — t a n — = - t a n — tan — = = y i y 2 (5) 5 5 5 5 V I 5J 5 J 2. Chifng minh rang 71 271 371 471 ^ S = t a n l " + tan5" + lan9" + t a n l 3 " + ... + t a n l 7 3 " + t a n l 7 7 " = 45. Tu^(4) ( 5 ) s u y r a tan —tan — t a n — t a n — = 5 => dpcm. Giai Bai 11. 1. Ta co: V3 + tana tana-V3 uti'} 1. ChiTng minh rang: tanatan(60" + a)tan(60" - a ) = tan3a, v d i gia thiet cac V 1 = tana + j= + -j= tang xet trong he thiJc deu c6 nghia. l-v3tana l + v3lana 2. A p diing phan 1 hay chiJng m i n h : tan a ( l - 3 tan^ a ) + (tan a + V3 )(1 + V3 tan a ) + (1 - V3 tan a)(tan a - ^/3) l-3tan^a a. tan3"tanl7"tan23"tan37"tan43"lan57"tan63"tan77"tan83" = tan27", TV b. t a n l " l a n 6 l " = t a n 3 " l a n 3 l " . 3-tan" a _ = 3tana — = 3 tan 3a => dpcm. Giai l-3tan a 1. T a c o : 2. A p dung phan 1/ Ian liTdt vdi . , , v3-tana v3+tana a = l " + 4"k (k = 0, 1,2, 3,..., 14) tanatan(60 - a)tan(60 + a ) = t a n a - l+V3tana 1-V3tana Ta co: t a n i " + t a n 6 l " + t a n l 2 l " = 3tan3" 3-lan^a tan5" + tan65" + tan 125" = 3tan 15" = tana — = tan3a => dpcm. l-3tan^a tan9" + tan69" + tan 129" - 3tan27" 2. tan53" + Ian 113" + tan 173" = 3 t a n l 5 9 " a. A p dung phan 1/ta c6: tan57" + t a n l 17" + t a n l 7 7 " = 3 t a n l 7 l " . V T = (tan3"tan57"tan63")(lan 17"tan43"tan77")(tan23"tan37"tan83") Cong tifng ve 15 diing thtfc tren va co: = tan9".tan5l".tan69" = tan27" => dpcm. S = 3(tan3" + tan 15" + lan27" + ... + Ian 159" + t a n l 7 l " )
WJy fiyt. 9UUM yMW MJuyMwy S^"*^ ~ M MwmM MMWMJ w w m Cty T/WH MTV D VVtt Kbmng VIft = 3((tan3" + tan63" + I a n 123") + ( I a n 15" + lan75" + tan 135") + 2. Dat ve'traicua diing thiJc can chiirng minh la S. Ta c6: (tan27" + lan87" + tan 147") + (tan39" + tan99" + tan 159") + S = (lan'5" + tan'55" + tan^65") + (tan^lO + lanl50 + tan^70) + (tan5l" + t a n l l l " + tanl7l")J. (tan'15" + tan'45" + tan'75) + (tan'20 + tan^40 + tan^80) + Lai ap dung phan 1/ta c6: '•'''^ (tan'25" + tan'35" + tan^85") + tan'3()" + tan-60. (1) S = 9(tan9" + tan45" + tan8l" + tanl 17" + tan 153") Tuf (1) va ap dung phan 1/ ta c6: = 9(tan9" + tan8l" + tanl 17" + tan 153") + 9 S = (9tan-15" + 6) + (9lan-30" + 6) + (9tan'45" + 6) + (9tan^60" + 6) + = 9(tan9" + tan8l"-tan63"-tan27") + (). (1) Lai c6: (9tan'75" + 6) + i + tan9"-.tan8l"- tan63"- lan27"= „^ i ^ ^ 1 = 9(tan' 15" + tan'45" + tan'75") + 33 + ^ + 9(tan'30" + tan'60") (2) cos9"cos8l" cos63"cos27" _ _ _ l 1 TCr (2) lai ap dung philn 1/ ta Ihu di/dc: sin 9 cos 9 sin 27 cos 27 1 :^f* S = 9(9tan'45 + 6) + 33 + - + 9 1 . 3 = 81 + 87+ - + 3 0 = 1 9 8 - . 2 2 2(sin54"-sinl8") 13 3 3 3 sin 18" sin 54 sin 18" sin 54" Do la dpcm. 4.cos36"sinl8" Bai 14. = 4. (2) sin 54" sin 18" 1. Chtfng minh rhng Thay (2) vao (1) va c6: S = 45 => dpcm. sinasin(60" + a)sin(60"- a) = l s i n 3 a , Bai 13. 