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第二十六章　解直角三角形
26. 4 解直角三角形的應用
第1課時　俯仰角、方向角
INCLUDEPICTURE"復習回顧.tif" INCLUDEPICTURE "D:\\課件\\九數JJ河北（北京）\\復習回顧.tif" \* MERGEFORMATINET INCLUDEPICTURE "D:\\課件\\九數JJ河北（北京）\\復習回顧.tif" \* MERGEFORMATINET
1．如圖，下列角中為俯角的是(　　)
A．∠1 B．∠2 C．∠3 D．∠4
INCLUDEPICTURE"image3385.tif" INCLUDEPICTURE "D:\\課件\\九數JJ河北（北京）\\image3385.tif" \* MERGEFORMATINET INCLUDEPICTURE "D:\\課件\\九數JJ河北（北京）\\image3385.tif" \* MERGEFORMATINET (第1題)　　　INCLUDEPICTURE"image3386.tif" INCLUDEPICTURE "D:\\課件\\九數JJ河北（北京）\\image3386.tif" \* MERGEFORMATINET INCLUDEPICTURE "D:\\課件\\九數JJ河北（北京）\\image3386.tif" \* MERGEFORMATINET (第2題)
2．直角梯形ABCD按如圖所示方式放置，AB，CD為水平線，BC⊥AB，如果∠BCA＝67°，那么從A處看向C處，點C在點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．如圖所示，下列說法錯誤的是(　　)
INCLUDEPICTURE"image3387.tif" INCLUDEPICTURE "D:\\課件\\九數JJ河北（北京）\\image3387.tif" \* MERGEFORMATINET INCLUDEPICTURE "D:\\課件\\九數JJ河北（北京）\\image3387.tif" \* MERGEFORMATINET
A．圖①中的方向角是南偏西20° B．圖②中的方向角是北偏西30°
C．圖③中的方向角是東北方向 D．圖④中的方向角是南偏西45°
INCLUDEPICTURE"預習效果檢測.tif" INCLUDEPICTURE "D:\\課件\\九數JJ河北（北京）\\預習效果檢測.tif" \* MERGEFORMATINET INCLUDEPICTURE "D:\\課件\\九數JJ河北（北京）\\預習效果檢測.tif" \* MERGEFORMATINET
1．如圖，在距旗桿15 m的A處，用測角儀測得旗桿頂點C的仰角為30°，已知測角儀AB的高為1 m，則旗桿CD的高為______m.
INCLUDEPICTURE"image3392.tif" INCLUDEPICTURE "D:\\課件\\九數JJ河北（北京）\\image3392.tif" \* MERGEFORMATINET INCLUDEPICTURE "D:\\課件\\九數JJ河北（北京）\\image3392.tif" \* MERGEFORMATINET
2．[2023大連期中]如圖，漁船在A處看到燈塔C在北偏東60°方向上，漁船向正東方向航行了12 km到達B處，在B處看到燈塔C在正北方向上，則A處與燈塔C的距離是__________．
INCLUDEPICTURE"image3397.tif" INCLUDEPICTURE "D:\\課件\\九數JJ河北（北京）\\image3397.tif" \* MERGEFORMATINET INCLUDEPICTURE "D:\\課件\\九數JJ河北（北京）\\image3397.tif" \* MERGEFORMATINET
INCLUDEPICTURE"課堂導學.tif" INCLUDEPICTURE "D:\\課件\\九數JJ河北（北京）\\課堂導學.tif" \* MERGEFORMATINET INCLUDEPICTURE "D:\\課件\\九數JJ河北（北京）\\課堂導學.tif" \* MERGEFORMATINET
INCLUDEPICTURE"W21.tif" INCLUDEPICTURE "D:\\課件\\九數JJ河北（北京）\\W21.tif" \* MERGEFORMATINET INCLUDEPICTURE "D:\\課件\\九數JJ河北（北京）\\W21.tif" \* MERGEFORMATINET
知識點1 仰角、俯角
INCLUDEPICTURE"L1.tif" INCLUDEPICTURE "D:\\課件\\九數JJ河北（北京）\\L1.tif" \* MERGEFORMATINET INCLUDEPICTURE "D:\\課件\\九數JJ河北（北京）\\L1.tif" \* MERGEFORMATINET [2023岳陽中考][2023年岳陽舉辦以“躍馬江湖”為主題的馬拉松賽事．如圖，某校數學興趣小組在A處用儀器測得賽場一宣傳氣球頂部E處的仰角為21.8°，儀器與氣球的水平距離BC為20米，且距地面高度AB為1.5米，則氣球頂部離地面的高度EC約是________米(結果精確到0.1米．參考數據：sin 21.8°≈0.371 4，cos 21.8°≈0.928 5，tan 21.8°≈0.400 0)．
