Bien Doi Luong Giac Hay

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L Trinh TngTHPT Tr ng Vng QNRN LUYN K NNG BIN ILNG GICA- CCVN V L THUYT .I- TM TC CNG THC LNG GICH THNG CC CNG THC LNG GIC:I- GC V CUNG LNG GIC:1. Cng thc quy i Raian: 180a ( a tnh bng , tnh bng rad)2. S o gc v cung lng gic theo v radian.s(ox, ot) = a0 + k3600 hoc s(ox, ot) = + k2, k Z. (vi00 a < 3600 , 00 < 2)Ti liu rn k nng bin i lng gic dng cho HS kh gii 10NC 106432 23 34 32 2sin 0122232132220 1 0cos 13222120 12221 0 1tan 0331 3P 3 1 0P0cotP3 1330331P0PL Trinh TngTHPT Tr ng Vng QNsAB= a0 + k3600 hoc sAB= + k2, k Z. ( vi 00 a < 3600 , 00 < 2)3. Cng thc tnh di cung: l = .R ( tnh bng rad)II.NHM CNG THC LNG GIC 1:1. Hng ng thc lng gic: sin2x + cos2x = 1

2 22 2sin x 1 cos xcos x 1 sin x 2211

t

t x xx xsin coscos sin 1+tan2x =21cos x cos2x = +211tan x cosx = t+211tan x 1+cot2x =21sin x sin2x = +211 cot x sinx = t+211 cot x tanx.cotx = 1 tanx = sinx 1cosx cotx cotx = cosx 1sinx tanxCh : Trong cc cng thc c cha du (t ) , vic chn du (+) hoc du () cn nhn xt gi tr ca cung x trn ng trn lng gic.2. Cung lin kt: x x2 x+ x2 + xsin sinx sinx cosx sinx cosxcos cosx cosx sinx cosx sinxtan tanx tanx cotx tanx cotxcot cotx cotx tanx cotx tanx3. Ch : a + b = 1800cosb = cosa sinb = sinaa + b = 2900cosb = sina sinb = cosaABCsin(B + C) = sinAcos(B + C) = cosAtan(B + C) = tanA+B C Asin cos2 2+B C Acos sin2 2+B C Atan cot2 2sin(x + k2) = sinxcos(x + k2) = cosxtan(x + k) = tanxcot(x + k) = cotxIII. NHM CNG THC LNG GIC 2:1.Cng thc cng:cos(a tb) = cosa.cosb m sina.sinb sin(a tb) = sina.cosb tsinb.cosatan(a tb) = tmtana tanb1tana.tanb2.Cng thc nhn:cos2a = cos2a sin2a = 2cos2a 1 = 1 2sin2a = +221tan a1tan asin2a = 2sina.cosa = +22tana1tan a;tan2a = 22tana1tan aTi liu rn k nng bin i lng gic dng cho HS kh gii 10NC 2L Trinh TngTHPT Tr ng Vng QN3.Cng thc h bc:21 cos2asin a2;+21 cos2acos a2;+21 cos2atan a1 cos2a4.Cng thc tnh theo t :at tan2+22tsina1t+221tcosa1t22ttana1t5. Cng thc bin i tch thnh tng:2cosa.cosb = cos(a + b) + cos(a b) 2sina.sinb = [ cos(a + b) cos(a b) ]2sina.cosb = sin(a + b) + sin(a b)6. Cng thc bin i tng thnh tch:+ + a b a bcosa cosb 2cos cos2 2+ a b a bcosa cosb 2sin sin2 2tana + tanb = aba bsin( )cos .cos++ + a b a bsina sinb 2sin cos2 2+ a b a bsina sinb 2cos sin2 2 tana tanb = aba bsin( )cos .cosH qu:cosx + sinx =2sin( x) 2cos( x)4 4 + cosx sinx = 2sin( x) 2cos( x)4 4 +III. H THC LNG TRONG ABC: 1. nh l hm s sin v cos:a b c2RsinA sinB sinC 2 2 2a b c 2bc.cosA + 2 2 2b a c 2ac.cosB + 2 2 2c a b 2ab.cosC + 2. Chuyn cnh sang gc: a = 2RsinA b = 2RsinBc = 2RsinC3. Chuyn gc sang cnh:asinA2R