4 1. Chu'ng minh: tan'a + lan'(60" - a) + tan^(60 + a) = 9tan\3a + 6. cosacos(6()" + a)cos(60" - a) = -i-cos3a. 2. Chu'ng minh: tanl5" + tan'lO" + tan'15" + ... + tan-80" + tan85" = 198^, 4 2. Chiirng minh rang Giai sin2"sin 18"sin22"sin38"sin42"sin58"sin62"sin78"sin82" = ^ ~^ 1. Ta c6: 1024 2 . . i^-tmaf (^/3 + lana)^ Giai VT = tan'a + (l + N/3tana)^ (l-v/3tana)^ 1. Ta c6: 1(V3 - l a n a ) ( l - 7 3 tana)]^ +|(V3 + lana)(l + >/3 t a n a ) f sinasin(6()" + a)sin(60" - a) = tan'a + (l-3lan-a)^ = sina(sin6()"cosa + sinacos6()")(sin60"cosa - sinacos60") [73(1 + t a n ' a ) - 4 t a n a f +[73(1 + tan" a) + 4tan^ a ] ' 3 = sina(sin'60"cos'a - sin'acos'60") = sina —cos2 1.2 a—sm a = tan'a + — 4 4 (l-3tan^a)^ 2 6(l + tan^a)^+32tan^a 9tan'^a + 45tan-a + 6 = l s i n a ( 3 - 3 s i n ^ a - s i n ^ a ) = l ( 3 s i n a - 4 s i n ' ' a ) = -i-sin3a. = tan a + — = — (l-3tan^a)- (l-3tan'^a)^ TiftJng tif cung co 9 t a n ' a ( 3 - t a n - a ) + 6 ( l - 3 l a n ^ a ) ' _ ^ tana(3-tan^ a 1 +6 cosacos(60" + a)cos(60" - a) = — cos3a dpcm. (l-3tan^a)^ l-3tan^a 4 = 9tan'3a + 6 => Do la dpcm. 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 Khal Cty Tiytlll nrvownKbang Vi^t TufCl) va apdung phan l/c6: V i s i n y ^ 0 , nen tir(2) c6: VT=: isin6" -sin54" -sin 66" = — (sin6"sin66"sin54") (2) 4 14 4 64 „ . 7: - . 71 271 - . 71 4J: . . 7t 67: 2S sin— = 2 s i n — c o s — + 2 s i n — c o s — + 2 s i n — c o s — Tijr (2) va lai ap dung phan 1/ ta c6: 7 7' 7 7 7 7 7 . 371 . 71 . 57: . 37: . .57: . 7t VT= — lsinl8" (3) = sin sin —+ sin sin — + s i n 7 : - s i n — - - s i n — (3) 64 4 7 7 7 7 7 7 ^ Tir(3)c6 S = - ^ . (4) Theo bai, ta c6 sinl8 = (4) rx,. / .S ^ V , • 2 ' ^ . 2 271 . 2 47: 7 V^-1 • dpcm. Thay ( 4 ) vao (2) va co: sin — + sin — -1- sin — = — Thay (4) vao (3) va c6 VT = • dpcm. 1024 3. T a c o : s i n — s i n — = sin — - s i n — B a i 15. ChiJng minh cac dang thufc sau: 7 7 7 7 1 1 ^ . iTi . 2 27t . 2 471 7 27: 1. ' 67: , 87: [ • 27C ' .71 .271 .471 2. sin — + s i n — + sin — = — cos — - cos = 1 -cos 1 - cos — Sin - sin— sin- 7 7 7 I V J 7 7 7 671 871 71 27t 471 1 c o s — = c o s — . (5) . 27X . 471 . 2 47: . 2 ^ 7 7 4. c o s — c o s — c o s — = — 3. sin — sin — = sm sin — 7 7 7 8 7 7 7 7 A p dung cong thu'c . 271 cosa = cos(27t - a ) , vay (5) dung => dpcm , 5. . . 27t . 47t 71 sin —sin — s i n — = 4l— 7: 6. cos—+ , 271 , 471 C 0 S - — + COS— = sin— 7 , 1 „ . 7t 71 27t 471 7 7 7 8 7 7 7 2 ^ . 8sin-COS —cos --cos — Sin - . 7: 27: 47: 7 7 7 4. COS—cos—COS- - (6) 7 7 7 8sin- Giai 7 1. Ta co: A p dung l i e n tiep cong thu'c sin2a = 2sinacosa, ta co: . 471 27t - . 3 7 1 71 .87: . 7 : Sin + sin 2sincos - sin-^— -sin— . 1 1 7 7 7 7 1 .271 .471 . 271 . 471 ^ . 71 7t . 471 .71 VP ( 6 ) = ^ = 1^-1- =5, dpcm. sin— sin— sin—sin- 2sin cos -sin sin-- „ . 7: . 7: 8 7 7 7 7 7 7 7 7 8sin 8sin 7 7 , . 