6.4 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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  Introduction
  ::导言

  

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  While running on your school's track, you notice a flagpole at the very end of the football field. The pole is 50 feet tall. Standing at the end of the track, you know that the distance between you and the flagpole is 350 feet. Since you are curious about such things, you want to determine the angle between the ground and the top of the flagpole from where you are standing. You have a firm grasp of trigonometric functions , but you' re still unsure how to solve for the angle of a trigonometric function . 
  ::在学校的赛道上运行时, 您注意到足球场尽头有一只旗杆。 杆高50英尺。 站在赛道的尽头, 您知道您和旗杆之间的距离是350英尺。 由于您对这些事情感到好奇, 您想要从您站的位置来决定地表和旗杆顶部之间的角 。 您有坚定的三角功能, 但是您仍然不确定如何解开三角函数的角度 。

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  Inverse Trigonometric Functions
  ::反逆三角函数

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  The word  inverse  is probably familiar to you. In mathematics, once you learn how to do an operation, you also learn how to "undo" it. For example, you may remember that addition and subtraction are considered inverse operations. Multiplication and division are also inverse operations. In learning about functions, you studied how to algebraically solve for the inverse of a given function. You also learned to graph a function and its inverse. When we apply the word inverse to the trigonometric ratios, we can find the acute angle measures within a right triangle. Normally, if you are given an angle and a side of a right triangle, you can find the other two sides using sine, cosine, or tangent. With the inverse trigonometric function , you can find the angle measure for the given two sides.
  ::反之词可能为您所熟悉 。 在数学中, 一旦学会如何操作, 您也会学会如何“ 取消” 操作 。 例如, 您可能记得, 增加和减法被认为是反向操作 。 乘法和分法也是反向操作 。 在学习函数时, 您会研究如何对给定函数的反向进行代数解析 。 您也会学会图形一个函数及其反向 。 当我们应用三重比时, 我们就可以在右三角内找到急性角度测量 。 通常, 如果您被给一个角度和右三角的侧面, 您可以用正弦、 cosine 或 切度来找到另外的两边 。 在反三重函数中, 您可以为给给定两边找到角测量角度 。

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  Inverse Tangent:  If you know the opposite side and adjacent side of an angle in a right triangle, you can use inverse tangent to find the measure of the angle. Inverse tangent is also called arctangent and is labeled  $\begin{array}{r}{\tan }^{-1}\end{array}$  or  arctan . The “-1” indicates inverse.
  ::反切线 : 如果您知道右三角形角的对面和相邻侧面, 您可以使用反切线来找到角的度量。 反切线也被称为弧切线, 标记为 tan-1 或 arctan 。 “ 1” 表示反向 。

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  Inverse Sine:  If you know the opposite side of an angle and the hypotenuse in a right triangle, you can use inverse sine to find the measure of the angle. Inverse sine is also called arcsine and is labeled  $\begin{array}{r}{\sin }^{-1}\end{array}$  or  arcsin .
  ::反弦 : 如果您知道角的对面和右三角形的下限, 您可以使用反弦来找到角的度量。 反弦也被称为弧素, 并被标为 sin-1 或 arcsin 。

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  Inverse Cosine:  If you know the adjacent side of an angle and the hypotenuse in a right triangle, you can use inverse cosine to find the measure of the angle. Inverse cosine is also called arccosine and is labeled  $\begin{array}{r}{\cos }^{-1}\end{array}$  or  arccos .
  ::逆余弦 : 如果您知道角的相邻侧面和右三角形的下限, 您可以使用反余弦来找到角的量度。 反余弦也被称为弧辛, 并被标为cos-1 或arcccos 。

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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 $\begin{array}{r}\mathrm{\angle }T\end{array}$ .  However , none of the angles are known. The inverse of the trigonometric function must be used to determine the measure of the angle. 
  ::从这个三角形中,我们知道如何确定 QC 和 QT 的所有六个三角函数。 但是, 没有一个角度是已知的 。 三角函数的反面必须用来决定角的度量 。

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  From any of these functions, we can also find the value of the angle using a  calculator. You might recall that $\begin{array}{r}\sin {30}^{\circ }=\frac{1}{2}\end{array}$ . The inverse function for this example would then be  $\begin{array}{r}{\sin }^{-1}\left(\frac{1}{2}\right)={30}^{\circ }.\end{array}$  On some c alculators you would press  $\begin{array}{r}{2}^{nd}\end{array}$  and  $\begin{array}{r}\sin \end{array}$ to get $\begin{array}{r}{\sin }^{-1}(\end{array}$ and then type in $\begin{array}{r}\frac{1}{2}\end{array}$ , close the parentheses, and press ENTER. 
  ::从任何这些函数中,我们也可以使用计算器找到角值。 您可以记得 sin3012。 这个例子的反函数将是 sin- 1( 12) =30 。 在一些计算器上, 您可以按 2 和 罪 来获得 sin - 1( 并在 12 中键入 12 , 关闭括号, 并按 ENTER 键 。

    

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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 makeshift conveyor belt made of bedsheets, which you'll 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,手工把干草移到梯子上需要太多的时间和精力。你决定设计一个用床单制成的临时传送带,你将把床单绑在阁楼的门上,固定在地上。如果阁楼的门在地上25英尺高,你还有30英尺的床单,你需要从什么角度把床单固定在地上?

