重难点12解三角形的最值和范围问题九大题型（原卷版）_11.4-2_2026-11-04

      
      
重难点12 解三角形的最值和范围问题【九大题型】

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¤ 1/2024·¥¦§¨©·ª1« △𝐴𝐵𝐶IJ𝐴、𝐵、𝐶—¬&#“­:𝑎、𝑏、𝑐,𝑐= 4,𝑎𝑏= 9q
(1)®sin𝐶=
2
3J8sin𝐴⋅sin𝐵&(u
(2)8 △𝐴𝐵𝐶$%&'¯(q
°7 1-1/2024·±²·ª³´µ1¶‹ 𝐴𝐵𝐶&·𝐴J𝐵J𝐶&¬#“­:𝑎J𝑏J𝑐J¸Y𝑏cos𝐴=
3−𝑎cos𝐵J2𝑎sin𝐶=
3.
(1)8𝐴.
(2)8 △𝐴𝐵𝐶$%&¹(*+.
°7 1-2/2024·º»·ª³´µ1‰¼J«C$·J"# 𝐴𝐵𝐶𝐷½¾𝐵J𝐷T«𝐴𝐶&¿ÀJ𝐴𝐵= 1J
𝐵𝐶= 2J △𝐴𝐶𝐷:` J”∠𝐴𝐵𝐶= 𝛼.
  
(1)Á𝛼=
π
3VJ8𝐴𝐶u
(2)Á𝛼°VJ8"# 𝐴𝐵𝐶𝐷$%&'¯(.
      
      
°7 1-3/2024·Â÷ª1¸Y △𝐴𝐵𝐶&·𝐴J𝐵J𝐶&¬#“­:𝑎J𝑏J𝑐JÄ 3𝑎= 2𝑐sin𝐴q
(1)8sin𝐶&(u
(2)®𝑐= 3J8 △𝐴𝐵𝐶$%𝑆&'¯(q
 2   #-&'()*+,
¤ 2/2024·"Å·ª1« △𝐴𝐵𝐶IJ·𝐴J𝐵J𝐶&¬#“­:𝑎J𝑏J𝑐JĽ¾2𝑐sin𝐵cos𝐴= 𝑏
(sin𝐴cos𝐵+ cos𝐴sin𝐵).
(1)8𝐴u
(2)® △𝐴𝐵𝐶&$%:16 3J𝐷:𝐴𝐶&ITJ8𝐵𝐷&'Æ(.
°7 2-1/2024·ÇÈ·ª³´µ1« △𝐴𝐵𝐶IJ𝐴J𝐵J𝐶—¬&#“­¶:𝑎J𝑏J𝑐JÄtan𝐴=
cos𝐵−sin𝐶
cos𝐶+sin𝐵q
(1)®𝐵=
π
6J8𝐶&¯Æq
(2)®𝑎= 2J8𝑏+ 𝑐&¹(*+q
      
      
°7 2-2/2024·ÉÊÉË·ª1«‹ △𝐴𝐵𝐶IJ· AJBJC &¬#“­: aJbJcJÄ𝑐= 𝑏sin
𝐴
2 +𝑎
cos𝐵q
(1)8 Au
(2)® D R#𝐵𝐶ÂPT/5ÌÍÎT1JÄ∠𝐴𝐵𝐷= ∠𝐵𝐴𝐷J8
𝐶𝐷
𝐵𝐷&¹(*+q
°7 2-3/2024·ÇÈÏзѪ1 △𝐴𝐵𝐶&·𝐴,𝐵,𝐶&¬#“­:𝑎J𝑏J𝑐J½¾
1−sin𝐴
cos𝐴=
sin𝐵
cos𝐵.
(1)8Ò
𝐴+ 2𝐵=
π
2u
(2)8𝑎2+𝑏2
𝑐2 &'Æ(.
 3   .-&'()*+,
¤ 3/2024·ÓÔÕ¦·Ñª1¶ △𝐴𝐵𝐶&·𝐴,𝐵,𝐶&¬#“­:𝑎,𝑏,𝑐J¸Y𝑐−𝑏= 2𝑐sin2𝐴
2
(1)Ö×Ø △𝐴𝐵𝐶& Ùu
(2)®𝑐= 1J8 △𝐴𝐵𝐶.-&'¯(.
°7 3-1/2024·"ÅÚÛ·ª³´µ1¸Y« △𝐴𝐵𝐶IJD : BC #&ITJÄ𝐴𝐷=
5q
(1)® △𝐴𝐵𝐶&$%:2Jcos∠𝐴𝐷𝐶=
5
5 J8𝐵u
(2)®𝐴𝐵2 +𝐴𝐶2 = 18J8 △𝐴𝐵𝐶&.-&'¯(q
      
