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第二十六章　解直角三角形
26. 4 解直角三角形的应用
第1课时　俯仰角、方向角
INCLUDEPICTURE"复习回顾.tif" INCLUDEPICTURE "D:\\课件\\九数JJ河北（北京）\\复习回顾.tif" \* MERGEFORMATINET INCLUDEPICTURE "D:\\课件\\九数JJ河北（北京）\\复习回顾.tif" \* MERGEFORMATINET
1．如图，下列角中为俯角的是(　　)
A．∠1 B．∠2 C．∠3 D．∠4
INCLUDEPICTURE"image3385.tif" INCLUDEPICTURE "D:\\课件\\九数JJ河北（北京）\\image3385.tif" \* MERGEFORMATINET INCLUDEPICTURE "D:\\课件\\九数JJ河北（北京）\\image3385.tif" \* MERGEFORMATINET (第1题)　　　INCLUDEPICTURE"image3386.tif" INCLUDEPICTURE "D:\\课件\\九数JJ河北（北京）\\image3386.tif" \* MERGEFORMATINET INCLUDEPICTURE "D:\\课件\\九数JJ河北（北京）\\image3386.tif" \* MERGEFORMATINET (第2题)
2．直角梯形ABCD按如图所示方式放置，AB，CD为水平线，BC⊥AB，如果∠BCA＝67°，那么从A处看向C处，点C在点A的(　　)
A．俯角67°方向 B．俯角23°方向 C．仰角67°方向 D．仰角23°方向
3．如图所示，下列说法错误的是(　　)
INCLUDEPICTURE"image3387.tif" INCLUDEPICTURE "D:\\课件\\九数JJ河北（北京）\\image3387.tif" \* MERGEFORMATINET INCLUDEPICTURE "D:\\课件\\九数JJ河北（北京）\\image3387.tif" \* MERGEFORMATINET
A．图①中的方向角是南偏西20° B．图②中的方向角是北偏西30°
C．图③中的方向角是东北方向 D．图④中的方向角是南偏西45°
INCLUDEPICTURE"预习效果检测.tif" INCLUDEPICTURE "D:\\课件\\九数JJ河北（北京）\\预习效果检测.tif" \* MERGEFORMATINET INCLUDEPICTURE "D:\\课件\\九数JJ河北（北京）\\预习效果检测.tif" \* MERGEFORMATINET
1．如图，在距旗杆15 m的A处，用测角仪测得旗杆顶点C的仰角为30°，已知测角仪AB的高为1 m，则旗杆CD的高为______m.
INCLUDEPICTURE"image3392.tif" INCLUDEPICTURE "D:\\课件\\九数JJ河北（北京）\\image3392.tif" \* MERGEFORMATINET INCLUDEPICTURE "D:\\课件\\九数JJ河北（北京）\\image3392.tif" \* MERGEFORMATINET
2．[2023大连期中]如图，渔船在A处看到灯塔C在北偏东60°方向上，渔船向正东方向航行了12 km到达B处，在B处看到灯塔C在正北方向上，则A处与灯塔C的距离是__________．
INCLUDEPICTURE"image3397.tif" INCLUDEPICTURE "D:\\课件\\九数JJ河北（北京）\\image3397.tif" \* MERGEFORMATINET INCLUDEPICTURE "D:\\课件\\九数JJ河北（北京）\\image3397.tif" \* MERGEFORMATINET
INCLUDEPICTURE"课堂导学.tif" INCLUDEPICTURE "D:\\课件\\九数JJ河北（北京）\\课堂导学.tif" \* MERGEFORMATINET INCLUDEPICTURE "D:\\课件\\九数JJ河北（北京）\\课堂导学.tif" \* MERGEFORMATINET
INCLUDEPICTURE"W21.tif" INCLUDEPICTURE "D:\\课件\\九数JJ河北（北京）\\W21.tif" \* MERGEFORMATINET INCLUDEPICTURE "D:\\课件\\九数JJ河北（北京）\\W21.tif" \* MERGEFORMATINET
知识点1 仰角、俯角
