æ£å¼¦å®ç
ç®å½
å®çæ¦è¿°
è¯æ
æä¹
æ©å±å±å¼
ç¼è¾æ¬æ®µ
å®çæ¦è¿°
ããå¨â³ABCä¸ï¼è§AãBãCæ对çè¾¹åå«ä¸ºaãbãcï¼åæ
ããa/sinA=b/sinB=c/sinC=2Rï¼R为ä¸è§å½¢å¤æ¥åçåå¾ï¼
ããæ£å¼¦å®ç(Sine theorem)ãï¼1ï¼å·²ç¥ä¸è§å½¢ç两è§ä¸ä¸è¾¹ï¼è§£ä¸è§å½¢
ããï¼2ï¼å·²ç¥ä¸è§å½¢ç两边åå
¶ä¸ä¸è¾¹æ对çè§ï¼è§£ä¸è§å½¢
ããï¼3ï¼è¿ç¨aï¼bï¼c=sinAï¼sinBï¼sinC解å³è§ä¹é´ç转æ¢å
³ç³»
ããç´è§ä¸è§å½¢çä¸ä¸ªéè§ç对边ä¸æè¾¹çæ¯å«åè¿ä¸ªè§çæ£å¼¦ã
ç¼è¾æ¬æ®µ
è¯æ
ããæ¥éª¤1
ããå¨éè§â³ABCä¸ï¼è®¾BC=aï¼AC=bï¼AB=cãä½CHâ¥ABå足为ç¹H
ããCH=a·sinB
ããCH=b·sinA
ããâ´a·sinB=b·sinA
ããå¾å°
ããa/sinA=b/sinB
ããåçï¼å¨â³ABCä¸ï¼
ããb/sinB=c/sinC
ããæ¥éª¤2.
ããè¯æa/sinA=b/sinB=c/sinC=2Rï¼
ããå¦å¾ï¼ä»»æä¸è§å½¢ABC,ä½ABCçå¤æ¥åO.
ããä½ç´å¾BD交âOäºD.
ããè¿æ¥DA.
ããå 为å¨ååæçåä¸ç´å¾æ对çåå¨è§æ¯ç´è§ï¼æ以â DAB=90度
ããå 为å¨ååæçåä¸å弧æ对çåå¨è§ç¸çï¼æ以â Dçäºâ ACB.
ããæ以c/sinC=c/sinD=BD=2R
ãã类似å¯è¯å
¶ä½ä¸¤ä¸ªçå¼ã
ç¼è¾æ¬æ®µ
æä¹
ããæ£å¼¦å®çæåºäºä»»æä¸è§å½¢ä¸ä¸æ¡è¾¹ä¸å¯¹åºè§çæ£å¼¦å¼ä¹é´çä¸ä¸ªå
³ç³»å¼ãä¹å°±æ¯ä»»æä¸è§å½¢çè¾¹è§å
³ç³»ã
ç¼è¾æ¬æ®µ
æ©å±
ä½å¼¦å®ç
ããä½å¼¦å®çæ¯æ示ä¸è§å½¢è¾¹è§å
³ç³»çéè¦å®çï¼ç´æ¥è¿ç¨å®å¯è§£å³ä¸ç±»å·²ç¥ä¸è§å½¢ä¸¤è¾¹å夹è§æ±ç¬¬ä¸è¾¹æè
æ¯å·²ç¥ä¸ä¸ªè¾¹æ±è§çé®é¢ï¼è¥å¯¹ä½å¼¦å®çå 以å形并éå½ç§»äºå
¶å®ç¥è¯ï¼å使ç¨èµ·æ¥æ´ä¸ºæ¹ä¾¿ãçµæ´»ã
ããä½å¼¦å®çæ§è´¨
ãã对äºä»»æä¸è§å½¢ï¼ä»»ä½ä¸è¾¹çå¹³æ¹çäºå
¶ä»ä¸¤è¾¹å¹³æ¹çååå»è¿ä¸¤è¾¹ä¸å®ä»¬å¤¹è§çä½å¼¦ç两å积ï¼è¥ä¸è¾¹ä¸ºa,b,c ä¸è§ä¸ºA,B,C ï¼å满足æ§è´¨ââ
ããa^2 = b^2 + c^2 - 2·b·c·cosA
ããb^2 = a^2 + c^2 - 2·a·c·cosB
ããc^2 = a^2 + b^2 - 2·a·b·cosC
ããcosC = (a^2 + b^2 - c^2) / (2·a·b)
ããcosB = (a^2 + c^2 - b^2) / (2·a·c)
ããcosA = (c^2 + b^2 - a^2) / (2·b·c)
ããï¼ç©çåå¦æ¹é¢çå¹³è¡å边形å®åä¸ä¹ä¼ç¨å°ï¼
ãã第ä¸ä½å¼¦å®çï¼ä»»æä¸è§å½¢å°å½±å®çï¼
ãã设â³ABCçä¸è¾¹æ¯aãbãcï¼å®ä»¬æ对çè§åå«æ¯AãBãCï¼åæ
ããa=b·cos C+c·cos Bï¼ b=c·cos A+a·cos Cï¼ c=a·cos B+b·cos Aã
ä½å¼¦å®ççè¯æ
ããå¹³é¢åéè¯æ³
ããâµå¦å¾ï¼æa+b=c (å¹³è¡å边形å®åï¼ä¸¤ä¸ªé»è¾¹ä¹é´ç对è§çº¿ä»£è¡¨ä¸¤ä¸ªé»è¾¹å¤§å°ï¼ â´c·c=(a+b)·(a+b) â´c^2=a·a+2a·b+b·bâ´c^2=a^2+b^2+2|a||b|Cos(Ï-θ)
ããï¼ä»¥ä¸ç²ä½å符表示åéï¼
ããåâµCos(Ï-θ)=-CosC
ããâ´c^2=a^2+b^2-2|a||b|cosθï¼æ³¨æï¼è¿éç¨å°äºä¸è§å½æ°çå
¬å¼ï¼
ããåæå¼ï¼å¾c^2=a^2+b^2-2*a*b*cosC
ããå³ cosC=(a^2+b^2-c^2)/2*a*b
ããåçå¯è¯å
¶ä»ï¼èä¸é¢çCosC=(c^2-b^2-a^2)/2abå°±æ¯å°CosC移å°å·¦è¾¹è¡¨ç¤ºä¸ä¸ã
ããå¹³é¢å ä½è¯æ³
ããå¨ä»»æâ³ABCä¸
ããåADâ¥BC.
