8.9 三角文字问题

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  三角词问题

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  Angle of Depression and Angle of Elevation
  ::萧条与升高之角

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  One application of the trigonometric ratios is to find lengths that you cannot measure. Very frequently, angles of depression and elevation are used in these types of problems.
  ::三角比的一个应用是找到无法测量的长度。 在这些类型的问题中经常使用抑郁和高位角度。

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  Angle of Depression: The angle measured down from the horizon or a horizontal line.
  ::萧条角:从地平线或水平线下测量的角度。

  

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  Angle of Elevation: The angle measured up from the horizon or a horizontal line.
  ::升降角度: 从地平线或水平线上测量的角度。

  

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  What if you placed a ladder 10 feet from a haymow whose floor is 20 feet from the ground? How tall would the ladder need to be to reach the haymow's floor if it forms a $\begin{array}{r}{30}^{\circ }\end{array}$ angle with the ground?
  ::如果你在离地面20英尺的干草地10英尺处设置了梯子呢?如果梯子与地面形成30平方格,那么要达到干草地地板的梯子要有多高呢?

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  Examples
  ::实例

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  Example 1
  ::例1

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  A math student is standing 25 feet from the base of the Washington Monument. The angle of elevation from her horizontal line of sight is $\begin{array}{r}{87.4}^{\circ }\end{array}$ . If her “eye height” is 5 ft, how tall is the monument?
  ::一位数学学生站在距离华盛顿纪念碑底部25英尺的地方。 从她水平视线的高度角度看是87.4。 如果她的“眼高”是5英尺,纪念碑有多高?

  

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  We can find the height of the monument by using the tangent ratio.
  ::我们可以使用正切比例找到纪念碑的高度。

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  $$\begin{array}{rl}\tan {87.4}^{\circ }& =\frac{h}{25}\\ h& =25\cdot \tan {87.4}^{\circ }=550.54\end{array}$$

  
  ::{\fn黑体\fs22\bord1\shad0\3aHBE\4aH00\fscx67\fscy66\2cHFFFFFF\3cH808080}什么? {\fn黑体\fs22\bord1\shad0\3aHBE\4aH00\fscx67\fscy66\2cHFFFFFF\3cH808080}什么? {\fn黑体\fs22\bord1\shad0\3aHBE\4aH00\fscx67\fscy66\2cHFFFFFF\3cH808080}什么? {\fn黑体\fs22\bord1\shad0\3aHBE\4aH00\fscx67\fscy66\2cHFFFFFF\3cH808080}什么 {\fn黑体\fs22\bord1\shad0\3aHBE\4aH00\fscx67\fscy66\2cHFFFFFF\3cH808080}什么?

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  Adding 5 ft, the total height of the Washington Monument is 555.54 ft.
  ::加上5英尺高,华盛顿纪念碑的总高度为555.54英尺。

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  Example 2
  ::例2

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  A 25 foot tall flagpole casts a 42 foot shadow. What is the angle that the sun hits the flagpole?
  ::25英尺高的旗杆投下了42英尺的阴影。太阳击中旗杆的角度是什么?

  

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  Draw a picture. The angle that the sun hits the flagpole is $\begin{array}{r}{x}^{\circ }\end{array}$ . We need to use the inverse tangent ratio.
  ::绘制图片。 太阳击中旗杆的角度是 x 。 我们需要使用反正切比例 。

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  $$\begin{array}{rl}\tan x& =\frac{42}{25}\\ {\tan }^{-1}\frac{42}{25}& \approx {59.2}^{\circ }=x\end{array}$$

  
  ::x=4225tan-1422559.2x

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  Example 3
  ::例3

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  Elise is standing on top of a 50 foot building and sees her friend, Molly. If Molly is 30 feet away from the base of the building, what is the angle of depression from Elise to Molly? Elise’s eye height is 4.5 feet.
  ::伊莉斯站在50英尺楼顶上,看到她的朋友莫莉。 如果莫莉离大楼底部30英尺,那么从伊莉斯到莫莉的萧条角度是什么? 艾莉斯的眼睛高度是4.5英尺。

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  Because of parallel lines, the angle of depression is equal to the angle at Molly, or $\begin{array}{r}{x}^{\circ }\end{array}$ . We can use the inverse tangent ratio.
  ::由于平行线的缘故, 抑郁的角等于 Molly 或 x 的角。 我们可以使用反正正切比 。

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  $$\begin{array}{r}{\tan }^{-1}\left(\frac{54.5}{30}\right)={61.2}^{\circ }=x\end{array}$$

  
  ::-1(54.530)=61.2x

  

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  Example 4
  ::例4

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  Mark is flying a kite and realizes that 300 feet of string are out. The angle of the string with the ground is $\begin{array}{r}{42.5}^{\circ }\end{array}$ . How high is Mark's kite above the ground?
  ::马克飞着风筝,意识到有300英尺的绳子被拔出。 带地面的绳子角度是42.5。 马克的风筝在地面上有多高?