      
°7 3-2/2024·ÜÝÞ߷Ѫ1« △𝐴𝐵𝐶IJ𝐴,𝐵,𝐶&¬#“­:𝑎,𝑏,𝑐JÄ𝑎cos𝐶+
3𝑐sin𝐴= 𝑏+ 𝑐.
(1)8𝐵&¹(*+u
(2)¸Y △𝐴𝐵𝐶·àá&âã6ä 3
2 J8 △𝐴𝐵𝐶.-&¹(*+.
°7 3-3/2024·åÝLæ·Pª1¸Y △𝐴𝐵𝐶&·𝐴,𝐵,𝐶&¬#“­R𝑎,𝑏,𝑐JÄ
𝑎
cos𝐶= 2𝑏.
(1)×Ø △𝐴𝐵𝐶& Ùu
(2)® △𝐴𝐵𝐶&çèáâã: 2J8 △𝐴𝐵𝐶.-&'¯(.
 4   &/&0
(1&'()*+,
¤ 4/2024··éêëH욷Pª1¶ △𝐴𝐵𝐶&·𝐴,𝐵,𝐶&¬#“­:𝑎,𝑏,𝑐q®𝑎=
3,𝑏= 2Jí𝐵+ 𝐶
&¹(*+R/    1
Aq 2π
3 , 5π
6
Bq 2π
3 ,π
Cq 5π
6 ,π
Dq π
2 , 5π
6
°7 4-1/2024··éêëH욷Ѫ1« △𝐴𝐵𝐶IJ𝐴!𝐵!𝐶&¬#“­:𝑎!𝑏!𝑐J®
1
𝑏2 +
5
4𝑎2 =
𝑐2
𝑎2𝑏2Jítan𝐴−
1
tan𝐶&'Æ(:/    1
Aq
1
3
Bq
2
3
Cq
2
9
Dq
1
9
°7 4-2/2024·îÈïð·Ñª1 △𝐴𝐵𝐶IJ𝐷:𝐵𝐶#&ITJ𝐴𝐷= 1.
      
      
(1)® △𝐴𝐵𝐶&$%:2 3JÄ∠𝐴𝐷𝐶=
2π
3 J8sin𝐶&(u
(2)®𝐵𝐶= 4J8cos∠𝐵𝐴𝐶&¹(*+q
°7 4-3/2024·¦ñ§òó·Pª1«‹ △𝐴𝐵𝐶IJ𝐴,𝐵,𝐶&¬#“­:𝑎,𝑏,𝑐JÄ2𝑏sin𝐴−3𝑎= 0q
(1)8𝐵&¯Æu
(2)8cos𝐴+ cos𝐶&¹(*+q
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¤ 5/2024·óÈôõ·ª1¸Y △𝐴𝐵𝐶 IJ𝐴= 120∘， 𝐷R𝐵𝐶&ITJÄ 𝐴𝐷= 1Jí △𝐴𝐵𝐶 $%
&'¯(/   1
Aq 3
Bq2 3
Cq1
Dq2
°7 5-1/2024·ö÷Çøùú·ª1¸Y △𝐴𝐵𝐶&·𝐴,𝐵,𝐶&¬#“­:𝑎,𝑏,𝑐JÄ𝑎=
3,𝐵𝐶#ÂI
û𝐴𝐷-: 1Jí𝑏𝑐'¯(:/    1
Aq
7
4
Bq
7
2
Cq 3
Dq2 3
°7 5-2/2024·ÓÔ\ü·Ñª1¶ △𝐴𝐵𝐶&·𝐴,𝐵,𝐶&¬#“­:𝑎,𝑏,𝑐J¸Y𝑐= 2,
1
tan𝐴+
1
tan𝐵+
1
tan𝐴tan𝐵
= 1qí △𝐴𝐵𝐶$%&'¯(:/    1
Aq1 +
2
Bq1 +
3
Cq2 2
Dq2 3
°7 5-3/2024·ýÇþ˷Ѫ1« △𝐴𝐵𝐶IJ AJBJC —¬&#“­: aJbJcJ®𝑎cos𝐶= 2𝑐cos
𝐴Jí
𝑏𝑐
𝑎2&'¯(:/    1
      
