题目内容
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= CG
GF=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
阅读快车系列答案
移到了点
．
B．Q＝30＋
D．Q＝30＋
B．
C．
D．
，则菱形ACEF的面积为 ．
与新函数的图象交于点C（1，a），点D是线段AC上一动点（不包括端点），过点D作x轴的平行线，与新函数图象交于另一点E，与双曲线交于点P．
