如图.我南海某海域A处有一艘捕鱼船在作业时突遇特大风浪.船长马上向我国渔政搜救中心发出求救信号.此时一艘渔政船正巡航到捕鱼船正西方向的B处.该渔政船收到渔政求救中心指令后前去救援.但两船之间有大片暗礁.无法直线到达.于是决定马上调整方向.先向北偏东60 º方向以每小时30海里的速度航行半小时到达C处.同时捕鱼船� 题目和参考答案——青夏教育精英家教网—

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如图，我南海某海域A处有一艘捕鱼船在作业时突遇特大风浪，船长马上向我国渔政搜救中心发出求救信号，此时一艘渔政船正巡航到捕鱼船正西方向的B处，该渔政船收到渔政求救中心指令后前去救援，但两船之间有大片暗礁，无法直线到达，于是决定马上调整方向，先向北偏东60 º方向以每小时30海里的速度航行半小时到达C处，同时捕鱼船低速航行到A点的正北1.5海里D处，渔政船航行到点C处时测得点D在南偏东53 º方向上．

(1)求CD两点的距离；

(2)渔政船决定再次调整航向前去救援，若两船航速不变，并且在点E处相会合，求∠ECD的正弦值． (参考数据：，，)

．解：(1)如图，过点C作CG⊥AB于点G，DF⊥CG于点F，··················································· 1分

则在Rt△CBG中，由题意知∠CBG=30°，

∴CG=BC==7.5， ···························································································· 2分

∵∠DAG=90°，

∴四边形ADFG是矩形，

∴GF= AD=1.5 ，

∴CF= CGGF=7.5－1.5=6，·································· 4分

在Rt△CDF中，∠CFD=90º，

∵∠DCF =53°，

∴cos∠DCF= ，

∴(海里)． ·························································································· 7分

答：CD两点距离为10海里． ······························································································· 8分

（2）如图，设渔政船调整方向后t小时能与捕渔船相会合，

由题意知CE=30t，DE=1.5×2×t=3t， ∠EDC=53°， ······························································· 9分

过点E作EH⊥CD于点H，则∠EHD=∠CHE=90º，

∴sin∠EDH=，

∴EH=EDsin53°=，····························································································· 11分

∴在Rt△EHC中，sin∠ECD=．

答：sin∠ECD=．············································································································· 12分

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