1.11 逆三角函数

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• 选择节反逆三角函数

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  反逆三角函数

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  One day after school you are trying out for the track team. Your school has a flag pole at the very end of its football field. The pole is 50 feet tall. While standing at the end of the track, you know that the distance between you and the flag pole is 350 feet. Being curious about such things, you decide to find the angle between the ground and the top of the flag pole from where you are standing.
  ::学校放学后一天,你会为球队尝试。你的学校在足球场的尽头有一个旗杆。 杆高50英尺。 当站在球场的尽头时, 你知道你和旗杆之间的距离是350英尺。 好奇这些事情, 你决定从你站的地方找到地与旗杆顶部之间的角 。

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  Can you solve this problem?
  ::你能解决这个问题吗?

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  Inverse Trigonometric Functions
  ::反逆三角函数

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  Consider the right triangle below.
  ::考虑下方的右三角形。

  

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  From this triangle, we know how to determine all six trigonometric functions for both $\begin{array}{r}\mathrm{\angle }C\end{array}$ and $\begin{array}{r}\mathrm{\angle }T\end{array}$ .
  ::从这个三角形上,我们知道如何确定 C和T的所有六个三角函数。

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   From any of these functions we can also find the value of the angle, using our graphing calculators. You might recall that $\begin{array}{r}\sin {30}^{\circ }=\frac{1}{2}\end{array}$ . If you type 30 into your graphing calculator and then hit the button, the calculator yields 0.5. (Make sure your calculator’s mode is set to degrees.)
  ::从任何这些函数中,我们也可以用我们的图形计算计算器找到角度的值。 您可能会记得 sin3012。 如果您在图形计算器中输入 30, 然后按下按钮, 计算器将产生 0.5 。 (确保您的计算器模式被设定为度 。)

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  Conversely, with the triangle above, we know the trig ratios, but not the angle. In this case the inverse of the trigonometric function must be used to determine the measure of the angle. These functions are located above the SIN, , and buttons on the calculator. To access one of these functions, press $\begin{array}{r}{2}^{nd}\end{array}$ and the appropriate button and the measure of the angle appears on the screen.
  ::反之, 与上方三角形相比, 我们了解三角比, 但不是角度。 在这种情况下, 三角函数的反函数必须用于确定角的度量 。 这些函数位于 SIN 上方, 以及计算器上的按钮 。 要访问其中一个函数, 请按 2 键, 以及屏幕上显示的角的大小 。

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  $\begin{array}{r}\cos T=\frac{24}{25}\to {\cos }^{-1}\frac{24}{25}=T\end{array}$ from the calculator we get
  ::我们从计算器得到的 COST=2425Cos- 1Q2425=T

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  For the problems below, use your calculator the find the angle measurement of the trig functions.
  ::对于下面的问题, 请使用您的计算器 查找三角函数的角度测量 。

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  1.  $\begin{array}{r}\sin x=0.687\end{array}$
  ::一罪=0.687

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  Plug into calculator.
  ::插进计算器

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  $\begin{array}{r}{\sin }^{-1}0.687={43.4}^{\circ }\end{array}$
  ::=43.4* * 缩略语 =43.4 *

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  2.  $\begin{array}{r}\tan x=\frac{4}{3}\end{array}$
  ::2. tanx=43

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  Plug into calculator.
  ::插进计算器

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  $\begin{array}{r}{\tan }^{-1}\frac{4}{3}={53.13}^{\circ }\end{array}$
  ::-143=53.13

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  Real-World Application: Conveyor Belt 
  ::真实世界应用程序: Conveyor Belt

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  You live on a farm and your chore is to move hay from the loft of the barn down to the stalls for the horses. The hay is very heavy and to move it manually down a ladder would take too much time and effort. You decide to devise a make shift conveyor belt made of bed sheets that you will attach to the door of the loft and anchor securely in the ground. If the door of the loft is 25 feet above the ground and you have 30 feet of sheeting, at what angle do you need to anchor the sheets to the ground?
  ::你住在农场里,你的工作是把干草从谷仓的阁楼移到马的马棚里。干草非常重,手动把干草从梯子上移下来需要花费太多的时间和精力。你决定设计一个由床单制成的轮椅传送带,床单将附在阁楼的门上,固定在地上。如果阁楼的门在地上25英尺高,有30英尺长的床单,你需要从什么角度把床单固定在地上?