      
Aq 3
Bq
3
2
Cq 3
2
Dq3
 6  9:0
8'(/*+1
¤ 6/2024·º»ÿÛ·ª³´µ1« △𝐴𝐵𝐶IJ· AJBJC —¬&#“­: aJbJcJÄ
sin2𝐶−sin𝐶sin𝐵
cos2𝐵−cos2𝐴= 1.
(1)8 A &¯Æu
(2)® △𝐴𝐵𝐶:‹ JT F : △𝐴𝐵𝐶&ĀyJ𝐴𝐹= 6J8𝐶𝐹+ 𝐵𝐹&¹(*+.
°7 6-1/2024·º»·ª³´µ1¸Y △𝐴𝐵𝐶&·𝐴,𝐵,𝐶&¬#“­:𝑎,𝑏,𝑐,(𝑐−3𝑏)sin𝐶= (𝑎−𝑏)
(sin𝐴+ sin𝐵)q
(1)8𝐴u
(2)® △𝐴𝐵𝐶:‹ JÄ𝑏= 6J8 △𝐴𝐵𝐶&.-𝑙&¹(*+q
°7 6-2/2024·¥¦āĂ·Pª1« △𝐴𝐵𝐶IJ·𝐴,𝐵,𝐶—¬&#“­R𝑎,𝑏,𝑐J $%:𝑆J®𝐷:
𝐴𝐶#ÂPTJ½¾𝐴𝐵⊥𝐵𝐷,𝐵𝐷= 2JÄ𝑎2 = −2 3
3 𝑆+ 𝑎𝑏cos𝐶.
(1)8𝐵u
(2)8
2
𝐴𝐷+
1
𝐶𝐷&¹(*+.
°7 6-3/2024·ălĄË·ª³´µ1‰¼J«"# 𝐴𝐵𝐶𝐷IJ∠𝐷𝐴𝐵=
π
2J𝐵=
π
6JÄ △𝐴𝐵𝐶&çèá
      
      
âã: 4.
(1)®𝐵𝐶= 4 2J𝐴𝐷= 2 2J8 △𝐴𝐶𝐷&$%u
(2)®𝐷=
2π
3 J8𝐵𝐶−𝐴𝐷&'¯(.
 7  9:;<0
8'(/*+1
¤ 7/2024·"ÅąĆ·ª³´µ1”‹ △𝐴𝐵𝐶&ć·𝐴,𝐵,𝐶&¬#“­:𝑎,𝑏,𝑐JÄ𝑐= 2,𝐵= 2𝐶J
í𝑎+ 𝑏&¹(*+: /    1
Aq(2,10)
Bq(2 + 2 2,10)
Cq(2 + 2 2,4 + 2 3) Dq(4 + 2 3,10)
°7 7-1/2024·±²·ª³´µ1¸Y △𝐴𝐵𝐶R‹ J· AJBJC —¬Ĉ&#“­: aJbJ
cq®𝑎2−𝑏2 = 𝑏𝑐Jí
𝑏
𝑎+𝑐&¹(*+R/    1
Aq
3
3 ,
2
2
Bq(2−3,1)
Cq(2−3, 2−1)
Dq( 2 + 1, 3 + 2)
°7 7-2/2023·±²·ª³´µ1¸Y △𝐴𝐵𝐶:‹ J;· AJBJC —¬&#“­:𝑎J𝑏J𝑐J
cos𝐵= cos2𝐴.
(1)8
𝑏
𝑎&¹(*+u
(2)®𝑎= 1J8 △𝐴𝐵𝐶.-&¹(*+.
°7 7-3/2024·±²·ª³´µ1¸Y △𝐴𝐵𝐶IJ𝐴J𝐵J𝐶—¬&#“­:𝑎，𝑏，𝑐，𝑆△𝐴𝐵𝐶= 𝑏2−𝑐2+16
4
⋅tan𝐶.
(1)8𝑎&(u
      
      
(2)®𝐷:ûĉ𝐵𝐶ÂPTĽ¾𝐵𝐷= 1,𝐷𝐴C“∠𝐵𝐴𝐶J8 △𝐴𝐵𝐶&$%&¹(*+.
 8  “>?@”8'(/*+1
¤ 8/23-24 PĊ·"ÅċČ·čĎ1‰¼J«C$"# 𝐴𝐵𝐶𝐷IJ𝐴𝐵⊥𝐵𝐶J∠𝐵𝐶𝐷= 60°J
∠𝐴𝐷𝐶= 150°J𝐵𝐸= 3𝐸𝐶J𝐶𝐷= 2 3
3 J𝐵𝐸=
3J®T F :# AD Â&ďTJí𝐸𝐹⋅𝐵𝐹&'Æ(:/    1
  