2 2 2b c acosA2bc+ 4. Cng thc din tch: a b c1 1 1 1 1 1S a.h b.h c.h bcsinA acsinB absinC2 2 2 2 2 2

abcS pr p(p a)(pb)(p c)4R , vi + + a b cp2 R: Bn knh ng trn ngoi tip, r: Bn knh ng trn ni tip ABC5. Cng thc ng trung tuyn v phn gic trong cc gc ca ABC:+ 2 2 22ab c am2 4+ 2 2 22ba c bm2 4+ 2 2 22ca b cm2 4(ma, mb, mc di trung tuyn)+a2bc Al cosb c 2+b2ac Bl cosa c 2+c2ab Cl cosa b 2 (la, lb, lc di phn gic)B. BI TP .VN 1. CC BI TP C BN V BIN I LNG GIC.1. Tnh gi trlng gic ca cung sau.1) sina = 35 vi0 < a < 22) tana = - 2vi< a < 3) cosa = 51 vi -2 < a < 0 4) sina = 31 vi a (2, )5) tana = 2 vi a (, 23)Ti liu rn k nng bin i lng gic dng cho HS kh gii 10NC 3L Trinh TngTHPT Tr ng Vng QN2. Chng minh cc ng thc sau:1) sin2x + tan2x = 21cos x - cos2x 2) tan2x - sin2x = tan2xsin2x 3) 22tan3 3 tantan 1 3tanx xx x4) 2 22 2cos sincot tanx xx x = sin2xcos2x 5) 2221(1 cot )( 1)cos1 tanxxx+ + = 16) cosx + cos(2/3 - x) + cos(2/3 - x) = 0 7) sin(a + b)sin(a - b) = sin2a -sin2b = cos2b - cos2a8) 2 22 2tan tan1 tan tana ba b = tan(a +b)tan(a - b) 9) cos3xsinx - sin3xcosx = 14sin4x10) cos sincos sinx xx x+= 1cos 2x- tan2x 11) sin 2 2sinsin 2 2sinx xx x+=-tan22x12) sin3xcos3x + sin3xcos3x = 34sin4x 13) sinx - sin2x +sin3x = 4cos32xcosxsin2x 14) sinx +2sin3x + sin5x = 4sin3xcos2x 15) 4 4 222sin cos coscos2(1 cos ) 2x x x xx + 3. Rt gn cc biu thc sau:1) A = sin(x + 52) - 3cos(x - 72) + 2sin(x + ) 2)B=( )11sin cos 5sin2 2x x x _ _ + + , ,3)( ) ( ) ( )os os 2 sin os2C c c c _ + + + + + , 4) D= 2cosa-3cos(+a)-5sin(/2-a)+cot(32- a) 5) cos( - a) - 2sin(3/2 + a) + tan(32- a ) + cot(2 - a)4. Chng minh rng cc biu thc sau khng ph thuc vo a.1) A = cos4a + cos2asin2a +sin2a 2) B = cos4a - sin4a + 2sin2a3) C = 2(sin6a + cos6a) - 3(sin4a + cos4a) 4) D = 1 cot1 cotaa+ - 2tan 1 a 5) E = 2sin 4 4cos a a + +4 2cos 4sin a a +6) F = cos2a + sin(300 + a)sin(300- a)7) G = sin6a + cos6a + 3sin2acos2a 8) H = 4 46 6sin cos 1sin cos 1a aa a+ + 9) m l mt s cho trc, chng minh rng nu: m.sin(a + b) = cos(a - b)Trong a - b k v m t 1 th biu thc: A = 11 sin 2 m a + 11 sin 2 m b (m l hng s khng ph thuc vo a, b ).5. Tnh cc biu thc i s.1) Tnh sin3a -cos3a bit sina -cosa = m2) Bit sina + cosa = m hy tnh theo m gi tr ca biu thc: A = 1 cos 2cot tan2 2aa a+ Ti liu rn k nng bin i lng gic dng cho HS kh gii 10NC 4L Trinh TngTHPT Tr ng Vng QN3) Bitcos( )cos( )a ba b+ = pq. Tnh