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www.hg8074.com:2019年一轮北师大版(理)数学教案:第3章 第7节 三角形中的几何计算、解三角形的实际应用举例

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第七节 三角形中的几何计算、解三角形的 实际应用举例 [考纲传真] 能够运用正弦定理、余弦定理等知识和方法解决一些与测量和 几何计算有关的实际问题. 1.仰角和俯角 在同一铅垂平面内的水平视线和目标视线的夹角, 目标视线在水平视线上方 时叫仰角,目标视线在水平视线下方时叫俯角.(如图 371①). ① 图 371 2.方位角和方向角 ② (1)方位角:从指北方向顺时针转到目标方向线的水平角,如 B 点的方位角 为 α(如图 371②). (2)方向角:相对于某正方向的水平角,如南偏东 30° 等. 1.(思考辨析)判断下列结论的正误.(正确的打“√”,错误的打“×”) (1)从 A 处望 B 处的仰角为 α,从 B 处望 A 处的俯角为 β,则 α,β 的关系为 α+β=180° .( ) ) π? ? (2)俯角是铅垂线与视线所成的角,其范围为?0,2?.( ? ? (3)方位角与方向角其实质是一样的,均是确定观察点与目标点之间的位置 关系.( ) (4)如图 372,为了测量隧道口 AB 的长度,可测量数据 a,b,γ 进行计 算.( ) 图 372 [答案] (1)× (2)× (3)√ (4)√ 2.(教材改编)海面上有 A,B,C 三个灯塔,AB=10 n mile,从 A 望 C 和 B 成 60° 视角,从 B 望 C 和 A 成 75° 视角,则 BC 等于( ) 【导学号:57962179】 A.10 3 n mile C.5 2 n mile D [如图,在△ABC 中, AB=10,∠A=60° , ∠B=75° ,∠C=45° , BC 10 ∴sin 60° =sin 45° , ∴BC=5 6.] 3.若点 A 在点 C 的北偏东 30° ,点 B 在点 C 的南偏东 60° ,且 AC=BC, 则点 A 在点 B 的( ) 【导学号:57962180】 A.北偏东 15° C.北偏东 10° B B.北偏西 15° D.北偏西 10° 10 6 B. 3 n mile D.5 6 n mile [如图所示,∠ACB=90° ,又 AC=BC,∴∠CBA=45° ,而 β=30° , ∴α=90° -45° -30° =15° ,∴点 A 在点 B 的北偏西 15° .] 4. 如图 373, 要测量底部不能到达的电视塔的高度, 选择甲、 乙两观测点. 在 甲、乙两点测得塔顶的仰角分别为 45° ,30° ,在水平面上测得电视塔与甲地连线 及甲、乙两地连线所成的角为 120° ,甲、乙两地相距 500 m,则电视塔的高度是 ( ) 图 373 【导学号:57962181】 A.100 2 m C.200 3 m B.400 m D.500 m D [设塔高为 x m,则由已知可得 BC=x m,BD= 3x m,由余弦定理可得 BD2=BC2+CD2-2BC· CDcos ∠BCD, 即 3x2=x2+5002+500x, 解得 x=500(m). ] 5.如图 374,已知 A,B 两点分别在河的两岸,某测量者在点 A 所在的河 岸边另选定一点 C,测得 AC=50 m,∠ACB=45° ,∠CAB=105° ,则 A,B 两 点的距离为( ) 图 374 A.50 3 m B.25 3 m C.25 2 m D.50 2 m AC D [因为∠ACB=45° , ∠CAB=105° , 所以∠B=30° .由正弦定理可知sin B= AB 50 AB ,即 = ,解得 AB=50 2 m.] sin C sin 30° sin 45° 测量距离问题 如图 375,从气球 A 上测得正前方的河流的两岸 B,C 的 俯角分别为 67° ,30° ,此时气球的高是 46 m,则河流的宽度 BC 约等于 ________m. (用四舍五入法将结果精确到个位. 参考数据: sin 67° ≈0.92, cos 67° ≈0.39,sin 37° ≈0.60,cos 37° ≈0.80, 3≈1.73) 图 375 60 [如图所示,过 A 作 AD⊥CB 且交 CB 的延长线于 D. 在 Rt△ADC 中,由 AD=46 m,∠ACB=30° 得 AC=92 m. 在△ABC 中,∠BAC=67° -30° =37° , ∠ABC=180° -67° =113° ,AC=92 m, 由正弦定理 AC BC = ,得 sin ∠ABC sin ∠BAC 92 BC 92 BC =sin 37° ,即sin 67° =sin 37° , sin 113° 92sin 37° 解得 BC= sin 67° ≈60(m).] [规律方法] 应用解三角形知识解决实际问题需要下列三步: (1)根据题意,画出示意图,并标出条件; (2)将所求问题归结到一个或几个三角形中(如本例借助方位角构建三角形), 通过合理运用正、余弦定理等有关知识正确求解; (3)检验解出的结果是否符合实际意义,得出正确答案. [变式训练 1] 江岸边有一炮台高 30 m,江中有两条船,船与炮台底部在同 一水平面上,由炮台顶部测得俯角分别为 45° 和 60° ,而且两条船与炮台底部连 线成 30° 角,则两条船相距________m. 【导学号:57962182】 10 3 [如图,OM=AOtan 45° =30(m), 3 ON=AOtan 30° = 3 ×30=10 3(m), 在△MON 中,由余弦定理得, MN= 3 900+300-2×30×10 3× 2 = 300=10 3(m).] 测量高度问题 (2015· 湖北高考)如图 376,一辆汽车在一条水平的公路上 向正西行驶,到 A 处时测得公路北侧一山顶 D 在西偏北 30° 的方向上, 行驶 600 m 后到达 B 处, 测得此山顶在西偏北 75° 的方向上, 仰角为 30° , 则此

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