   

  

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  From the picture, we need to use the inverse sine function.
  ::从情况看,我们需要使用反正正弦功能。

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  $$\begin{array}{rl}\sin \theta & =\frac{25feet}{30feet}\\ \sin \theta & =0.8333\\ {\sin }^{-1}(\sin \theta )& ={\sin }^{-1}0.8333\\ \theta & ={56.4}^{\circ }\end{array}$$

  
  ::25英尺30英尺 0.8333sin -1(sin)=sin -10.833356.4

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  The sheets should be anchored at an angle of $\begin{array}{r}{56.4}^{\circ }\end{array}$ .
  ::床单应固定在56.4的角上。

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  Examples
  ::实例

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  Example 1
  ::例1

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  Earlier, you were asked to find the angle between the ground and the top of the flag pole from where you are standing.
  ::早些时候,有人要求你从你站的地方 找到地与旗杆顶部之间的角。

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  Using your knowledge of inverse trigonometric functions, you can set up a tangent relationship to solve for the angle:
  ::使用对反三角函数的知识,您可以建立正切关系来解决角度问题:

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  $$\begin{array}{r}\tan \theta =\frac{50}{350}\\ \theta ={\tan }^{-1}\frac{50}{350}\\ \theta \approx {8.13}^{\circ }\end{array}$$

  
  ::50350 1503508.13

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  Example 2
  ::例2

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  Find the angle measure for the trig function below.
  ::查找下面三角函数的角度量。

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  $\begin{array}{r}\sin x=0.823\end{array}$
  ::sinux=0.823 x=0.823

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  Plug into calculator.
  ::插进计算器

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  $\begin{array}{r}{\sin }^{-1}0.823\approx {55.39}^{\circ }\end{array}$
  :一九九九年八月八日八日三月五日) (一九九九年九月九日) (一九九九年八月八日八日二十三日) (五零三年三月三日) (一九九九年八月三十日)

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  Example 3
  ::例3

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  Find the angle measure for the trig function below.
  ::查找下面三角函数的角度量。

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  $\begin{array}{r}\cos x=-0.112\end{array}$
  ::COSx0.112

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   Plug into calculator.
  ::插进计算器

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  $\begin{array}{r}{\cos }^{-1}-0.112\approx {96.43}^{\circ }\end{array}$
  ::96.43

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  Example 4
  ::例4

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  Find the angle measure for the trig function below.
  ::查找下面三角函数的角度量。

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  $\begin{array}{r}\tan x=0.2\end{array}$
  ::tanx=0.2

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  Plug into calculator.
  ::插进计算器

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  $\begin{array}{r}{\tan }^{-1}0.2\approx {11.31}^{\circ }\end{array}$
  ::11.31\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\

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  Review
  ::回顾

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  Use inverse trigonometry to find the angle measure of angle A for each angle below. Round your answers to the nearest integer.
  ::使用逆三角测量来查找下方每个角度的角度 A 角角的角度量。 将您的答案四舍五入到最接近的整数 。

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  1. $\begin{array}{r}\sin A=0.839\end{array}$
     ::A=0.839
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  2. $\begin{array}{r}\cos A=0.19\end{array}$
     ::COsA=0.19
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  3. $\begin{array}{r}\tan A=0.213\end{array}$
     ::泰纳=0.213
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  4. $\begin{array}{r}\csc A=1.556\end{array}$
     ::cscA=1.556
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  5. $\begin{array}{r}\sec A=2.063\end{array}$
     ::秒A=2.063
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  6. $\begin{array}{r}\cot A=2.356\end{array}$
     ::comtA=2.356
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  7. $\begin{array}{r}\csc A=8.206\end{array}$
     ::cscA=8.206
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  8. $\begin{array}{r}\sin A=0.9994\end{array}$
     ::A=0.9994
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  9. $\begin{array}{r}\cot A=1.072\end{array}$
     ::A=1.072
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  10. $\begin{array}{r}\cos A=0.174\end{array}$
      ::COsA=0.174
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  11. $\begin{array}{r}\tan A=1.428\end{array}$
      ::泰纳=1.428
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  12. $\begin{array}{r}\csc A=2.92\end{array}$
      ::cscA=2.92
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  13. A 70 foot building casts an 100 foot shadow. What is the angle that the sun hits the building? 
      ::70英尺的建筑物投下了100英尺的阴影。太阳击中大楼的角度是什么?
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  14. Over 2 miles (horizontal), a road rises 300 feet (vertical). What is the angle of elevation? (Note that 1 mile = 5280 feet)
      ::2英里以上(横向),一条公路上升300英尺(垂直)。高处的角度是什么? (注意1英里=5280英尺)
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  15. Whitney is sailing and spots a shipwreck 100 feet below the water. She jumps from the boat and swims 250 feet to reach the wreck. What is the angle of depression from the boat to the shipwreck? 
      ::惠特尼正在航行,在水下100英尺处发现一艘沉船,她从船上跳下,游了250英尺到沉船。

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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 。