37: . 47: ail*-; c , ^ ,27: 37: , , 7: , ^ , do sin — = sin — ^\ dpcm. J. l a nhan thay y; — ; — thoa man phi/dng trinh: 7 7 2. Ta c6: cos'4x = cos^3x. (7) , 271 , 4?: 87t Dat y = cos'x > 0, thi (7) co dang: . . 2 27r . 7 47: 1-cos - 1-cos 1-cos — (2cos'2x - 1)^ = (4cos''x - 3cosx)' sin — + sin —-I-sin — = 7 7 7 2 [2(2cos^x - 1)' - IJ- = cos^x(4cos^x - 3)^ 27: 47t Tu'(8)suyra: 1_1 67: c o s — -t- c o s — + cos — (1) 87: do c o s — = cos 67: 2 2 7 7 7 7 7 64y^ - 144y' + l()4y^ - 25y + 1 = 0 „ 27: 47: 67: o 64(y - 1) 1 5 . 23 3 r D a t S = c o s — + COS — + c o s — (2) y y +_y = 0. (9) 7 7 7 4^ 8^ 64
B6I duaing hpc sbib gkil Lu^ng gUc - Fhmn Htiy lOuU Cty TNHH MTVDWHKhang VIft 87t 127t 187t r37i 71 > ^71 7t^ r57t TlA I I Do cos^ cos^ cos^ ^ deu khac nhau khdc 1, nen phirong irinh cos- COS + cos — - = cos + cos 1 + cos • 35 35 35 5; ^7 5, 7 7 5J ,3 1 5711 7t . 37t . 57C .71 7t 37t . 71 : 0 hay 64y' - 80y' + 24y - 1 = 0. cos — + 7 COS 7 H COS — 7 cos h Sin — + sin sin — sin — 5 5 7 7 1) Nhan y, = cos y ' y 2 -^'^^ ~'^y^= '-"os — la nghiem o 71 „ . 7t = S| cos—+ S, Sin — (1) 5 Taco: » 71 37: 57: ^ . 37: . 57: . 7: . 2 . 2 271 . 2 37: f 2 271^ ,37r^ vdiSi = cos—+ cos— + C O S — ; ST = s i n — + sin sin — sin —sin — s i n — 1- cos 1- cos" — 7 7 7 ^ 7 7 7 7 7 7 I 1) v 7 )V 7 J do sin-y ;t 0, ncn la c6: , = (i-yi)(i-y2)(i-y3) = 1 - (y, + y. + y.i) + (y.y: + y2y3 + yiyi) - yiy2y3- (lO) ^ . 7:_ T • ^ 7 : 7 : 37: 7: 57: 2sin — S | = 2sin —cos— + 2sin —cos— + 2sin —cos— 80 7 7 7 7 7 7 7 yi + y 2 + y 3 = 64 . 2 7 : .47: . 2 7 : . 67: . 4T: . 67: . 7: = sin — + sin sin — + sin s i n — = sin — = sin — 24 7 7 7 7 7 7 7 Ap dung djnh li Vict, ta c6: yiy2 + y2y3 + y3yi = (11) 64 (2) yiy2y3 = 64 ^ , . , ^ . 47: . 27: . 7: Ta lai co: S-, = sin — + sin sin—>() 7 7 7 % Thay (11) v^odO) va co: sin^-sin^ — sin^ — = — 7 7 7 64 ^2 . -> 4n . 2 27: . 2 ^ • 47: . 2?: ^ . 47: . 7: . . 2 • t =>S7=sin' — + sin — + sin —+ 2sin—.sin 2sin—sm 2sin —sm — . 71 . 271 . 371 4l 7 7 7 7 7 7 7 7 7 Sin—sin—sin— - — dpcm. , 87: 47: , 27: 7 7 7 8 I-COS +1-COS +1-COS -) c o T 7 7 7 2T: 67: 57: 37: 37: 7: 27t ^ . sin — , = — ^ + COS 7 COS 7 + cos COS 7 + cos 7 COS— ^ ^ ^ 71 2TC 471 7 1 6. Ta c6:cos—+ C0S— + cos— 7 7 7 2 = 1 i ' cos — T: + COS 37: + COS 57: — 3 = - + -S, = - 1 2^2 2 2 4 7 7 7 ; 71 27t 47t - 71 1 o COS — + cos — + COS — = 2 COS Vi S2 > 0, ncn CO ST = (3) 7 7 7 7 2 47t 671 Thay (2) ( 3 ) vao ( 1 ) vii c6 dpcm. 7t 271 47t 1 271 O - C O S — + COS + COS = O COS + COS + COS — 2 Bai 17. Cho n la so nguycn du'dng. Chiang minh 7 7 7 2 7 7 7 TT 37t 57t 1 C O S — + C O S + C O S — = — (12) 7 2 7 7 2 + \/2TV2^+^^^^^? ~ 2cos — ^ . Thco bai 4, thi (12) dung => dpcm. n clii u ca n Bai 16. Chdrng minh Giai 871 1271 187t 1 n 4l . Ti Ta chrfng minh bhng nguycn ly quy nap loan hoc: cos — + COS + cos = — C O S — + — sm —. 35 35 35 2 5 2 5 Vdi n = 1, la c6 -Jl = 2 . - ^ = 2cos— = 2cos—^ . Giai 2 4 2'^'