Aq1
Bq
15
16
Cq
31
32
Dq2
°7 8-1/2023·ÓÔĐđ󷪳´µ1¸YCĒ"# 𝐴𝐵𝐶𝐷IJ ∠𝐴𝐷𝐶= 60°J𝐸J𝐹“­:#𝐴𝐵J𝐵𝐶
&ITJ®𝐷𝐸⋅𝐷𝐹= 13Jí"# 𝐴𝐵𝐶𝐷$%&'¯(:/    1
Aq2
Bq2 3
Cq4
Dq4 3
°7 8-2/2023·±²·ª³´µ1«6ē △𝐴𝐵𝐶IJ AJBJC —¬Ĉ&#: aJbJcJ𝐵= 𝐶=
π
6J
𝑎= 2 3JP R △𝐴𝐵𝐶çèáÂPTJí𝑃𝐴⋅𝑃𝐵+ 𝑃𝐵⋅𝑃𝐶+ 𝑃𝐶⋅𝑃𝐴&¹(*+R/    1
Aq[−3,23]
Bq[−1,33]
Cq[−2,30]
Dq[−4,20]
°7 8-3/2024·ÇÈÝĔ·ª1‰¼J«ĕ OAB IJâã𝑂𝐴= 4J∠𝐴𝑂𝐵= 90°JC «âã OB ÂJD
«âã OA ÂJE Rĕ ĖÂ&ďT/5ÌėÎT1JíCĒ"# BCDE &.-&¹(*+R/   1
Aq(8,12]
Bq(8 2,12]
Cq(8,8 2]
Dq(4,8 2]
      
      
 9  BC$DEF&'(/*+1,
¤ 9/2023·¥ÝĘę·ª1¸Y𝑒1!𝑒2:ĚěDEJ|𝑒1−𝑒2| =
3JĜĝDE𝑎½¾|𝑎−2𝑒2| = 1Jí|𝑒1−𝑎|
&'Æ(:/    1
Aq 7
Bq 7−1
Cq 3
Dq 3−1
°7 9-1/23-24 ·¦ñKË·čĎ1«Ğ 𝐴𝐵𝐶𝐷IJ𝐴𝐵= 2,∠𝐵𝐴𝐷= 60∘,𝐸R𝐵𝐶&ITJ𝐹R𝐶𝐷Â
PT/5B𝐶J𝐷U\1J𝐷𝐸B𝐴𝐹ğä𝐺Jí𝐴𝐺⋅𝐷𝐺&¹(*+R/    1
Aq 0, 2
3
Bq 0, 4
3
Cq(0,2)
Dq(0,3)
°7 9-2/2024·ălĠË·ª³´µ1¸YCĒ"# ABCD IJ𝐴𝐵= 2J𝐵𝐶= 4J𝐵=
2π
3 J®ġ C :
áy&áB¬û BD ĢàJP Rá C Â&PTJí𝐵𝐷⋅𝐶𝑃−𝐶𝐵&'Æ(R/    1
Aq8−2 3
Bq4 + 2 3
Cq12−4 3
Dq6 + 2 3
°7 9-3/2023·ălģĤ·Ñª1« △𝐴𝑂𝐵IJ¸Y|𝑂𝐵| =
2J|𝐴𝐵| = 1J∠𝐴𝑂𝐵= 45°J®𝑂𝑃= 𝜆𝑂𝐴+𝜇
𝑂𝐵JÄ𝜆+ 2𝜇= 2J𝜇∈[0,1]Jí𝑂𝐴«𝑂𝑃Â&ĥĦDE:𝑚𝑒/𝑒:B𝑂𝑃ħD&ĚěDE1Jí m &¹(*
+R/    1
Aq −
2
2 ,1
Bq
2
2 ,1
Cq −
2
2 ,1
Dq
2
2 ,1
P!Ě
1q/2024·ÇĨĩÜĪ·ª³´µ1« △𝐴𝐵𝐶IJ AJBJC &¬#“­: aJbJcJ®𝑎= 1J𝑏cos𝐴= 1 +
cos𝐵Jí# b &¹(*+:/    1
Aq(0,1)
Bq(1,2)
Cq(0,2)
Dq(2,3)
2q/2024·ÓÔ\ü·ª³´µ1¸Y △𝐴𝐵𝐶𝐴、𝐵、𝐶&¬#“­:𝑎、𝑏、𝑐½¾
2𝑏
𝑎−𝑐=
sin𝐴+sin𝐶
sin𝐵
Jí𝐵&
'¯(:/    1
Aq
π
6
Bq
π
4
Cq
π
3
Dq
2π
3
3q/2024·ÉÊÊī·ª³´µ1¸Y«ħPC$·&ćT AJBJC ½¾|𝐴𝐵| = 2J|
𝐶𝐴
|𝐶𝐴| −𝐶𝐵
|𝐶𝐵|| ≥1Jí|𝐴𝐶+ 𝐵𝐶|
&¹(*+R/    1
Aq[0,1]
Bq[0,2]
Cq[0, 3]
Dq[0,2 3]
4q/2024·¥Ý·ª1« △𝐴𝐵𝐶IJ AJBJC —¬&#“­: aJbJcq®
𝑎
cos𝐴+
𝑏
cos𝐵=
3𝑐
cos𝐶Jítan𝐴+
      
