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15hdav. com:正弦余弦定理应用举例教学课件PPT_图文

例6 一艘海轮从A出发,沿北偏东75°的方向航行67.5n mile 后到达海岛B,然后从B出发,沿北偏东32°的方向航行54.0n mile后到达海岛C.如果下次航行直接从A出发到达C,此船应该 沿怎样的方向航行,需要航行多少距离(角度精确到0.1°,距 离精确到0.01n mile)? 解:在⊿ABC中,∠ABC= 180°-75°+32°=137°, 根据余弦定理, AC ? ? AB 2 2 ? BC 2 2 ? 2 AB ? BC cos ? ABC ? 67 . 5 ? 54 . 0 ? 2 ? 67 . 5 ? 54 . 0 cos 137 ? 113 . 15 根据正弦定理, BC sin ? CAB ? AC sin ? ABC BC sin ? ABC AC ? sin ? CAB ? ? 54 . 0 sin 137 113 . 15 ? 0 . 3255 , 所以,∠CAB=19.0°, 75°-∠CAB=56.0°. 答:此船应该沿北偏东56.0°的方向航行,需要航行113.15n mile. 例7 在⊿ABC中,根据下列条件,求三角形的面积S(精确到 0.1cm? ) (1)已知a=14.8cm,c=23.5cm,B=148.5°; 1 2 解: (1 ) 应用 S ? S ? 1 2 ca sin B , 得 ? 2 ? 23 . 5 ? 14 . 8 ? sin 148 . 5 ? 90 . 9 ( cm ) (2)已知B=62.7°,C=65.8°,b=3.16cm; ( 2 ) 根据正弦定理, S ? 1 2 bc sin A ? ? b sin B b 2 ? c sin C , ,c ? b sin C sin B , 1 2 sin C sin A sin B ? A ? 180 S ? 1 2 ? ( B ? C ) ? 180 ? 2 ? ( 62 . 7 ? 65 . 8 ) ? 51 . 5 , ? ? ? ? ? 3 . 16 ? sin 65 . 8 sin 51 . 5 sin 62 . 7 ? ? 4 . 0 ( cm ). 2 (3)已知三边的长分别为a=41.4cm,b=27.3cm,c=38.7cm. ( 3)根据余弦定理的推论 cos B ? sin B ? 应用 S ? S ? 1 2 c ? a ?b 2 ca 1 ? cos 1 2 2 2 2 2 2 ,得 38 . 7 ? 41 . 4 ? 27 . 3 2 ? 38 . 7 ? 41 . 4 1 ? 0 . 7697 2 2 2 ? ? 0 . 7679 B ? ? 0 . 6384 ca sin B , 得 2 ? 38 . 7 ? 41 . 4 ? 0 . 6384 ? 511 . 4 ( cm ). 例8 在某市进行城市环境建设中,要把一个三角形的区域改造成 市内公园,经过测量得到这个三角形区域的三条边长分别为 68m,88m,127m,这个区域的面积是多少(精确到0.1cm? )? 解:设a=68m,b=88m,c=127m,根据余弦定理的推论, cos B ? sin B ? 应用 S ? S ? 1 2 c ? a ?b 2 ca 1 ? 0 . 7532 1 2 2 2 2 ? 2 127 2 ? 68 2 ? 88 2 2 ? 127 ? 68 ? 0 . 6578 . ? 0 . 7532 , ca sin B , 得 2 ? 127 ? 68 ? 0 . 6578 ? 2840 . 38 ( m ). 2840 . 38 m . 2 答:这个区域的面积是 例9 2 在 ? ABC 中,求证: 2 a ?b ( 1) 2 c 2 ? 2 sin 2 2 A ? sin sin 2 2 B ; C ( 2) a ? b ? c ? 2 ( bc cos A ? ca cos B ? ab cos C ). 在任一? ABC 中,求证: a (sin B ? sin C ) ? b (sin C ? sin A ) ? c (sin A ? sin B ) ? 0 在⊿ABC中,若B=60°,2b=a+c,试 判断⊿ABC的形状。

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