Cty Timn IVTVDVVH Khang VIft B6I daang hpc sinh gidl Lupng gidc - rhan Huy Khal + Gia su" dang thtfc can chrfng minh da dung den n = k, tiJc la ta c6: Giai sin a cos a sin^a-cos^a ^ cos 2 a ^2 + V2 + ... + ^ 2 ^2cos^• (1) Tacd: t a n a - c o t a = = = -2 = -2cot2a cosa sina sinacosa sin2a k da'u can V a y tana = cota - 2cot2a (1) + X e t k h i n = k + 1, ta c6: A p dung (1) lien tie'p ta cd cac dang Ihu'c sau: V2 + V2+T+W= 2 + 7 2 + V2 + ... + N/2 .(2) 1 a 1 a , — t a n — = —cot cota , j ; k +1 cla'u can k da'u ca n • 2 2 2 2 j - i;. T h a y ( l ) v a o ( 2 ) va c6: 1 a 1 a 1 a — tan—r- = —r-cot — — c o t — 22 2^ 2' 2' 2 2 VT (2)= 12+ 2 cos = . 2 1 + cos - ,k+l ,k + l 1 a 1 a 1 a — tan — = c o t — - — c o t - r - 23 2^ 2^ 2^ 2^ 2^ (3) 1 a 1 a 1 a V a y tiT (3) suy ra dang thiJc can chiJng minh ciing dung v d i n = k + 1. 2 n - i -tan 2" 'r = 2"~'rCOt 2 " '; 2"^^r r C O t -2""^ ' I , !^• T h c o nguyen ly quy nap toan hoc suy ra dieu phai chiJng minh. l a 1 a 1 a B a i 18. Cho n la so' nguyen du'dng. Chufng minh x\ng 2 n - tan 2 — " =— 2" col 2" 2""' r t^ot 2""' - Uji'': ^/2.^/2 + ^/2.^/2 + ^/2 + ^/2...^/2 + ^/2 + ... + ^ = i . . V- 1 ot 1 a Cong ttfng vc n dang Ihu'c tren, ta CO: > — t a n — = — c o t cota n da'u can sm .n+l 2^ 2" 2" D d la dpcm. Giai B a i 20. ChuTng minh rang v d i m o i so' nguyen du'dng n ta cd: A p dung bai 17, ta c6 V T ( v e ' t r a i dang thiJc can chi^ng minh) . ^a - . ^ a T2 • ct „ n " i • 1 ot 1 • 3" . a y . VT = 2 cos- 2 cos sm — - f 3sm — + 3 sm" — + ... + 3 sm' — = — s m a + — s m — ' 2 cos 2 cos - 3 3^ 3-^ 3" 4 4 3" .n + l 2' 2-^ 1' Giai 2 COS , cos , cos V ...COS - - sm ,n+l A p dung cdng thiJc: sin3x = 3sinx - 4 s i n \ ta cd: 2 2 2 2"'*' (1) •n sin''x = — s i n x - — s i n 3 x . (1) sm •,n+] 4 4 A p dung l i e n t i c p (1), ta cd cac dang ihu'c sau: A p diing lien tie'p cong thiJc sin2a = 2sinacosa, tiif (1) ta cd: . ^a 3 . a 1 . 71 sm sm — = —sm sma ^ 1 3 4 3 4 VT = dpcm. , . , a 3^ . a 3 . a . . u d ^t^s^n./i r sm in + l Sin 3sm—- = —sm — — s m — .n + l 32 4 B a i 19. Chtfug minh rang v d i m o i n nguyen du'dng ta cd ,2 . ^ a 3-V a 3^ . a 3 s i n — = —sin — -sin— •ni,-|. 1 a 1 a 1 a 1 a 1 a 3.3 4 3.3 4 32 W'^ — tan — + — t a n — + —rtan -tan — = — c o t — - c o t a , 2 2 2^ 2^ 2^ ""' 2^ "' 2" ' 2n 2" 2" ( v d i gia thie't la cac bieu thi^c tang va cotang cd mat trong b i e u thiJc deu cd ,n-i . ^ a 3" . a 3"^' . a 3 sm' — = — s m sm nghla) •^n 4 3n 4 3n I • -I ' C