INCLUDEPICTURE"image3398.tif" INCLUDEPICTURE "D:\\課件\\九數JJ河北（北京）\\image3398.tif" \* MERGEFORMATINET INCLUDEPICTURE "D:\\課件\\九數JJ河北（北京）\\image3398.tif" \* MERGEFORMATINET
變式1如圖，某測量隊采用無人機技術測量無法直達的A，B兩處的直線距離，已知在無人機的鏡頭O處測得A，B兩處的俯角分別為45°和50°，無人機的飛行高度OC為238米，點A，B，C在同一直線上，求AB的長度(結果保留整數，參考數據：sin 50°≈0.77，cos 50°≈0.64，tan 50°≈1.19)．
INCLUDEPICTURE"image3409.tif" INCLUDEPICTURE "D:\\課件\\九數JJ河北（北京）\\image3409.tif" \* MERGEFORMATINET INCLUDEPICTURE "D:\\課件\\九數JJ河北（北京）\\image3409.tif" \* MERGEFORMATINET
　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　
知識點2 方向角INCLUDEPICTURE"L2.tif" INCLUDEPICTURE "D:\\課件\\九數JJ河北（北京）\\L2.tif" \* MERGEFORMATINET INCLUDEPICTURE "D:\\課件\\九數JJ河北（北京）\\L2.tif" \* MERGEFORMATINET 如圖，一艘輪船在A處測得燈塔P位于其北偏東60°方向上，輪船沿正東方向航行30海里到達B處后，測得燈塔P位于其北偏東30°方向上，此時輪船與燈塔P的距離是(　　)
INCLUDEPICTURE"image3410.tif" INCLUDEPICTURE "D:\\課件\\九數JJ河北（北京）\\image3410.tif" \* MERGEFORMATINET INCLUDEPICTURE "D:\\課件\\九數JJ河北（北京）\\image3410.tif" \* MERGEFORMATINET
A．15 海里 B．30海里 C．45海里 D．30 海里
變式2如圖，一艘海輪在位于燈塔P的北偏東64°方向上，距離燈塔200海里的A島，它沿正南方向航行一段時間后，到達位于燈塔P的東南方向上的B島，求A，B兩島之間的距離．(結果保留整數．參考數據：sin 64°≈0.899，cos 64°≈0.438，tan 64°≈2.050)
INCLUDEPICTURE"image3418.tif" INCLUDEPICTURE "D:\\課件\\九數JJ河北（北京）\\image3418.tif" \* MERGEFORMATINET INCLUDEPICTURE "D:\\課件\\九數JJ河北（北京）\\image3418.tif" \* MERGEFORMATINET
第2課時　坡度、坡角及其他問題
INCLUDEPICTURE"復習回顧.tif" INCLUDEPICTURE "D:\\課件\\九數JJ河北（北京）\\復習回顧.tif" \* MERGEFORMATINET INCLUDEPICTURE "D:\\課件\\九數JJ河北（北京）\\復習回顧.tif" \* MERGEFORMATINET
1．[2023濟南期中]如圖，電線桿AB直立在水平的地面BC上，AC是電線桿AB的一根拉線，測得BC＝5，∠ACB＝52°，則拉線AC的長為(　　)
INCLUDEPICTURE"image3423.tif" INCLUDEPICTURE "D:\\課件\\九數JJ河北（北京）\\image3423.tif" \* MERGEFORMATINET INCLUDEPICTURE "D:\\課件\\九數JJ河北（北京）\\image3423.tif" \* MERGEFORMATINET
A. B. C．5·cos 52° D.
2．[2023杭州期中]小王是一名攝影愛好者，新入手一臺無人機用于航拍．在一次航拍時，數據顯示，從無人機A看建筑物頂部B的仰角為45°，看底部C的俯角為60°，無人機A到該建筑物BC的水平距離AD為10米，如圖所示，求該建筑物BC的高度．(結果保留根號)
INCLUDEPICTURE"image3434.tif" INCLUDEPICTURE "D:\\課件\\九數JJ河北（北京）\\image3434.tif" \* MERGEFORMATINET INCLUDEPICTURE "D:\\課件\\九數JJ河北（北京）\\image3434.tif" \* MERGEFORMATINET
INCLUDEPICTURE"預習效果檢測.tif" INCLUDEPICTURE "D:\\課件\\九數JJ河北（北京）\\預習效果檢測.tif" \* MERGEFORMATINET INCLUDEPICTURE "D:\\課件\\九數JJ河北（北京）\\預習效果檢測.tif" \* MERGEFORMATINET
1．我們通常把坡面的________h和________l的比叫做坡面的坡度(或坡比)，坡面與水平面的夾角α叫做__________．顯然，tan α＝.