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  It might help to draw a picture. Then write and solve a trig equation.
  ::画一张图片也许有用,然后写和解答三重方程。

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  $$\begin{array}{rl}\sin {42.5}^{\circ }& =\frac{x}{300}\\ 300\cdot \sin {42.5}^{\circ }& =x\\ x& \approx 202.7\end{array}$$

  
  ::-=YTET -伊甸园字幕组=-伊甸园字幕组=-伊甸园字幕组=-伊甸园字幕组=- 翻译:

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  The kite is about $\begin{array}{r}202.7\end{array}$ feet off of the ground.
  ::风筝离地面大约202.7英尺

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  Example 5
  ::例5

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  A 20 foot ladder rests against a wall. The base of the ladder is 7 feet from the wall. What angle does the ladder make with the ground?
  ::20英尺的梯子靠在墙上,梯子的底部离墙7英尺。 梯子与地面有什么关系?

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  It might help to draw a picture.
  ::画幅画也许有帮助

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  $$\begin{array}{rl}\cos x& =\frac{7}{20}\\ x& ={\cos }^{-1}\frac{7}{20}\\ x& \approx {69.5}^{\circ }\end{array}$$

  
  ::cosx=720x=cos-1720x_69.5

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  Review
  ::回顾

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  1. Kristin is swimming in the ocean and notices a coral reef below her. The angle of depression is $\begin{array}{r}{35}^{\circ }\end{array}$ and the depth of the ocean, at that point is 250 feet. How far away is she from the reef? Round your answer to the nearest tenths place.
     ::Kristin在海洋中游泳,发现她下面有珊瑚礁。低沉的角是35英寸,海洋的深度是250英尺。她离珊瑚礁还有多远?你回答的答案是最近的十分之一。
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  2. The Leaning Tower of Piza currently “leans” at a $\begin{array}{r}{4}^{\circ }\end{array}$ angle and has a vertical height of 55.86 meters. How tall was the tower when it was originally built? Round your answer to the nearest tenth.
     ::Piza的落叶塔目前“利宁塔”,角度为4,垂直高度为55.86米。该塔最初建造时有多高?您回答的答案是最近的十分之一。

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  Use what you know about right triangles to solve for the missing angle. If needed, draw a picture. Round all answers to the nearest tenth of a degree.
  ::使用您所知道的右三角以解析缺失角度。 如果需要, 请绘制图片。 将所有答案都回圆到一个度的十分之一 。

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  3. A 75 foot building casts an 82 foot shadow. What is the angle that the sun hits the building?
     ::75英尺的建筑物投下了82英尺的阴影。太阳击中大楼的角度是什么?
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  4. Over 2 miles (horizontal), a road rises 300 feet (vertical). What is the angle of elevation? (Note there are 5280 feet in a mile)
     ::2英里以上(横向),一条公路上升300英尺(垂直)。高角是什么? (注意一英里有5 280英尺)
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  5. A boat is sailing and spots a shipwreck 650 feet below the water. A diver jumps from the boat and swims 935 feet to reach the wreck. What is the angle of depression from the boat to the shipwreck?
     ::一艘船正在航行,在水下650英尺处发现一艘沉船,潜水员从船上跳下935英尺游到沉船。
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  6. Standing 100 feet from the base of a building, Sam measures the angle to the top of the building from his eye height to be $\begin{array}{r}{50}^{\circ }\end{array}$ . If his eyes are 6 feet above the ground, how tall is the building?
     ::山姆将距离建筑物底部100英尺处的角从他的眼睛高度测量到建筑物顶部50英尺。如果他的眼睛在地面上6英尺处,那么大楼有多高?
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  7. Over 4 miles (horizontal), a road rises 200 feet (vertical). What is the angle of elevation?
     ::4英里以上(横向),一条公路上升200英尺(垂直)。高处的角是什么?
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  8. A 90 foot building casts an 110 foot shadow. What is the angle that the sun hits the building?
     ::90英尺的建筑物投下了110英尺的阴影。太阳击中大楼的角度是什么?
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  9. Luke is flying a kite and realizes that 400 feet of string are out. The angle of the string with the ground is $\begin{array}{r}{50}^{\circ }\end{array}$ . How high is Luke's kite above the ground?
     ::卢克飞着风筝,意识到有400英尺的绳子被拔出。 与地面的绳子角是50英寸。 卢克的风筝在地面上有多高?
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  10. An 18 foot ladder rests against a wall. The base of the ladder is 10 feet from the wall. What angle does the ladder make with the ground?
      ::梯子的底部离墙有10英尺,梯子与地面有什么关系?

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  Review (Answers)
  ::回顾(答复)

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  Click to see the answer key or go to the Table of Contents and click on the Answer Key under the 'Other Versions' option.
  ::单击可查看答题键, 或转到目录中, 单击“ 其他版本” 选项下的答题键 。

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  Resources
  ::资源