Bdi duang hpc sinh gloi Lupng gUc - Fhan Huy Khal 1 1 1 Cong tufng v c ' n - 1 dang thuTc Iren va co: 4- cos" -, 4 sm^ - 4- sm^ , ^ . k - i • 1a > 3 La^ sin-i- = 3k —41s i •n a + — 3" . a 4 sin — 3" 2-' 2^ 2-' Do la dpcm. I 1 An 2 a . n - l ,-2 « 4" sin^ " Bai 21. Chtfng m i n h rang v d i m o i so nguycn di/dng n ta c6: 4 cos 2 n 4 sin -in-l 2" 1 a 1 a 1 2 S = —tan — + ...+ -tan— Cong tfifng ve n dang thufc trcn, ta c 6 : —r-tan-1'} sin^a 4 n , i „ 2 a 3 2 2 2" (5) S.--2 Giai Sin a 4n,i„2 a 2" 1 a ^2 Vc'Ji m o i k = 1, 2,...,n, la c6: -Ian (1) f 1V 2 a Thay (5) vao (2) (3) va c6: S = 1- cos sin^a 4nsin2« ^ .4. 2" (ap dung cong thi?c tan'x = — \. Do la dpcm. cos" X Bai 22. ChuTng minh r i n g v d i m o i n nguyC-n dU'dng, ta c6: A p dung l i e n tiep (1), ta c6: " 2^"'cos2^ 2 2""' 1 I 1 1 • + ... + • ^l-cos2'^"' 1-COS2 1-COS2"*' 4"cos^ « 4 4^ 4" 4 cos 4 cos V 2 2' 2" Giai V d i m o i k = 1, 2,,...,n, la c 6 : 2^"cos2' 2'^'(l +c o s 2 ' ) - 2 ' ^ ' 2''^'(1 + cos2'^) - 2 ^ " ' U f 1V = s,-. = s, 1- 4 (2) 1-COS2 k+l 1 - C 0 S 2 k+l l-(2cos^2'' -1) 1- ,k + l 2^^'(l + c o s 2 S 2'^^'(l + cos2'^) 1 1 1 2-2cos-2k l - c o s 2 ^ ^ ' ~ 2(l + c o s 2 ' ' ) ( I - c o s 2 ^ ) l-cos2''^' d day S, =• --t-...-f- (3) -> a 2 ct ,kii 4 cos" 4" cos V 4"cos^ " 2" (1) 2 2^ l-cos2' - c o s 2 k+l Dc d a n g Chiang m i n h diftfc c o n g thufc sau: Tu" (1) suy ra 1 ^ 1 1_ (4) n ,k + l ^ 4cos^x sin"2x 4sin"x k=l l-cos2'^ l-cos2'^^' Trong (4) Ian Itfdt thay x b i n s ta c6 cac dang thii'c sau: •,n+\ 2 2^ 2"^ 2" 2' 2' 2' + ...+ 1 1 1-COS2 i-cos2' l-cos2^ l-cos2"\ 1-COS2" 1-C0S2n+l 4cos2" sin^a 4,i„2 a •,11+1 1-COS2 l-cos2"^' •I 1 1 1 Do la dpcm. 4\os2" 4sin2" 42sin2 4 2' 2 2^
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 : tanP+ tan 0, ne'u X = 2kn Un+l = 12- = tan (3) 12 X 2n + i 1 - t a n p tan 12 Sn = s i n x + s i n 2 x +...+ s i n n x - < cos cos X , d day k e Z. 2 2 , neu X ^ 2kn (n 7 1 ^ . X K e t h d p ( 1 ) , (3) suy r a Uj = tan —+— U3 = tan ssin 2 U 12 6 12 Giai TH d o b a n g q u y n a p suy r a : u ^ = tan - + ( n - l ) — (4) + R 6 r a n g ne'u x = 2kn => sinx = s i n 2 x = s i n n x = 0 , d o d o h i e n n h i e n S„ = 0. 6 '12 X X B a y gicf tir ( 4 ) , ta d i de'n: + NeuX jt 2kn => — ^ l s i n — ;^ 0. "2012 = tan ^ + 2 0 1 1 ^ = tan - + 16771 + — X X X X 6 12 {6 12 D o d o t a c o : 2sin — S„ = 2sin —sinx + 2sin — s i n 2 x + ... + 2 s i n —sinnx . (1) f9n' 2 " 2 2 2 ^371^ n = tan = tan - - tan — A p dung cong thtfc: 2sinasinb = cos(a - b) - cos(a + b), ta co tiT (1) ll2y V 4 J 4 ^ . x^ X 3x 3x 5x 2n-l 2n + 1 B a i 2 5 . (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) 2sin —S„ = cos c o s — + cos c o s — + ... + cos x - cos x 2 2 2 2 2 2 2 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 t r u y h o i nhuTsau: X 2n + l cos -cos X U() = 2; v„ = 1 ^ • X ^ X 2n + 1 2 2 dpcm. 2u„v„ 2 s m —S„ ==cos cos x => Sn = 2 sin ^ 2 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 ) Tim Un v a Vp. Giai u, =- 3 Ta co: Uo = 2 - — = ; v,, = 1, C h o d a y so' x a c d i n h nhu" sau: 1 7t cos , n = l,2,... 