2．如圖所示．
INCLUDEPICTURE"image3436.tif" INCLUDEPICTURE "D:\\課件\\九數JJ河北（北京）\\image3436.tif" \* MERGEFORMATINET INCLUDEPICTURE "D:\\課件\\九數JJ河北（北京）\\image3436.tif" \* MERGEFORMATINET
(第2題)　
(1)若斜坡的坡度是1∶，則坡角α＝__________；
(2)若斜坡的坡角是45°，則坡比是__________；
(3)若斜坡長是12米，坡高6米，則坡比是________；
(4)特別注意，坡度不是一個度數，而是一個________，是坡角的__________值．
3．[2023泉州期中]如圖，在一坡度i＝1∶的斜面上，一木箱沿斜面向上推進了10米，則木箱升高了________米．
INCLUDEPICTURE"image3439.tif" INCLUDEPICTURE "D:\\課件\\九數JJ河北（北京）\\image3439.tif" \* MERGEFORMATINET INCLUDEPICTURE "D:\\課件\\九數JJ河北（北京）\\image3439.tif" \* MERGEFORMATINET
(第3題)　
INCLUDEPICTURE"課堂導學.tif" INCLUDEPICTURE "D:\\課件\\九數JJ河北（北京）\\課堂導學.tif" \* MERGEFORMATINET INCLUDEPICTURE "D:\\課件\\九數JJ河北（北京）\\課堂導學.tif" \* MERGEFORMATINET
INCLUDEPICTURE"W21.tif" INCLUDEPICTURE "D:\\課件\\九數JJ河北（北京）\\W21.tif" \* MERGEFORMATINET INCLUDEPICTURE "D:\\課件\\九數JJ河北（北京）\\W21.tif" \* MERGEFORMATINET
知識點1 坡度和坡角
INCLUDEPICTURE"L1.tif" INCLUDEPICTURE "D:\\課件\\九數JJ河北（北京）\\L1.tif" \* MERGEFORMATINET INCLUDEPICTURE "D:\\課件\\九數JJ河北（北京）\\L1.tif" \* MERGEFORMATINET 如圖，水庫大壩的橫斷面是梯形，壩頂CD的寬是8米，壩高(DE，CF)為30米，斜坡AD的坡度為∶3，斜坡CB的坡度為3∶2，則壩底AB的寬為________米．
INCLUDEPICTURE"image3452.tif" INCLUDEPICTURE "D:\\課件\\九數JJ河北（北京）\\image3452.tif" \* MERGEFORMATINET INCLUDEPICTURE "D:\\課件\\九數JJ河北（北京）\\image3452.tif" \* MERGEFORMATINET
變式1[2023沈陽期中]如圖，與斜坡CE垂直的太陽光線照射立柱AB(與水平地面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°.請解決下列問題，如果結果有根號請保留根號．
(1)求點D到地面的距離；
(2)求立柱AB的高為多少米．
INCLUDEPICTURE"image3457.tif" INCLUDEPICTURE "D:\\課件\\九數JJ河北（北京）\\image3457.tif" \* MERGEFORMATINET INCLUDEPICTURE "D:\\課件\\九數JJ河北（北京）\\image3457.tif" \* MERGEFORMATINET
　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　
知識點2　 其他問題
INCLUDEPICTURE"L2.tif" INCLUDEPICTURE "D:\\課件\\九數JJ河北（北京）\\L2.tif" \* MERGEFORMATINET INCLUDEPICTURE "D:\\課件\\九數JJ河北（北京）\\L2.tif" \* MERGEFORMATINET 如圖所示的是某高速公路邊水平地面上的交通警示牌，經測量得到如下數據：AB＝8 m，BC＝4 m，∠DAC＝30°，∠EBC＝45°，∠DCA＝90°，則警示牌DE的高度約為________．(結果精確到0.1 m．參考數據：≈1.41，≈1.73)
INCLUDEPICTURE"image3467.tif" INCLUDEPICTURE "D:\\課件\\九數JJ河北（北京）\\image3467.tif" \* MERGEFORMATINET INCLUDEPICTURE "D:\\課件\\九數JJ河北（北京）\\image3467.tif" \* MERGEFORMATINET
變式2[2023濟南期中]如圖，某一時刻太陽光從教室窗戶射入房間內，與地面的夾角∠DPC＝30°，已知窗戶的高度AF＝2 m，窗臺的高度CF＝1 m，CP＝ m，AD為窗外水平遮陽篷．
(1)求BC的長；
(2)求遮陽篷AD的寬度(≈1.732，結果精確到0.1 m)．
INCLUDEPICTURE"image3472.tif" INCLUDEPICTURE "D:\\課件\\九數JJ河北（北京）\\image3472.tif" \* MERGEFORMATINET INCLUDEPICTURE "D:\\課件\\九數JJ河北（北京）\\image3472.tif" \* MERGEFORMATINET