INCLUDEPICTURE"L1.tif" INCLUDEPICTURE "D:\\课件\\九数JJ河北（北京）\\L1.tif" \* MERGEFORMATINET INCLUDEPICTURE "D:\\课件\\九数JJ河北（北京）\\L1.tif" \* MERGEFORMATINET [2023岳阳中考][2023年岳阳举办以“跃马江湖”为主题的马拉松赛事．如图，某校数学兴趣小组在A处用仪器测得赛场一宣传气球顶部E处的仰角为21.8°，仪器与气球的水平距离BC为20米，且距地面高度AB为1.5米，则气球顶部离地面的高度EC约是________米(结果精确到0.1米．参考数据：sin 21.8°≈0.371 4，cos 21.8°≈0.928 5，tan 21.8°≈0.400 0)．
INCLUDEPICTURE"image3398.tif" INCLUDEPICTURE "D:\\课件\\九数JJ河北（北京）\\image3398.tif" \* MERGEFORMATINET INCLUDEPICTURE "D:\\课件\\九数JJ河北（北京）\\image3398.tif" \* MERGEFORMATINET
变式1如图，某测量队采用无人机技术测量无法直达的A，B两处的直线距离，已知在无人机的镜头O处测得A，B两处的俯角分别为45°和50°，无人机的飞行高度OC为238米，点A，B，C在同一直线上，求AB的长度(结果保留整数，参考数据：sin 50°≈0.77，cos 50°≈0.64，tan 50°≈1.19)．
INCLUDEPICTURE"image3409.tif" INCLUDEPICTURE "D:\\课件\\九数JJ河北（北京）\\image3409.tif" \* MERGEFORMATINET INCLUDEPICTURE "D:\\课件\\九数JJ河北（北京）\\image3409.tif" \* MERGEFORMATINET
　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　
知识点2 方向角INCLUDEPICTURE"L2.tif" INCLUDEPICTURE "D:\\课件\\九数JJ河北（北京）\\L2.tif" \* MERGEFORMATINET INCLUDEPICTURE "D:\\课件\\九数JJ河北（北京）\\L2.tif" \* MERGEFORMATINET 如图，一艘轮船在A处测得灯塔P位于其北偏东60°方向上，轮船沿正东方向航行30海里到达B处后，测得灯塔P位于其北偏东30°方向上，此时轮船与灯塔P的距离是(　　)
INCLUDEPICTURE"image3410.tif" INCLUDEPICTURE "D:\\课件\\九数JJ河北（北京）\\image3410.tif" \* MERGEFORMATINET INCLUDEPICTURE "D:\\课件\\九数JJ河北（北京）\\image3410.tif" \* MERGEFORMATINET
A．15 海里 B．30海里 C．45海里 D．30 海里
变式2如图，一艘海轮在位于灯塔P的北偏东64°方向上，距离灯塔200海里的A岛，它沿正南方向航行一段时间后，到达位于灯塔P的东南方向上的B岛，求A，B两岛之间的距离．(结果保留整数．参考数据：sin 64°≈0.899，cos 64°≈0.438，tan 64°≈2.050)
INCLUDEPICTURE"image3418.tif" INCLUDEPICTURE "D:\\课件\\九数JJ河北（北京）\\image3418.tif" \* MERGEFORMATINET INCLUDEPICTURE "D:\\课件\\九数JJ河北（北京）\\image3418.tif" \* MERGEFORMATINET
第2课时　坡度、坡角及其他问题
INCLUDEPICTURE"复习回顾.tif" INCLUDEPICTURE "D:\\课件\\九数JJ河北（北京）\\复习回顾.tif" \* MERGEFORMATINET INCLUDEPICTURE "D:\\课件\\九数JJ河北（北京）\\复习回顾.tif" \* MERGEFORMATINET
1．[2023济南期中]如图，电线杆AB直立在水平的地面BC上，AC是电线杆AB的一根拉线，测得BC＝5，∠ACB＝52°，则拉线AC的长为(　　)
INCLUDEPICTURE"image3423.tif" INCLUDEPICTURE "D:\\课件\\九数JJ河北（北京）\\image3423.tif" \* MERGEFORMATINET INCLUDEPICTURE "D:\\课件\\九数JJ河北（北京）\\image3423.tif" \* MERGEFORMATINET