2 3 l + (V3-2)u„ 2U(|V() ^ 2 T i m U2()i2. u, = U ( | + V „ 1 . 1 2 n cos +1 cos Giai Uo v„ 3 6 dr.: 1 - cos ' 2-^/3 ^1 = V u i ^ = - Ta c6: tan- = 2-V3. cos 12 V3 1+ - V 2 + V3 2u,V| 2 1 U,+V, 1 1 271 t n 2 Ui = t a n — (1) ' ' - + — cos - + C 0 S - cos — c o s — U, V, 6 6 6 12 1 =7^ V, 71 Tit d o t h e o e a c h x a c d i n h d a y ta c o : u,, + t a n — 2^1 =- _^ 12_ 71 7t (2) cos COS — 1 - u _ tan — 6 12 ,iiH " 12 N e u d a t u„ = t a n p , t h i ttr (2) c o :
2U2V2 2 1 V . ^ ^. cosa + cosP+cosy sina + smB+sinv "3 = u, + v. 1 1 n n B a i 1. Cho — — — — = = m. 1+ COS COS COS cos —- cos(a + p + y) sin(a + p + Y) + cos COS 12 6 12 24 Uj 6 12 ChiJng minh rang cos(a + P) + cos(p + y) + cos(y + a) = m. V, ==• 17t Giai 7t 7t COS COS COS Dat a + p + y = S ; u = P + y ; v = y + a ; w = a + p. i- 6 12 24 K h i do de thay a = S-u;P = S-v;y = S-w. % > 1 (1) Ta c6: c o s a cos(S-u) cosScosu + sinSsinu n n 71 n 2 cos - cos , cos , ...cos ~ - cos cosS cosS cosS 2.3 2^3 2\ 2"-'.3 2".3 = cosu + sinu.tanS, Bkng qui nap la c6: (1) 1 (2f sina sin(.S-u) sinScosu-sinucosS v., - • cos cos r ...cos sinS sinS .sinS 2.3 2^3 2".3 = cosu - sinu.cotS. (2) (ban doc x i n tiC nghicm lai) Ti/dng tur(2), ta c6: Tu'Ong lu" nhu'cach giai trong bai 17, la c6 he thu"c sau: cosP sinp = cosv + sinv.tanS; = cosv - sinv.cotS (2) X X n sinx cos S sinS cos—cos — . . . cos — = . 2 2^ 2" 2" s i n ' ' cosy siny = cosw + sinw.tanS; = cosw - sinw.colS (3) 2" cosS sinS . n TO (1) (2) (3) suy ra sin /:j Ti/ do suy ra: cos cos——...cos = = = •' cosa + cos P + cosy = (cosu + cosv + cosw) + (sinu + sinv + sinw)tanS (4) 2.3 2^.3 2".3 2" sin 2^'s.n cosS 2".3 2".3 sina + sinP + siny = (cosu + cosv + cosw) - (sinu + sinv + sinw)cotS (5) sinS 2 3 I . 2 V3 , n Thay lai vao (1), va c6: u^ = Trtr tiTng ve (4) (5) va tif giii thiel ta c6: hay u„ = tan V3cos-- 3 2".3 (sinu + sinv + sinw)(tanS + cotS) = 0. (6) 2". 3 V i |tanS + cot S| = |tan S| + jcol S| > 2 , 2"^'-\/3 TT Con lhay viio (2), i h i v,, = sin nen noi rieng tanS + cotS ^ 0, vay tu" (6) suy ra , 2".3 sinu + sinv + sinw = 0. (7) ; TO(7), (4)vagiathietsuyra § 3. D A N G THirC LUONG GIAC C6 DIEU KIEN cosu + cosv + cosw = m o cos(P + y) + cos(y + a) + cos(a + p) = m. Ccic bai loan trong muc nay c6 dang sau day: Gia suT cac dai lifdng trong he Do la dpcm. IhiJc can chiyng minh thoa man mot dieu k i e n nao do. Ta phai chi^ng minh he B a i 2. Cho cosa + cosP + co.sy = 0 thuTc da cho la dung. Dieu kien cho triTdc c6 the cho difcKi dang hinh hoc, dai Chiirng minh rang cosacospcosy = (cos3a + cos3p + cos3y). so,,.. PhiTctng phap giai cac bai loan nay la sir van dung kheo leo ke't hdp Giai giCTa ciic phep bien lUOng giac cO ban va i r i e l de siir dung cac dieu k i e n da Ta CO: cho trong de bai. cos3a + cos3p + cos3y = 4cos'a + 4cos'p + 4cos''y - 3(cosa + cosp + cosy) = 4(cos-^a + cos^'p + cosV) • - - (1)