tana.tanb4) Bit sina + sinb = 2sin(a + b) vi (a + b) k2 tnh tan2a.tan2b 5) Tnh sin2x nu: 5tan2x - 12tanx - 5 = 0(4 < x < 2)6. Khng dng my tnh hy tnh gi tr cc biu thc :1) A = cos200cos400cos600cos8002) B = cos7.cos47.cos573) C = sin60.sin420.sin660.sin7804) Tnh: E = sin50.sin150sin250.sin350. ...... sin850 5) Tnh: F = sin18.sin318.sin518.sin718. sin918 6) A = sin370.cos530 + sin1270.cos39707) A = tan1100 + cot2008) Tnh sin150 v cos1508) A = tan20o.tan40o.tan60o.tan80ob) B = 12sin10o- 2sin70o,M = cos5 - cos25 c) C = sin416 + sin4316 + sin4516 + sin4716d) D = tan212 + tan2312 + tan2512e) E = tan9o - tan27o - tan63o + tan81o. f) F = cos616 + cos6316+ cos6516+ cos6716g) G1 = sin18o.cos18o; G2 = sin36o.cos36oh) H = cos27 + cos47 + cos67 i) I = sin5 + sin235 + sin6 + cos135k) K = cos5 + cos25 + cos35 + cos459. Vi a k(k Z) chng minh:a) cosa.cos2a.cos4a...cos16a = sin3232.sinaab) cosa.cos2a.cos4a....cos2na = 11sin 22 sinnnaa++10. Tnh: A = cos20o.cos40o.cos60o. 11. Tnh: A = sin6o.sin42o.sin66o.sin78o.12. Tnh: A = cos7. cos47. cos57.13. Tnh: cos65. cos265. cos465. cos865. cos1665. cos3265.14.Tnh: sin18.sin318.sin518.sin718. sin918. 15. Tnh: cos15.cos215.cos315.cos415....cos715.16. Tnh: sin5o. sin15o .sin25o... sin85o.17. Tnh: 96 3 .sin48.cos48. cos24. cos12. cos6.18. Tnh: 16.sin10o.sin30o.sin50o.sin70o. 19. Tnh: sin10o.sin20o.sin30o....sin80o.20. Tnh: cos9o. cos27o. cos45o. cos63o. cos81o. cos99o. cos117o. cos135o. cos153o. cos171o.Ti liu rn k nng bin i lng gic dng cho HS kh gii 10NC 5L Trinh TngTHPT Tr ng Vng QN21. Tnh: A = cos5 + cos25 B = cos5 + cos357. Ch cc cng thc sau:1) 4sinx.sin(3 - x)sin(3 + x) = sin3x 2) 4cosx.cos(3 - x)cos(3 + x) = cos3x3) tanx.tan(3 - x)tan(3 + x) = tan3x 4) cosa.cos2a.cos4a .......... cos2na = 11sin 2 .2 sinnnaa++ 5) tnhS = cosa - cos(a + x) + cos(a +2x) +......+(-1)n. cos(a +nx).th nhn 2 v vi2cos2x nu cos2x 0.8.Cc bi tp khc:1. Chng minh rng:a) cos15 sin15cos15 sin15o oo o+=3 b) sin75 cos75cos75 sin75o oo o+ = 132. Rt gn cc biu thc sau:a) A = sin3x.sin3x + cos3x.cos3x b) B = 1 cossinxx+[1 + 22(1 cos )sinxx]c) C = cos3x.cos3x - sin3x.sin3x3. Chng minh rng :a) 4.cosx.cos(3 - x).cos(3 + x) = cos3x. b) 4.sinx.sin(3 - x).sin(3 + x) = sin3x.c) tanx.tan(3 - x).tan(3 + x) = tan3x. p dng tnh: A = sin20o.sin40o.sin80o. B = cos10o.cos20o.cos30o....cos80o.C = tan20o.tan40o.tan60o.tan80o.4. Chng minh rng :a) sin6x + cos6x = 58+ 38cos2x b) tanx = 1 cos 2sin 2xxp dng tnh: A = sin6(24) + cos6(24) B = tan2(12) + tan2(3.12) + tan2(5.12)5. Chng