26. 4 解直角三角形的應用
第1課時　俯仰角、方向角
INCLUDEPICTURE"復習回顧.tif" INCLUDEPICTURE "D:\\課件\\九數JJ河北（北京）\\復習回顧.tif" \* MERGEFORMATINET INCLUDEPICTURE "D:\\課件\\九數JJ河北（北京）\\復習回顧.tif" \* MERGEFORMATINET
1．C　2.D　3.B
INCLUDEPICTURE"預習效果.tif" INCLUDEPICTURE "預習效果.tif" \* MERGEFORMAT INCLUDEPICTURE "D:\\課件\\九數JJ河北（北京）\\預習效果.tif" \* MERGEFORMATINET INCLUDEPICTURE "D:\\課件\\九數JJ河北（北京）\\預習效果.tif" \* MERGEFORMATINET INCLUDEPICTURE "D:\\課件\\九數JJ河北（北京）\\預習效果.tif" \* MERGEFORMATINET
1．16　2.8 km
INCLUDEPICTURE"課堂導學.tif" INCLUDEPICTURE "課堂導學.tif" \* MERGEFORMAT INCLUDEPICTURE "D:\\課件\\九數JJ河北（北京）\\課堂導學.tif" \* MERGEFORMATINET INCLUDEPICTURE "D:\\課件\\九數JJ河北（北京）\\課堂導學.tif" \* MERGEFORMATINET INCLUDEPICTURE "D:\\課件\\九數JJ河北（北京）\\課堂導學.tif" \* MERGEFORMATINET
例1　9.5
變式1.解：由題意可得∠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的長度約為438米．
例2　B
變式2.解：過點P作PC⊥AB于點C.由題意，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兩島之間的距離約為267海里．
第2課時　坡度、坡角及其他問題
INCLUDEPICTURE"復習回顧.tif" INCLUDEPICTURE "D:\\課件\\九數JJ河北（北京）\\復習回顧.tif" \* MERGEFORMATINET INCLUDEPICTURE "D:\\課件\\九數JJ河北（北京）\\復習回顧.tif" \* MERGEFORMATINET
1．B
2．解：由題意可知，∠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．
∴該建筑物BC的高度為(10 ＋10)米．
INCLUDEPICTURE"預習效果.tif" INCLUDEPICTURE "預習效果.tif" \* MERGEFORMAT INCLUDEPICTURE "D:\\課件\\九數JJ河北（北京）\\預習效果.tif" \* MERGEFORMATINET INCLUDEPICTURE "D:\\課件\\九數JJ河北（北京）\\預習效果.tif" \* MERGEFORMATINET INCLUDEPICTURE "D:\\課件\\九數JJ河北（北京）\\預習效果.tif" \* MERGEFORMATINET
1．豎直高度；水平寬度；坡角
2．(1)30°　(2)1∶1　(3)1∶　(4)比值；正切
3．5
INCLUDEPICTURE"課堂導學.tif" INCLUDEPICTURE "課堂導學.tif" \* MERGEFORMAT INCLUDEPICTURE "D:\\課件\\九數JJ河北（北京）\\課堂導學.tif" \* MERGEFORMATINET INCLUDEPICTURE "D:\\課件\\九數JJ河北（北京）\\課堂導學.tif" \* MERGEFORMATINET INCLUDEPICTURE "D:\\課件\\九數JJ河北（北京）\\課堂導學.tif" \* MERGEFORMATINET
例1　(30 ＋28)
變式1.解：(1)如圖，過點D作DH⊥BF于點H，則DH的長即為點D到地面的距離．
∵在Rt△CDH中，∠DHC＝90°，
∴DH＝CD·sin ∠DCH＝CD·sin 30°＝8×＝4(米)．
答：點D到地面的距離是4米．
(2)如圖，延長AD交BF于點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:\\課件\\九數JJ河北（北京）\\image3843.tif" \* MERGEFORMATINET INCLUDEPICTURE "D:\\課件\\九數JJ河北（北京）\\image3843.tif" \* MERGEFORMATINET INCLUDEPICTURE "D:\\課件\\九數JJ河北（北京）\\image3843.tif" \* MERGEFORMATINET
例2　2.9 m
變式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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