      
tan𝐶&'Æ(R/    1
Aq
4
3
Bq
8
3
Cq2 3
Dq4
5q/2024·¥Ý·ª³´µ1«‹ △𝐴𝐵𝐶IJ AJBJC —¬&#“­: aJbJcJ½¾𝑏3+𝑐3
𝑏+𝑐= 𝑎2q®
𝑎= 2 3Jí𝑏2 + 𝑐2&¹(*+:/    1
Aq(12,24]
Bq(20,24]
Cq[12,24]
Dq[20,24]
6q/2024·ÇȷѪ1« △𝐴𝐵𝐶IJ®sin𝐴= 2cos𝐵cos𝐶Jícos2𝐵+ cos2𝐶&¹(*+:/    1
Aq 1, 6
5
Bq 1,
2+1
2
Cq 6
5 ,2
Dq
2+1
2
,2
7q/2024·±²·Ñª1«△ABC IJ· AJBJC —¬&#“­: aJbJcJ2𝑎cos𝐴= 𝑏cos𝐶+ 𝑐cos𝐵JÄ
𝑎= 4sin𝐴Jí△ABC .-&'¯(:/    1
Aq4 2
Bq6 2
Cq4 3
Dq6 3
8q/2024·îÈĬÛ·ª1:ĭĮPį˜İıIJ ijJ½¾ĴĵĶķĸHĹĺ&Ļ8J2023 ļĽIJľĿ
«IJŀŁłŃńl”ĭ“ŅņŇ”.‰¼J«ĕ “ŅņŇ”𝑂𝑃𝑄IJňʼnŊP• ķĸįŋ𝑂𝐴𝐵J¸
Yĕ &âã𝑂𝑃= 3Jáy∠𝑃𝑂𝑄=
π
3J𝐴Rĕ ĖÂ&ďTJ𝐵Râã𝑂𝑄Â&ďTJ𝐴𝐵//𝑂𝑃Jí △𝑂𝐴𝐵
$%&'¯(:/    1
Aq3 3
4
Bq
3
4
Cq3 3
5
Dq
3
5
Ñ!Ł
9q/2024·ÇĨÝñ·Ñª1¸Y △𝐴𝐵𝐶·𝐴J𝐵J𝐶&¬#“­:𝑎J𝑏J𝑐J𝑂: △𝐴𝐵𝐶&UyJcos𝐴=
1
5J
𝐴𝑂= 2Jí/    1
Aq𝐴𝑂=
1
3𝐴𝐵+
1
3𝐴𝐶
Bq𝐴𝐵⋅𝐴𝐶≤3
Cq △𝐴𝐵𝐶&$%&'¯(:3 6
Dq𝑎&'Æ(:2 5
10q/2024·åݷѪ1« △𝐴𝐵𝐶IJ𝐴,𝐵,𝐶—¬&#“­:𝑎,𝑏,𝑐JÄ𝑐= 𝑏(2cos𝐴+ 1)JíĊŌtō`
&F/    1
Aq𝐴= 2𝐵
      
