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www.9inokia.com:2019届江苏高三数学一轮复习讲义+课时作业第四章三角函数与解三角形第24讲正弦定理、余弦定理

第 24 讲 考试要求 级要求). 正弦定理、余弦定理 1.正弦定理、余弦定理(B 级要求);2.运用定理解决解三角形问题(B 诊 断 自 测 1.思考辨析(在括号内打“√”或“×”) (1)三角形中三边之比等于相应的三个内角之比.( (2)在△ABC 中,若 sin A>sin B,则 A>B.( ) ) ) (3)在△ABC 的六个元素中,已知任意三个元素可求其他元素.( (4)当 b2+c2-a2>0 时,三角形 ABC 为锐角三角形.( a+b-c a (5)在△ABC 中,sin A= .( sin A+sin B-sin C ) ) ) (6)在三角形中,已知两边和一角就能求三角形的面积.( a b 解析 (2)△ABC 中,sin A>sin B,结合sin A=sin B=2R 得 a>b,故 A>B. a b c (5)△ABC 中,sin A=sin B=sin C=2R, a+b-c 2R(sin A+sin B-sin C) a 故 = =2R=sin A. sin A+sin B-sin C sin A+sin B-sin C (6)已知两边和一角,就能由余弦定理或正弦定理解出这个三角形的其它角与边, 1 故据面积公式 S△ABC=2absin C 即可求三角形的面积. 答案 (1)× (2)√ (3)× (4)× (5)√ (6)√ 2.(教材改编)在△ABC 中,a=2,A=30°,C=45°,则△ABC 的面积 S△ABC= ________. asin B 2×sin 105° 解析 ∵b= sin A = = 6+ 2, sin 30° 1 2 ∵S△ABC=2absin C=( 6+ 2)× 2 = 3+1. 答案 3+1 3.(2017· 苏、锡、常、镇调研二)在△ABC 中,角 A,B,C 的对边分别是 a,b,c, 若满足 2bcos A=2c- 3a,则角 B 的大小是________. 解析 由余弦定理,2bcos A=2c- 3a, b2+c2-a2 即 2b· 2bc =2c- 3a, ∴b2+c2-a2=2c2- 3ac, 即 a2+c2-b2= 3ac, a2+c2-b2 3ac 3 ∴cos B= 2ac = 2ac = 2 , π 又 B∈(0,π),∴B= 6 . 答案 π 6 4.(2017· 南通、扬州、泰州、淮安三调)在锐角三角形 ABC 中,AB=3,AC=4. 若△ABC 的面积为 3 3,则 BC 的长是________. 1 3 解析 因为 b=4,c=3,由 S△ABC=2bcsin A=6sin A=3 3,解得 sin A= 2 . 因 为 是 在 锐 角 三 角 形 ABC 中 , 所 以 cos A = 1 1-sin2A = 2 ? ? π ?或求出锐角A= ,再求cos A=1?.在锐角三角形 ABC 中,由余弦定理得 a2=b2 3 2? ? 1 +c2-2bccos A=16+9-2×4×3×2=13,所以 a= 13,即 BC= 13. 答案 13 5.(2018· 淮安质检)已知在△ABC 中,内角 A,B,C 所对边长分别为 a,b,c,若 π A= 3 ,b=2acos B,c=1,则△ABC 的面积等于________. π 解析 由正弦定理得 sin B=2sin A·cos B,故 tan B=2sin A=2sin 3 = 3, π π 又 B∈(0,π),所以 B= 3 ,又 A= 3 ,所以△ABC 是正三角形,所以 S△ABC 1 1 3 3 =2bcsin A=2×1×1× 2 = 4 . 答案 3 4 知 识 梳 理 1.正弦定理、余弦定理 在△ABC 中,若角 A,B,C 所对的边分别是 a,b,c,R 为△ABC 外接圆半径, 则 定理 正弦定理 a b c sin A=sin B=sin C=2R (1)a=2Rsin A, b=2Rsin__B,c=2Rsin__C; a b c (2)sin A=2R,sin B=2R,sin C=2R; 变形 (3)a∶b∶c=sin__A∶sin__B∶sin__C; (4)asin B = bsin A , bsin C = cos A = b2+c2-a2 2bc ; b2+c2-a2 cos A= 2bc ; cos B= c2+a2-b2 ; 2ca 余弦定理 a2=b2+c2-2bccos__A; 内容 b2=c2+a2-2cacos__B; c2=a2+b2-2abcos__C a2+b2-c2 cos C= 2ab 2.在△ABC 中,已知 a、b 和 A 时,解的情况如下 A 为锐角 A 为钝角或直角 图形 关系式 解的个 数 a=bsin A 一解 bsin A<a<b 两解 a≥b 一解 a>b 一解 3.三角形常用面积公式 1 (1)S=2a·ha(ha 表示边 a 上的高); 1 1 1 (2)S=2absin C=2acsin__B=2bcsin__A; 1 (3)S=2r(a+b+c)(r 为三角形内切圆半径). 考点一 利用正弦定理、余弦定理解三角形 【例 1】 (2018· 连云港、徐州、宿迁调研)如图,在△ABC 中,已知点 D 在边 AB 4 5 上,AD=3DB,cos A=5,cos∠ACB=13,BC=13. (1)求 cos B 的值; (2)求 CD 的长. 4 解 (1)在△ABC 中,cos A=5,A∈(0,π ), 所以 sin A= 1-cos2A= 12 同理可得 sin∠ACB=13. 所以 cos B=cos[π -(A+∠ACB)]=-cos(A+∠ACB) =sin Asin ∠ACB-cos Acos∠ACB ?4?2 3 1-?5? =5. ? ? 3 12 4 5 16 =5×13-5×13=65. BC 13

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