minh rng: a) sin4x = 3 1 1cos 2 cos 48 2 8x x + b) sin8x + cos8x = 35 7 1cos 4 cos64 16 16x x + +p dng tnh A = sin8(24) + cos8(24) B = sin4(16) + sin4(3.16) + sin4(5.16) + sin4(7.16)6. Tnh: cos(27) + cos(47) + cos(67)22. Tnh cos(5) + cos(25) + cos(35) + cos(45) 7. Cho: sin2a + sin2b = 2sin2(a + b). Tnh: tana.tanb. 24. CMR: 0 00 0sin75 cos75sin75 cos75+ =13 Ti liu rn k nng bin i lng gic dng cho HS kh gii 10NC 6L Trinh TngTHPT Tr ng Vng QNVN 2. BI TON LNG GIC TRONG TAM GIC.I. CC KIN THC C BN. + A + B + C = + a b < c < a + b+ a2 = b2 + c2 - 2a.b.cosC +2sin sin sina b cRA B C + S = 1 1. .sin ( ) .2 2 4a aabca h ab C pr p a rR S =( )( )( ) pp a p b p c Trong : p = 2a b c + + r: bn knh ng trn ni tipra: bn knh ng trn bng tip trong gc A.+ ng trung tuyn :ma2 = 2 2 22 4b c a + mb2 = 2 2 22 4a c b + mc2 = 2 2 22 4b a c ++ ng phn gic:la = 2 .cos2Abcb c +lb = 2 .cos2Baca c +la = 2 .cos2Caba b ++ M rng nh lsin v cosin:cotA = 2 2 24b c as+ cotB = 2 2 24a c bs+ cotC = 2 2 24a b cs+ II-BI TP : CHNG MINHNG THC C BN TRONG TAM GIC.1. sinA + sinB + sinC = 4cos2A.cos2B.cos2C. 2. sin2A + sin2B + sin2C = 4sinA.sinB.sinC.3. sin3A+sin3B+sin3C = -4cos32Acos32Bcos32C. 4.sin4A+sin4B+sin4C = -4sin2A.sin2B.sin2C.5. cosA + cosB + cosC = 1+ 4sin2A.4sin2B.4sin2C. 6. cos2A+cos2B+cos2C = -1-4cosA.cosB.cosC.7. cos3A+cos3B+cos3C =1- 4sin32Asin32Bsin32C. 8. tanA + tanB + tanC = tanA.tanB.tanC.9. cos4A+cos4B+cos4C = -1+ 4cos2Acos2Bcos2C. 10. tan2A +tan2B + tan2C = tan2A.tan2B.tan2C.11. cotA.cotB + cotB.cotgC + cotC.cotA = 1 12. tan2Atan2B+ tan2Btan2C+ tan2Ctan2A=113. cot2A+cot2B+ cot2C= cot2Acot2Bcot2C.14. cos2A + cos2B + cos2C = 1 -2cosA.cosB.cosC.15. cos22A + cos22B + cos22C = 1 + 2cos2A.cos2B.cos2C.16. 2am+ 2bm+ 2cm= 34(a2 + b2 + c2).17. la = 2 .cos2Abcb c + = 2bc. . .( ) b c p p a . 18. r = p.tan2Atan2Btan2C = sin sin2 2cos2B CaA.19. R = C4.cos .cos .cos2 2 2pA B. 20.r = 4R.cos2A. cos2B. cos2C.Ti liu rn k nng bin i lng gic dng cho HS kh gii 10NC 7L Trinh TngTHPT Tr ng Vng QNIII. CC BI TON V NG THC TRONG TAM GIC.1. Chng minh rngdin tch tam gic c th tnh theo cc cng thc sau:S = 2 2( ).sin .sin2.sin( )a b A BA B = 14(a2sin2B + b2sin2A) = p2.tan2Atan2Btan2C = 2R2.sinA.sinB.sinC.2. Chng minh cc ng thc sau:a) a.sin(B - C) + b.sin(C - A) + c.sin(A - B) = 0b) (b - c)cot2A +(c - a)cot2B + (a - b)cot2C = 0.c) (b2 - c2)cotA +(c2 - a2)cotB+(a2 - b2)cotC = 0.d) 2p = (a + b)cosC + (a + c)cosB+(a + b)cosC.e) sin2B C = b cacos2A. f) cos2B C = b ca+sin2A. g) b.cosB + c.cosC = a.c