      
Bq®𝑎=
3𝑏Jí △𝐴𝐵𝐶:Q 
Cq® △𝐴𝐵𝐶:‹ J
1
tan𝐵−
1
tan𝐴&'Æ(: 1
Dq® △𝐴𝐵𝐶:‹ Jí
𝑐
𝑎&¹(*+:
2
2 , 2 3
3
11q/2024·¥¦Ŏŏ·ª1¸Y △𝐴𝐵𝐶&ć· AJBJC &¬#“­R aJbJcJ$%: 3
4
(𝑎2 + 𝑐2−𝑏2)JíĊŌŐ@`&R/    1
Aqcos𝐴cos𝐶&¹(*+R −1
2 , 1
4
Bq®𝐷:#𝐴𝐶&ITJÄ𝐵𝐷= 1Jí △𝐴𝐵𝐶&$%&'¯(: 3
3
Cq® △𝐴𝐵𝐶R‹ Jí
𝑎
𝑐&¹(*+R 1
2 ,2
Dq®𝐵&C“û𝐵𝐸B#𝐴𝐶ĢğäT𝐸JÄ𝐵𝐸=
3Jí𝑎+ 4𝑐&'Æ(: 10
!őŒ
12q/2024·¦ñ·ª1« △𝐴𝐵𝐶IJ𝑎,𝑏,𝑐“­R𝐴,𝐵,𝐶&¬#JÄ𝑎+ 𝑐= 2𝑏Jí𝐵&¹(*+:      
q
13q/2024·îÈÓœ·ª³´µ1« △𝐴𝐵𝐶IJ·𝐴,𝐵,𝐶—¬&#“­:𝑎,𝑏,𝑐J®𝑏= 2J
2𝑎
cos𝐶=
2
cos𝐵+
𝑐
cos𝐶J
í2𝑎+ 𝑐&'¯(:      
.
14q/2024·ÇĨŔı·Pª1« △𝐴𝐵𝐶IJ¸Y𝐴𝐵= 2J𝐵𝐶= 3JT P « △𝐴𝐵𝐶·JĽ¾𝐶𝑃= 2J
∠𝐴𝑃𝐶+ ∠𝐴𝐵𝐶= πJí"# 𝐴𝐵𝐶𝑃$%&'¯(:      
.
"!Gŕ
15q/2024·óÊŖŗ·ª³´µ1« △𝐴𝐵𝐶IJ𝐴,𝐵,𝐶—¬&#“­:𝑎,𝑏,𝑐.¸Y𝐴𝐵⋅𝐴𝐶−𝐵𝐴⋅𝐵𝐶= 𝜆𝐴𝐵
2
(1)®𝜆= 1J×Ø △𝐴𝐵𝐶& Ùu
(2)®𝜆=
1
2J8tan(𝐵−𝐴)&'¯(.
16q/2024·ÇĨŔı·ª³´µ1« △𝐴𝐵𝐶IJ¸Y𝐴J𝐵J𝐶—¬&#“­:𝑎J𝑏J𝑐J𝑎sin2𝐵
2 +𝑏sin2𝐴
2 =
3𝑎𝑏
2(𝑎+𝑏+𝑐)q
      
      
(1)8𝐶&¯Æu
(2)® △𝐴𝐵𝐶:‹ J8
𝑎+𝑏
𝑐&¹(*+q
17q/2024·UŘřI·ª³´µ1¸Y △𝐴𝐵𝐶&·𝐴,𝐵,𝐶&¬#“­:𝑎,𝑏,𝑐JĽ¾ 3𝑐
𝑎−sin𝐵= tan𝐴⋅cos
𝐵.
(1)8𝐴&¯Æu
(2)® △𝐴𝐵𝐶:‹ Ä𝑎= 2 6J8 △𝐴𝐵𝐶$%&¹(*+.
18q/2024·"Åݒ·ª³´µ1« △𝐴𝐵𝐶IJ
sin𝐶
sin𝐴+sin𝐵=
sin𝐴−sin𝐵
sin𝐵+sin𝐶.
(1)8𝐴u
(2)®𝐵𝐶= 3J8 △𝐴𝐵𝐶.-&'¯(.
19q/2024·îÈŚś·ª³´µ1«‹ △𝐴𝐵𝐶I.·𝐴J𝐵J𝐶—¬&#“­R𝑎J𝑏J𝑐J¸Y𝑎−2𝑐cos
𝐵= 𝑐.
(1)8Ò
𝐵= 2𝐶u
(2)8sin𝐵+ 2 3cos2𝐶&¹(*+.