5.3 弧度和度的角

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  弧度和度的角

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  Introduction
  ::导言

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  A California vegetable grower is considering new options to irrigate crops. He needs to find a method that is both sustainable and economical. One possible choice is the center pivot irrigation system, a method widely used in the Great Plains region of the United States. The grower needs to learn how to convert the current drip irrigation system to a center pivot system. He needs critical information to answer his question about how to calculate the area of the fields to be irrigated with a center pivot irrigation system.
  ::加利福尼亚蔬菜种植者正在考虑灌溉作物的新选择。 他需要找到一种既可持续又经济的方法。 一个可能的选择是中央支流灌溉系统,这是在美国大平原地区广泛使用的一种方法。 种植者需要学会如何将目前的滴灌系统转换为中央支流灌溉系统。 他需要关键信息来回答他关于如何计算用中央支流灌溉系统灌溉的农田面积的问题。

  

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  When setting the mechanism that turns the 400-foot irrigation pipe, the grower can set the rotation angle and speed. The system is set to start in a direction that is due east and rotate in a counterclockwise fashion, so the angle of rotation is in standard position . The grower found a formula on the internet to calculate the irrigated area, but the formula did not make sense.  The formula requires that when he enters the rotation angle, he must first multiply it by 0.0175. Now the grower has more questions: "Where did 0.0175 come from? How is this formula created? Why  must I enter the angle  in this way?"
  ::当设定旋转400英尺灌溉管道的机制时, 种植者可以设定旋转角度和速度。 系统将开始于一个向东的方向, 并逆时针旋转, 因此旋转角度处于标准位置 。 种植者在互联网上找到了计算灌溉面积的公式, 但公式不合理 。 公式要求当他进入旋转角度时, 他必须先乘以0.0175。 现在种植者有更多的问题 : “ 0.0175 是从哪里来的? 这个公式是如何创建的? 为什么我必须以此方式进入角度 ? ”

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  Definition of Radian Measure
  ::辐射措施的定义

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  A radian is a unit of measuring angles that is based on the properties of circles. One   radian angle is defined to be the central angle where the subtended arc length, which is  the part of the circle in between the two rays that make the central angle ,  is the same length as the radius.  
  ::弧度是一个基于圆形特性的测量角度单位。一个弧度角度被定义为中心角,其下角弧长度是形成中心角的两个射线之间的圆形部分,其长度与半径相同。

  

  
  

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  Another way to think about radians is through the circumference of a circle. The circumference of a circle with radius $\begin{array}{r}r\end{array}$  is $\begin{array}{r}2\pi r\end{array}$ . Just over 6 radii (exactly $\begin{array}{r}2\pi \end{array}$  radii) would stretch around any circle. 
  ::另一种思考弧度的方法是通过圆圈环绕。圆圈半径为 r 的环绕值为 2 °r 。 仅略高于 6 弧度( 直径为 2 radi) 的环绕度将围绕任何圆圈延伸 。

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  Conversions between Degrees and Radians
  ::度与弧度之间的转换

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  Note that  $\begin{array}{r}{360}^{\circ }=2\pi \text{ radians}\end{array}$ , so  $\begin{array}{r}\frac{{360}^{\circ }}{2\pi }=\frac{{180}^{\circ }}{\pi }\end{array}$  is the proportion used to convert  from  radians to degrees.
  ::注意3602弧度, 也就是3602180, 是用于从弧度转换成度的比例 。

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  Conversions from Radians to Degrees
  ::从弧度转换为度

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  $$\begin{array}{r}\theta \text{ radians}\cdot \frac{{180}^{\circ }}{\pi \text{ radians}}={\theta }^{\circ }\end{array}$$

     
  ::弧度 180 弧度

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  Similarly,  $\begin{array}{r}{360}^{\circ }=2\pi \text{ radians}\end{array}$ , so  $\begin{array}{r}\frac{2\pi }{{360}^{\circ }}=\frac{\pi }{{180}^{\circ }}\end{array}$  is the proportion used to convert from degrees to radians.  
  ::同样,360 22 弧度, 所以 2360 180是用于从度转换成弧度的比例 。

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     Conversion from Degree s to Radian s
  ::从度转换为弧度

  $$\begin{array}{r}{\theta }^{\circ }\cdot \frac{\pi \text{ radians}}{{180}^{\circ }}=\theta \text{ radians}\end{array}$$

   

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  Thus, the conversion factor to convert from degrees to radians is  $\begin{array}{r}\frac{\pi }{{180}^{\circ }}\end{array}$  and from radians to degrees is  $\begin{array}{r}\frac{{180}^{\circ }}{\pi }\end{array}$ .
  ::因此,从度向弧度转换为弧度的换算系数为 180 ,从弧度向度转换为 180 。

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  Learn, Play, and Explore with Radians: 
  ::学习、玩耍和探索与Radians:

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  Note that if an angle has no units listed, it is assumed to be in radians. 
  ::请注意,如果角度没有列出单位,则假定为弧度。

    

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  Learn, Play, and Explore with Hours in the Day: 
  ::学习、游戏、白昼的探索、

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  Examples
  ::实例

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  Example 1
  ::例1

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  Convert $\begin{array}{r}{150}^{\circ }\end{array}$  into radians. 
  ::将150转换成弧度

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  Solution: 
  ::解决方案 :

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  $$\begin{array}{r}{150}^{\circ }\cdot \frac{\pi }{{180}^{\circ }}=\frac{15\pi }{18}=\frac{5\pi }{6}\text{ radians}\end{array}$$

