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Luonggiac-Chuong10 - C HNG X H E TH C L N G TRONG TAM GIA C...

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CHÖÔNG X: HEÄ THÖÙC LÖÔÏNG TRONG TAM GIAÙC I. ÑÒNH LYÙ HAØM SIN VAØ COSIN Cho ABC Δ coù a, b, c laàn löôït laø ba caïnh ñoái dieän cuûa ± ± ± A, B,C, R laø baùn kính ñöôøng troøn ngoaïi tieáp ABC Δ , S laø dieän tích ABC Δ thì = = = = + = + = + = + = + = + 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 a b c 2R sin A sinB sinC a b c 2bccosA b c 4S.cotg b a c 2accosB a c 4S.cotgB c a b 2abcosC a b 4S.cotg A C Baøi 184 Cho ABC Δ . Chöùng minh: 2 2 A 2B a b bc = = + Ta coù: 2 2 2 2 2 2 2 a b bc 4R sin A 4R sin B 4R sinB.sinC = + = + ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) = = = ⇔ − + = + = = + = > = = π − = 2 2 sin A sin B sinBsinC 1 1 1 cos2A 1 cos2B sinBsinC 2 2 cos2B cos2A 2sinBsinC 2sin B A sin B A 2sinBsinC sin B A sin A B sinBsinC sin A B sinB do sin A B sinC 0 A B B A B B loaïi A 2B Caùch khaùc: = + = + + = 2 2 sin A sin B sinBsinC (sin A sinB)(sin A sinB) sinBsinC A B A B A B A B 2cos sin .2sin cos sinBsinC 2 2 2 2 ( ) ( ) ( ) ( ) ( ) ( ) + = = + = > = = π − = sin B A sin A B sinBsinC sin A B sinB do sin A B sinC 0 A B B A B B loaïi A 2B
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Baøi 185: Cho ABC Δ . Chöùng minh: ( ) 2 2 2 sin A B a b sinC c = Ta coù = 2 2 2 2 2 2 2 2 2 a b 4R sin A 4R sin B c 4R sin C ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) = = + = = + = = + = > 2 2 2 2 2 2 2 1 1 1 cos2A 1 cos2B sin A sin B 2 2 sin C sin C 2sin A B sin B A cos2B cos2A 2sin C 2sin C sin A B .sin A B sin A B sinC sin C do sin A B sinC 0 Baøi 186: Cho ABC Δ bieát raèng A B 1 tg tg 2 2 3 = Chöùng minh a b 2c + = Ta coù : = = A B 1 A B A B tg tg 3sin sin cos cos 2 2 3 2 2 2 2 A B do cos 0,cos 0 2 2 > > ( ) A B A B A 2sin sin cos cos sin sin 2 2 2 2 2 2 A B A B A B cos cos cos 2 2 2 A B A B cos 2cos * 2 2 = + + ⇔ − = + = B Maët khaùc: ( ) a b 2R sinA sinB + = + ( ) ( ) ( ) + = + + = = + = = A B A B 4Rsin cos 2 2 A B A B 8Rsin cos do * 2 2 4Rsin A B 4RsinC 2c Caùch khaùc: ( ) + = + = a b 2c 2R sin A sinB 4RsinC
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+ = + + = = = A B A B C C 2sin cos 4sin cos 2 2 2 2 A B C A B A B cos 2sin 2cos dosin cos 2 2 2 2 C 2 + = = A B A B A B A cos cos sin sin 2cos cos 2sin sin 2 2 2 2 2 2 2 A B A B 3sin sin cos cos 2 2 2 2 B 2 = A B 1 tg tg 2 2 3 Baøi 187: Cho ABC Δ , chöùng minh neáu taïo moät caáp soá coäng thì cotgA,cotgB,cotgC 2 2 2 a ,b ,c cuõng laø caáp soá coäng. Ta coù: ( ) + = cot gA,cot gB,cot gClaøcaápsoá coäng cot gA cot gC 2cot gB * Caùch 1: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) [ ] ( ) ( ) ( ) 2 2 2 2 2 2 2 2 2 2 2 2 2 sin A C 2cosB Ta coù: * sin B 2sinAsinCcosB sinAsinC sinB sin B cos A C cos A C cos A C sin B cos A C cos A C cos A C 1 sin B cos B cos2A cos2C 2 1 sin B 1 sin B 1 2sin A 1 2sin C 2 2sin B sin A sin C + = = = − + + ⎤ ⎡ ⎦ ⎣ = + + = + = + = + 2 2 2 2 2 2 2 2 2 2 2 2 2b a c 4R 4R 4R 2b a c a ,b ,c laø caâùp soá coäng = + = + Caùch 2: ( ) = + = + = + + = + + = = + + + + = = + 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 Ta coù:a b c 2abcosA 1 a b
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