A. B. C．5·cos 52° D.
2．[2023杭州期中]小王是一名摄影爱好者，新入手一台无人机用于航拍．在一次航拍时，数据显示，从无人机A看建筑物顶部B的仰角为45°，看底部C的俯角为60°，无人机A到该建筑物BC的水平距离AD为10米，如图所示，求该建筑物BC的高度．(结果保留根号)
INCLUDEPICTURE"image3434.tif" INCLUDEPICTURE "D:\\课件\\九数JJ河北（北京）\\image3434.tif" \* MERGEFORMATINET INCLUDEPICTURE "D:\\课件\\九数JJ河北（北京）\\image3434.tif" \* MERGEFORMATINET
INCLUDEPICTURE"预习效果检测.tif" INCLUDEPICTURE "D:\\课件\\九数JJ河北（北京）\\预习效果检测.tif" \* MERGEFORMATINET INCLUDEPICTURE "D:\\课件\\九数JJ河北（北京）\\预习效果检测.tif" \* MERGEFORMATINET
1．我们通常把坡面的________h和________l的比叫做坡面的坡度(或坡比)，坡面与水平面的夹角α叫做__________．显然，tan α＝.
2．如图所示．
INCLUDEPICTURE"image3436.tif" INCLUDEPICTURE "D:\\课件\\九数JJ河北（北京）\\image3436.tif" \* MERGEFORMATINET INCLUDEPICTURE "D:\\课件\\九数JJ河北（北京）\\image3436.tif" \* MERGEFORMATINET
(第2题)　
(1)若斜坡的坡度是1∶，则坡角α＝__________；
(2)若斜坡的坡角是45°，则坡比是__________；
(3)若斜坡长是12米，坡高6米，则坡比是________；
(4)特别注意，坡度不是一个度数，而是一个________，是坡角的__________值．
3．[2023泉州期中]如图，在一坡度i＝1∶的斜面上，一木箱沿斜面向上推进了10米，则木箱升高了________米．
INCLUDEPICTURE"image3439.tif" INCLUDEPICTURE "D:\\课件\\九数JJ河北（北京）\\image3439.tif" \* MERGEFORMATINET INCLUDEPICTURE "D:\\课件\\九数JJ河北（北京）\\image3439.tif" \* MERGEFORMATINET
(第3题)　
INCLUDEPICTURE"课堂导学.tif" INCLUDEPICTURE "D:\\课件\\九数JJ河北（北京）\\课堂导学.tif" \* MERGEFORMATINET INCLUDEPICTURE "D:\\课件\\九数JJ河北（北京）\\课堂导学.tif" \* MERGEFORMATINET
INCLUDEPICTURE"W21.tif" INCLUDEPICTURE "D:\\课件\\九数JJ河北（北京）\\W21.tif" \* MERGEFORMATINET INCLUDEPICTURE "D:\\课件\\九数JJ河北（北京）\\W21.tif" \* MERGEFORMATINET
知识点1 坡度和坡角
INCLUDEPICTURE"L1.tif" INCLUDEPICTURE "D:\\课件\\九数JJ河北（北京）\\L1.tif" \* MERGEFORMATINET INCLUDEPICTURE "D:\\课件\\九数JJ河北（北京）\\L1.tif" \* MERGEFORMATINET 如图，水库大坝的横断面是梯形，坝顶CD的宽是8米，坝高(DE，CF)为30米，斜坡AD的坡度为∶3，斜坡CB的坡度为3∶2，则坝底AB的宽为________米．
INCLUDEPICTURE"image3452.tif" INCLUDEPICTURE "D:\\课件\\九数JJ河北（北京）\\image3452.tif" \* MERGEFORMATINET INCLUDEPICTURE "D:\\课件\\九数JJ河北（北京）\\image3452.tif" \* MERGEFORMATINET
变式1[2023沈阳期中]如图，与斜坡CE垂直的太阳光线照射立柱AB(与水平地面BF垂直)形成的影子，一部分落在地面上，另一部分落在斜坡上，若BC＝2米，CD＝8米，斜坡的坡角∠ECF＝30°.请解决下列问题，如果结果有根号请保留根号．
(1)求点D到地面的距离；