  
  ::1501801518=56弧度

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  Note: Make sure the degree units cancel. 
  ::注:确保度单位取消。

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  Example 2
  ::例2

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  Convert $\begin{array}{r}\frac{\pi }{6}\end{array}$  radians into degrees. 
  ::将 6 弧度转换为度。

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  Solution:
  ::解决方案 :

   

  $$\begin{array}{r}\frac{\pi }{6}\cdot \frac{{180}^{\circ }}{\pi }=\frac{{180}^{\circ }}{6}={30}^{\circ }\end{array}$$

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  Note: Often the $\begin{array}{r}\pi \end{array}$ s will cancel. 
  ::注意: 通常 s会取消 。

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  Example 3
  ::例3

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  Estimate the angle measure  $\begin{array}{r}(6\pi {)}^{\circ }\end{array}$  by substituting 3.14 for  $\begin{array}{r}\pi \end{array}$ . 
  ::估计角度量(6) * , 以 3. 14 替代 。

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  Solution: 
  ::解决方案 :

  $$\begin{array}{r}6\left(3.14\right)={19}^{\circ }\end{array}$$

   

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  Example 4
  ::例4

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  Convert $\begin{array}{r}(6\pi {)}^{\circ }\end{array}$  into radians. 
  ::将 (6) 转换成弧度 。

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  Solution:
  

  $$\begin{array}{r}(6\pi {)}^{\circ }\cdot \frac{\pi }{{180}^{\circ }}=\frac{6{\pi }^2}{180}=\frac{{\pi }^2}{30}\approx 0.329\text{ radians}\end{array}$$

  
  ::解答: (6) 180621802300. 329弧度

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  Example 5 
  ::例5

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  Return to the problem from the Introduction of this section. The grower had a few questions: " Where did 0.0175 come from? How is this formula created? Why  must I enter the angle  in this way?"
  ::回到本节导言中的问题。种植者有几个问题:“0.0175来自何方?这个公式是如何形成的?为什么我必须从这个角度进入?”

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  Solution:
  ::解决方案 :

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  If the angle of rotation is in degrees, the conversion to radians would be done by multiplying by  $\begin{array}{r}\frac{\pi }{180}\approx \mathrm{0.0175.}\end{array}$  The formula found on the internet must have required that the angle be converted to radians.
  ::如果旋转角度为度,则转换成弧度的方法是乘以1800.0175。 互联网上找到的公式必须要求将角转换成弧度。

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  Summary
  ::摘要

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    A radian is defined as the central angle where the subtended arc length is the same length as the radius.
    ::弧度的定义是子弧长度与半径相同长度的中心角。

    
    ::弧度的定义是子弧长度与半径相同长度的中心角。
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    The conversion factor to convert from degrees to radians is  $\begin{array}{r}\frac{\pi }{{180}^{\circ }}\end{array}$  .
    ::从度向弧度转换为弧度的转换系数为 180 。

    
    ::从度向弧度转换为弧度的转换系数为 180 。
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    The conversion factor to convert from radians to degrees is  $\begin{array}{r}\frac{{180}^{\circ }}{\pi }\end{array}$ .
    ::从弧度转换为度的转换系数为 180 。

    
    ::从弧度转换为度的转换系数为 180 。

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  Review 
  ::回顾

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  Find the radian measure of each angle:
  ::查找每个角度的弧度量度 :

  1.  $\begin{array}{r}{120}^{\circ }\end{array}$

  2. $\begin{array}{r}{300}^{\circ }\end{array}$

  3. $\begin{array}{r}{90}^{\circ }\end{array}$

  4. $\begin{array}{r}{330}^{\circ }\end{array}$

  5. $\begin{array}{r}{270}^{\circ }\end{array}$

  6.  $\begin{array}{r}{45}^{\circ }\end{array}$

  7. $\begin{array}{r}(5\pi {)}^{\circ }\end{array}$

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  Find the degree measure of each angle:
  ::查找每个角度的度量 :

  8.  $\begin{array}{r}\frac{7\pi }{6}\end{array}$

  9.  $\begin{array}{r}\frac{5\pi }{4}\end{array}$

  10.  $\begin{array}{r}\frac{3\pi }{2}\end{array}$

  11.  $\begin{array}{r}\frac{5\pi }{3}\end{array}$

  12. $\begin{array}{r}\pi \end{array}$

  13. $\begin{array}{r}\frac{\pi }{6}\end{array}$

  14. 3

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  15. Explain why, if you are given an angle in degrees and you multiply it by  $\begin{array}{r}\frac{\pi }{180},\end{array}$ you will get the same angle in radians.
  ::15. 解释一下,如果给您一个角度以度乘以 180, 为何在弧度上得到相同的角度。

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  16. Suppose you are working on a science lab in which your professor asks you to turn a knob   $\begin{array}{r}{75}^{\circ }\end{array}$  on the detector you are using. Unfortunately, you have been working in radians for a while, so you are having trouble remembering how far to turn the knob. How would you translate the instructions in degrees to radians? 
  ::16. 假设你在一个科学实验室工作,你的教授请你在实验室里对正在使用的探测器开一个75°C的开关。 不幸的是,你在弧度上工作了一段时间,所以你记不清要翻转点有多远。你如何将指示的度译为弧度?

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  Review ( Answers) 
  ::回顾(答复)

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  Please see the Appendix.
  ::请参看附录。