(do cosa + cosp + cosy = 0) V i X | - X 2 ^ kTT => D = s i n ( X | - X 2 ) 0 A p dung hang dang thiJc, la c6: V a y he ( 1 ) (2) co nghiem duy nhat: cos^a + cos'p + cosV = (cosa + cosP + cosy)^ - 3cosacosp(cosa + cosP) D,, 0 cosx a = (do D ^ 0) - 3cosPcosy(cosP + cosy) - 3cosycosa(cosy + cosa) - 6cosacosPcosy. (2) D 0 C0SX2 D V I cosa + cosP + cosy - 0, nen ?| -t . » l i l t smx cosa + cosP - -cosy; cosp + cosy = - cosa; cosycosa = - c o s p . (3) D. smx^ Thay (3) vao (2) di den cos'a + cos^P + cos y = 3cosacosPcosy. (4) TiTdng tir b = = = 0. Lai thay (4) vao (1), va co: cosacospcosy = ^ ( c o s 3 a + cos3p + cos3y). 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, " Do la dpcm. vdi gia Ihiet tan(a + P) va tanp co nghia. acoscv + bcos3 = 0 B a i 3. Cho Giai a cos((\ 42) + b cos(3 + a ) = 0, v d i 43 ;^ k i r , k G Z. V i e l lai gia thiel du-citi dang sau: sinf(a + P) + p] = 2 s i n [ ( a + P) - p] (1) ChiJng minh rang l"(x) = acos(x + a) + bcos(P + a) = 0, Vx. T i r ( l ) , la co: Giai sin(a + P)cosp + sinPcos(a + P) = 2sin(a + P)cosP - 2sinPcos(a + p) Tir gia Ihiet: acos(a + cp) + bcos(p + cp) = 0, ta co: => 3sinPcos(a + p) = sin(a + P)cosp. acosacoscp - asinasincp + bcosPcoscp - bsinPsincp = 0 Do lan(a + P) va tanP co nghia nen cos(a + P) ;^ 0; cosp ^ 0. => coscp(acosa + bcosP) - sin(p(asina + bsinP) = 0. (1) Chia ca hai ve cho cos(a + p)cosP va co: tan(a + P) 3tanP => dpcm. Tif gia I h i e l acosa + bcosP = 0, va sincp ^ 0 (do 9 k n , k e Z), nen tijf (1) suy ra: asina + bsinP = 0. (2) VcHi m o i x, la co: B a i 6. Cho cos(2a + P) = 1. ChlTng minh he thiTc tan(a + p) - tana = 2 t a n - 2 (•(x) = acosxcosa - asinxsina + bcosxcosp - bsinxsinP vdi gia thiet cac tang co mat trong he thiJc deu co nghia. ;, = (acosa + bcosP)cosx - (asina + bsinP)sinx. (3) Giai Do acosa + bcosP = 0 (gt) va ihco (1) i h i asina + bsinP = 0, tijf do tir (3) suy Tir cos(2a + p) = I 2 a + p = 2k7r, k G Z ra f(x) = 0, Vx e R. =:^(a + P) + a = 2k7t, k e Z (1) Do la dpcm. Tir ( I ) suy ra l a n ( a + P) = - t a n a . , B a i 4. Cho f(x) = asinx + bcosx. => tan(a + P) - tana = - 2 t a n a (2) Biet rang r(x,) = r(x2) = 0; v d i X | - X2 ;t kTt Lai tir 2 a + p = 2k7i ChiJng minh rang r(x) = 0, Vx e R. Giai => a + - = k n , k e Z. (3) 2 Ta co: r ( X | ) = asinx 1 + bcosx 1 = 0 r ( X 2 ) = asinxn + bcosx: = 0. . T i r (3) suy ra tana = - tan —. (4) 2 X e t he phuUng trinh an a, b , ff^A: iT ' asinx, + b c o s x , = 0 (1) Thay (4) vao (2) va co tan(a + P) - tana =: 2 t a n ^ => d p c m ! asinxj + bcosxj = 0 (2) tanp 1 + Ta co: D = sinx, cosx = s m X | COSX2 — s i n x 2 cosx, B a i 7, Cho ^'"^ = i . ChUng minh = Lz^^R^^. = ^ ^'' - sin(2a + P) m m+ n m - n .jH smx^ cosx- = sin(X| - X j ) .