(2)求立柱AB的高为多少米．
INCLUDEPICTURE"image3457.tif" INCLUDEPICTURE "D:\\课件\\九数JJ河北（北京）\\image3457.tif" \* MERGEFORMATINET INCLUDEPICTURE "D:\\课件\\九数JJ河北（北京）\\image3457.tif" \* MERGEFORMATINET
　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　
知识点2　 其他问题
INCLUDEPICTURE"L2.tif" INCLUDEPICTURE "D:\\课件\\九数JJ河北（北京）\\L2.tif" \* MERGEFORMATINET INCLUDEPICTURE "D:\\课件\\九数JJ河北（北京）\\L2.tif" \* MERGEFORMATINET 如图所示的是某高速公路边水平地面上的交通警示牌，经测量得到如下数据：AB＝8 m，BC＝4 m，∠DAC＝30°，∠EBC＝45°，∠DCA＝90°，则警示牌DE的高度约为________．(结果精确到0.1 m．参考数据：≈1.41，≈1.73)
INCLUDEPICTURE"image3467.tif" INCLUDEPICTURE "D:\\课件\\九数JJ河北（北京）\\image3467.tif" \* MERGEFORMATINET INCLUDEPICTURE "D:\\课件\\九数JJ河北（北京）\\image3467.tif" \* MERGEFORMATINET
变式2[2023济南期中]如图，某一时刻太阳光从教室窗户射入房间内，与地面的夹角∠DPC＝30°，已知窗户的高度AF＝2 m，窗台的高度CF＝1 m，CP＝ m，AD为窗外水平遮阳篷．
(1)求BC的长；
(2)求遮阳篷AD的宽度(≈1.732，结果精确到0.1 m)．
INCLUDEPICTURE"image3472.tif" INCLUDEPICTURE "D:\\课件\\九数JJ河北（北京）\\image3472.tif" \* MERGEFORMATINET INCLUDEPICTURE "D:\\课件\\九数JJ河北（北京）\\image3472.tif" \* MERGEFORMATINET
26. 4 解直角三角形的应用
第1课时　俯仰角、方向角
INCLUDEPICTURE"复习回顾.tif" INCLUDEPICTURE "D:\\课件\\九数JJ河北（北京）\\复习回顾.tif" \* MERGEFORMATINET INCLUDEPICTURE "D:\\课件\\九数JJ河北（北京）\\复习回顾.tif" \* MERGEFORMATINET
1．C　2.D　3.B
INCLUDEPICTURE"预习效果.tif" INCLUDEPICTURE "预习效果.tif" \* MERGEFORMAT INCLUDEPICTURE "D:\\课件\\九数JJ河北（北京）\\预习效果.tif" \* MERGEFORMATINET INCLUDEPICTURE "D:\\课件\\九数JJ河北（北京）\\预习效果.tif" \* MERGEFORMATINET INCLUDEPICTURE "D:\\课件\\九数JJ河北（北京）\\预习效果.tif" \* MERGEFORMATINET
1．16　2.8 km
INCLUDEPICTURE"课堂导学.tif" INCLUDEPICTURE "课堂导学.tif" \* MERGEFORMAT INCLUDEPICTURE "D:\\课件\\九数JJ河北（北京）\\课堂导学.tif" \* MERGEFORMATINET INCLUDEPICTURE "D:\\课件\\九数JJ河北（北京）\\课堂导学.tif" \* MERGEFORMATINET INCLUDEPICTURE "D:\\课件\\九数JJ河北（北京）\\课堂导学.tif" \* MERGEFORMATINET
例1　9.5
变式1.解：由题意可得∠OAC＝45°，∠OBC＝50°，∠ACO＝∠BCO＝90°.
∵在Rt△AOC中，∠OAC＝45°，
∴∠AOC＝45°，∴AC＝OC＝238米．
∵在Rt△OBC中，∠OBC＝50°，tan ∠OBC＝，
∴tan 50°＝≈1.19，∴BC≈200米，
∴AB＝AC＋BC≈238＋200＝438(米)．
∴AB的长度约为438米．
例2　B
变式2.解：过点P作PC⊥AB于点C.由题意，得∠A＝64°，∠B＝45°.