   

  

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  Using the picture, we need to use the inverse sine function:
  ::使用图片,我们需要使用反正正弦函数:

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

  
  ::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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  Learn, Play, and Explore with Inverse Trig Functions: 
  ::学习、玩耍和探索与反三边函数:

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  Examples
  ::实例

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  Example 1
  ::例1

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  Determine  the angle measure for the trig function below.
  ::确定以下三角函数的角度量。

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

  
  ::sinux=0.823 x=0.823

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  Solution:
  ::解决方案 :

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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 2
  ::例2

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  Determine  the angle measure for the trig function below.
  ::确定以下三角函数的角度量。

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  $$\begin{array}{r}\cos x=-0.112\end{array}$$

  
  ::COSx0.112

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  Solution:
  ::解决方案 :

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

  
  ::96.43

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  Example 3
  ::例3

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  Determine  the angle measure for the trig function below.
  ::确定以下三角函数的角度量。

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

  
  ::tanx=0.2

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  Solution:
  ::解决方案 :

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

  
  ::11.31\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\

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  Example 4
  ::例4

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  Recall from the Introduction that you noticed a 50-foot-tall flag pole at the very end of the football field . While standing at the end of the track, you are a distance of 350 feet from the flagpole. Determine  the angle between the ground and the top of the flagpole from where you are standing. 
  ::从导言中回顾,你注意到足球场尽头有一支50英尺高的旗杆。站在赛道尽头的距离是离船旗杆350英尺。从站着的地方确定地与船旗杆顶部之间的角。

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  Solution:
  ::解决方案 :

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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 5
  ::例5

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  Determine  the measures of the unknown angles in the triangle shown. Round answers to the nearest degree .
  ::确定所显示三角形中未知角度的度量。将回答圆到最接近的度。

  

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  Solution:
  ::解决方案 :

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  $$\begin{array}{r}x={\cos }^{-1}\left(\frac{13}{20}\right)\approx {49}^{\circ }\\ y={\sin }^{-1}\left(\frac{13}{20}\right)\approx {41}^{\circ }\end{array}$$

  
  ::x=cos- 1(1320) 49y=sin- 1(1320) 41

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  Example 6
  ::例6

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  Solve the triangle for all missing angles and side lengths. Round side lengths to the nearest tenth and angles to the nearest degree.
  ::解开三角形, 用于所有缺失的角度和侧长。 圆边长度为最接近的十分之一, 角度为最接近的高度 。

  

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  Solution:
  ::解决方案 :

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  $$\begin{array}{r}m\mathrm{\angle }A={\cos }^{-1}\left(\frac{17}{38}\right)\approx {63}^{\circ }\\ m\mathrm{\angle }B={\sin }^{-1}\left(\frac{17}{38}\right)\approx {27}^{\circ }\\ a=\sqrt{{38}^2-{17}^2}\approx 34.0\end{array}$$

  
  ::mA=cos-1(1738) 63mB=sin_1(1738) 27a=382 –1723400)

   

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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.
  ::使用逆三角测量法为下方每个角度寻找角度 A 角角的角度量。

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  1.  $\begin{array}{r}\sin A=0.839\end{array}$
  ::一罪A=0.839

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  2.  $\begin{array}{r}\cos A=0.19\end{array}$
  ::2. COsA=0.19

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  3.  $\begin{array}{r}\tan A=0.213\end{array}$
  ::3. tanA=0.213

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  4.  $\begin{array}{r}\csc A=1.556\end{array}$
  ::4. cscA=1.556

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  5.  $\begin{array}{r}\sec A=2.063\end{array}$
  ::5. 秒A=2.063

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  6.  $\begin{array}{r}\cot A=2.356\end{array}$
  ::6. COtA=2.356

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  Find the measures of the unknown acute angles. Round measures to the nearest degree.
  ::查找未知急性角度的测量。 圆形测量到最接近的程度 。

  7.

   

  8.

  

  9.

  

  10. 

  

  11.

  

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  Solve the following right triangles for all missing angles and side lengths. Round angle measures to the nearest degree and side lengths to the nearest tenth.
  ::为所有缺失的角度和侧边长度解决以下右三角形。圆角测量到最近的纬度和侧边长度至最近的十分。

  12.

  

  13.

  

  14.

  

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  15. A 70-foot building casts an 100-foot shadow. At what angle does the sun hit the building?
  ::15. 一座70英尺高的建筑物投下100英尺高的阴影,太阳从哪个角度击中大楼?

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  16. 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? (Note this is the angle from the horizontal line of the water down.)
  ::16. 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,游到250英尺处到达沉船,从船到沉船的压抑角度是什么? (注意这是从水下水平线的角。 )

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  17. A boat is docked at the end of a 10-foot pier. The boat leaves the pier and drops anchor 230 feet away, 3 feet straight out from shore (which is perpendicular to the pier). What was the angle  ( $\begin{array}{r}x\end{array}$ ) of the boat from a line drawn from the end of the pier through the foot of the pier, as shown below?
  ::17. 一艘船舶停靠在10英尺码头的尽头,离码头有230英尺远,离岸直达3英尺处(直达码头),从码头尽头从码头尽头到码头脚处划出一条线,其角度(x)为何,如下文所示?

  

   

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  Review (Answers)
  ::回顾(答复)

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  Please see the Appendix.
  ::请参看附录。