B61 duSng bpc ainb gfol Lupng g U c - Fhan Huy Khal Cty TimnmvDVVHKbaag Vlft Giai Giai DiTa d i n g thtfc can chiJng minh ve dang tifcJng diTdng sau: Dat t = tana. :>( J sin a sin (3 , . . ^, , tan a + tan P , / t a n a + tan 3 cos a cos (3 sin(a + 9) cos(ft + p) Tur gia thiet ta co: = 3tana (m + n ) t a n a m - n (m + n)tanacosacos3 (m-n)costtcosP 1 -tanottanP %• , ..... ... . , , * i=> tanp + t = 3t( 1 - t.tanP) => tanp( 1 + 3t^) = 3t 3t >'f"l)'t =>tan3 = - — y . (1) (m - n)taji(a + P) = (m + n)tana. (1) l+3t^ .)-•••; Tijrgia thiet ta c6: Taco: 2sin2p = 2 ^'^"^ . (2) msinp = nsin(2a + P) => msin[(a + p) - a ] = nsin[(a + P) + a ] l + tan^0 =:> msin(a + P)cosa - msinacos(a + P) = nsin(a + P)cosa + nsinacos(a + p) Thay (2) vao (1) r o i rut gon ta d i den: => (m - n)sin(a + p)cosa = (m + n)sinacos(a + P). (2) Chia ca hai vc cua (2) cho cosacos(a + P) ta c6: 2sm23= f ' + f • (3) ^ ' QtVlOt^+l (m - n)tan(a + p) = (m + n)tana. ... , . . ^ 2tan(a + 3) 2tana • V a y (1) dung => dpcm. Mat khac: sm[2(a + P)] + .sm2a = — + r—. (4) 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 l + tan^(a + 3) 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 = . (5) Bai 8. Cho t a n - = 4 t a n — . Chu'ng minh: tan^^—3sin(\ 9tVl0t^ +1 5 — 3cosft Tur (3) (5) suy ra dpcm. Giai n^'ttx /^u s i n ( x - a ) a cos(x-a) a, , , _ Bai 10. C h o — -= -; - = — , vc)i ab, + a|b ;t 0. Dat t = t a n — , tir gia thiet ta co tan - 4t. sin(x —p) b cos(x —3) b, 2 2 (5 (V Chu'ng minh r^ng: cos(a - P) = . ,3 tan - tan . ab| + a , b Khido t a n ^ = — 2 _ ^ ^ ^ ^ _ 3 t _ . (,) Giai ' 2 l + tan•^an" l+ ^r l + 4t- aa| ^ ^ sin(x-a) cos(x-a) 2 2 , aa, + b b , _ bb, _ s i n ( x - 3 ) co.s(x-3) 2 tan 1 d CO. — — —:— — A p dung cong thufc: sina = ^— = r-, ta c6: ab|+a|b ^_^'^±_ sm(x-a) ^ cos(x-a) 1 + tan^" l+ t2# b b, sin(x-3) cos(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) 3 sin a (2) sin(2x - 2a) + sin(2x - 23) 5 —3cosa 2 s i n [ 2 a - ( a + 3)] 2 sin[2x - (a + 3)] cos(a - 9) = cos(a - P) => dpcm. T i r ( l ) (2) suy ra dpcm. 2 s i n [ 2 x - ( a + 3)] Bai 9. Cho tan(a + P+ = 3tana. Bai 11. Cho cos(cp - a ) = a; sin((p - P) = b. Chtirng minh: sin(2a + 2P) + sin2a = 2sin2p. Chu'ng minh r i n g : a^ - 2absin(a - P) + b^ = cos^(a - P).