∵在Rt△APC中，∠ACP＝90°，∠A＝64°，
sin A＝，cos A＝，
∴PC＝PA·sin A≈200×0.899＝179.8(海里)，
AC＝PA·cos A≈200×0.438＝87.6(海里)．
∵在Rt△BPC中，∠BCP＝90°，∠B＝45°，
∴BC＝PC≈179.8(海里)，
∴AB＝AC＋BC≈87.6＋179.8＝267.4≈267(海里)，
∴A，B两岛之间的距离约为267海里．
第2课时　坡度、坡角及其他问题
INCLUDEPICTURE"复习回顾.tif" INCLUDEPICTURE "D:\\课件\\九数JJ河北（北京）\\复习回顾.tif" \* MERGEFORMATINET INCLUDEPICTURE "D:\\课件\\九数JJ河北（北京）\\复习回顾.tif" \* MERGEFORMATINET
1．B
2．解：由题意可知，∠BAD＝45°，∠CAD＝60°，AD⊥BC，
∴∠ABD＝∠BAD＝45°，∴BD＝AD＝10米．
∵在Rt△ACD中，tan ∠CAD＝，
即tan 60°＝＝，
∴CD＝10 米，∴BC＝CD＋BD＝(10 ＋10)米．
∴该建筑物BC的高度为(10 ＋10)米．
INCLUDEPICTURE"预习效果.tif" INCLUDEPICTURE "预习效果.tif" \* MERGEFORMAT INCLUDEPICTURE "D:\\课件\\九数JJ河北（北京）\\预习效果.tif" \* MERGEFORMATINET INCLUDEPICTURE "D:\\课件\\九数JJ河北（北京）\\预习效果.tif" \* MERGEFORMATINET INCLUDEPICTURE "D:\\课件\\九数JJ河北（北京）\\预习效果.tif" \* MERGEFORMATINET
1．竖直高度；水平宽度；坡角
2．(1)30°　(2)1∶1　(3)1∶　(4)比值；正切
3．5
INCLUDEPICTURE"课堂导学.tif" INCLUDEPICTURE "课堂导学.tif" \* MERGEFORMAT INCLUDEPICTURE "D:\\课件\\九数JJ河北（北京）\\课堂导学.tif" \* MERGEFORMATINET INCLUDEPICTURE "D:\\课件\\九数JJ河北（北京）\\课堂导学.tif" \* MERGEFORMATINET INCLUDEPICTURE "D:\\课件\\九数JJ河北（北京）\\课堂导学.tif" \* MERGEFORMATINET
例1　(30 ＋28)
变式1.解：(1)如图，过点D作DH⊥BF于点H，则DH的长即为点D到地面的距离．
∵在Rt△CDH中，∠DHC＝90°，
∴DH＝CD·sin ∠DCH＝CD·sin 30°＝8×＝4(米)．
答：点D到地面的距离是4米．
(2)如图，延长AD交BF于点G.
∵在Rt△CDG中，∠CDG＝90°，
∴CG＝＝＝＝(米)，
∴BG＝CG＋BC＝米．
∵在Rt△CDG中，∠CDG＝90°，∠DCG＝30°，
∴∠DGC＝90°－∠DCG＝90°－30°＝60°.
∵在Rt△ABG中，∠ABG＝90°，
∴AB＝BG·tan ∠AGB＝·＝16＋2 (米)．
INCLUDEPICTURE"image3843.tif" INCLUDEPICTURE "image3843.tif" \* MERGEFORMAT INCLUDEPICTURE "D:\\课件\\九数JJ河北（北京）\\image3843.tif" \* MERGEFORMATINET INCLUDEPICTURE "D:\\课件\\九数JJ河北（北京）\\image3843.tif" \* MERGEFORMATINET INCLUDEPICTURE "D:\\课件\\九数JJ河北（北京）\\image3843.tif" \* MERGEFORMATINET
例2　2.9 m
变式2.解：(1)在Rt△BCP中，∵PC＝ m，∠BPC＝30°，
∴BC＝PC·tan 30°＝×＝(m)．
(2)∵AF＝2 m，CF＝1 m，∴AC＝3 m.
∵BC＝ m，∴AB＝3－＝(m)．
易知AD∥PC，∴∠ADB＝∠BPC＝30°.
在Rt△ADB中， ∵tan ∠ADB＝tan 30°＝＝，
∴AD＝ m≈0.9 m.

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