ããâ Cæ对ç边为cï¼â Bæ对ç边为bï¼â Aæ对ç边为a
ããåæBD=cosB*cï¼AD=sinB*cï¼DC=BC-BD=a-cosB*c
ããæ ¹æ®å¾è¡å®çå¯å¾ï¼
ããAC^2=AD^2+DC^2
ããb^2=(sinB*c)^2+(a-cosB*c)^2
ããb^2=(sinB*c)^2+a^2-2ac*cosB+(cosB)^2*c^2
ããb^2=(sinB^2+cosB^2)*c^2-2ac*cosB+a^2
ããb^2=c^2+a^2-2ac*cosB
ããcosB=(c^2+a^2-b^2)/2ac
ä¸è§å½¢é¢ç§¯å
¬å¼
ãã1.海伦-秦ä¹é¶å
¬å¼:
ãã设P=(a+b+c)/2
ããSâ³ABC=â[P(P-a)(P-b)(P-c)]
ãã解éï¼å设æä¸ä¸ªä¸è§å½¢ï¼è¾¹é¿åå«ä¸ºaãbãcï¼ä¸è§å½¢çé¢ç§¯Så¯ç±ä»¥ä¸å
¬å¼æ±å¾ï¼
ããS=â[p(p-a)(p-b)(p-c)]
ããèå
¬å¼éçp为åå¨é¿ï¼
ããp=(a+b+c)/2
ãã2.Sâ³ABC=(ab/2)·sinC=(bc/2)·sinA=(ac/2)·sinB=abc/(4R)[R为å¤æ¥ååå¾]
ãã3.Sâ³ABC=ah/2
æ£å¼¦å®ççåå½¢å
¬å¼
ãã(1) a=2RsinA, b=2RsinB, c=2RsinC;
ãã(2) sinA : sinB : sinC = a : b : c;
ããï¼æ¡ä»¶åä¸ï¼
ããå¨ä¸ä¸ªä¸è§å½¢ä¸ï¼åè¾¹ä¸å
¶æ对è§çæ£å¼¦çæ¯ç¸çï¼ä¸è¯¥æ¯å¼é½çäºè¯¥ä¸è§å½¢å¤æ¥åçç´å¾å·²ç¥ä¸è§å½¢æ¯ç¡®å®çï¼å©ç¨æ£å¼¦å®ç解ä¸è§å½¢æ¶ï¼å
¶è§£æ¯å¯ä¸çï¼å·²ç¥ä¸è§å½¢ç两边åå
¶ä¸ä¸è¾¹ç对è§ï¼ç±äºè¯¥ä¸è§å½¢å
·æä¸ç¨³å®æ§ï¼æ以å
¶è§£ä¸ç¡®å®ï¼å¯ç»åå¹³é¢å ä½ä½å¾çæ¹æ³åâ大边对大è§ï¼å¤§è§å¯¹å¤§è¾¹âå®çåä¸è§å½¢å
è§åå®çå»èè解å³é®é¢
ãã(3)ç¸å
³ç»è®ºï¼
ããa/sinA=b/sinB=c/sinC=(a+b)/(sinA+sinB)=(a+b+c)/(sinA+sinB+sinC)
ããc/sinC=c/sinD=BD=2Rï¼R为å¤æ¥ååå¾ï¼
ããï¼4ï¼è®¾R为ä¸è§å¤æ¥ååå¾ï¼å
¬å¼å¯æ©å±ä¸ºï¼a/sinA=b/sinB=c/sinC=2R,å³å½ä¸å
è§ä¸º90°æ¶ï¼æ对ç边为å¤æ¥åçç´å¾ãçµæ´»è¿ç¨æ£å¼¦å®çï¼è¿éè¦ç¥éå®çå 个åå½¢
ããsinA=a/2R,sinB=b/2R,sinC=c/2R
ããasinB=bsinA,bsinC=csinB,asinC=csinA
ããï¼5ï¼a=bsinA/sinB sinB=bsinA/a
温馨提示:答案为网友推荐,仅供参考