1.16 共有角

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  交界形角

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  While playing a game with friends, you use a spinner that looks like this:
  ::在和朋友玩游戏时,你用一个像这样的旋翼人:

  

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  As you can see, the angle that the spinner makes with the horizontal is $\begin{array}{r}{60}^{\circ }\end{array}$ . Is it possible to represent the angle any other way?
  ::您可以看到, 旋转器在水平上产生的角是 60 。 是否可以以任何其他方式代表角 ?

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  Coterminal Angles
  ::交界形角

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    Consider the angle $\begin{array}{r}{30}^{\circ }\end{array}$ , in standard position .
  ::以标准位置考虑角度 30 。

  

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  Now consider the angle $\begin{array}{r}{390}^{\circ }\end{array}$ . We can think of this angle as a full rotation $\begin{array}{r}({360}^{\circ })\end{array}$ , plus an additional 30 degrees.
  ::现在考虑角度 390\\\\\\\\\\\\\\\\\\\\\\\"390\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\可以把这个角度 完全旋转(360\\\\\\\\\\\\\\\\\\\\增加30度。

  

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  Notice that $\begin{array}{r}{390}^{\circ }\end{array}$ looks the same as $\begin{array}{r}{30}^{\circ }\end{array}$ . Formally, we say that the angles share the same terminal side. Therefore we call the angles co-terminal . Not only are these two angles co-terminal, but there are infinitely many angles that are co-terminal with these two angles. For example, if we rotate another $\begin{array}{r}{360}^{\circ }\end{array}$ , we get the angle $\begin{array}{r}{750}^{\circ }\end{array}$ . Or, if we create the angle in the negative direction (clockwise), we get the angle $\begin{array}{r}-{330}^{\circ }\end{array}$ . Because we can rotate in either direction, and we can rotate as many times as we want, we can continuously generate angles that are co-terminal with $\begin{array}{r}{30}^{\circ }\end{array}$ .
  ::请注意 390 {} 看上去与 30 {} 30} 相同。 形式上, 我们说角度是相同的终点边。 因此, 我们称角度为共同终点。 不仅这两个角度为共同终点, 而且还有无限多的角度与这两个角度是共同终点。 例如, 如果我们再旋转360 }, 我们就会得到 750 } 。 或者, 如果我们在负方向创建角度( 时针) , 我们就会得到角度 - 330 } 。 因为两个方向都可以旋转, 我们可以旋转多少次, 我们可以不断生成与 30 } 共同终点的角 。

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  Identifying Co-Terminal Angles 
  ::确定共同终点

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  For the following questions, determine if the angle is co-terminal with  $\begin{array}{r}{45}^{\circ }\end{array}$ . 
  ::对于下列问题,确定角度是否与45++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

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

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  No, it is not co-terminal with $\begin{array}{r}{45}^{\circ }\end{array}$
  ::不,它不是 共同终结与45

  2.  $\begin{array}{r}{405}^{\circ }\end{array}$

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  Yes, $\begin{array}{r}{405}^{\circ }\end{array}$ is co-terminal with $\begin{array}{r}{45}^{\circ }\end{array}$ .
  ::是的 405是45的同义词

  3.  $\begin{array}{r}-{315}^{\circ }\end{array}$

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  Yes, $\begin{array}{r}-{315}^{\circ }\end{array}$ is co-terminal with $\begin{array}{r}{45}^{\circ }\end{array}$ .
  ::是的, -315 -315 -45是共同终结的。

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  Examples
  ::实例

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  Example 1
  ::例1

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  Earlier, you were asked if  it is possible to represent the angle any other way.
  ::早些时候,有人问您是否可以以任何其他方式代表角度。

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  You can either think of $\begin{array}{r}{60}^{\circ }\end{array}$ as $\begin{array}{r}{420}^{\circ }\end{array}$ if you rotate all the way around the circle once and continue the rotation to where the spinner has stopped, or as $\begin{array}{r}-{300}^{\circ }\end{array}$ if you rotate clockwise around the circle instead of counterclockwise to where the spinner has stopped.
  ::如果您在圆圈周围旋转一次,然后继续旋转到旋转器停止的位置,您可以将60 视为420;如果在圆圈周围旋转时顺时针而不是逆时针旋转到旋转器停止的位置,您也可以将60 视为420。

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  Example 2
  ::例2

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   Find a coterminal angle to $\begin{array}{r}{23}^{\circ }\end{array}$
  ::查找到 23 的共界角度@ info: whatsthis

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   A coterminal angle would be an angle that is at the same terminal place as $\begin{array}{r}{23}^{\circ }\end{array}$ but has a different value. In this case, $\begin{array}{r}-{337}^{\circ }\end{array}$ is a coterminal angle.
  ::共同终点角将是一个与 23 相同的终点角, 但具有不同价值 。 在这种情况下, 337 是一个共同终点角 。

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  Example 3
  ::例3

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  Find a coterminal angle to $\begin{array}{r}-{90}^{\circ }\end{array}$
  ::查找到 -90 的共界角度@ info: whatsthis

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  A coterminal angle would be an angle that is at the same terminal place as $\begin{array}{r}-{90}^{\circ }\end{array}$ but has a different value. In this case, $\begin{array}{r}{270}^{\circ }\end{array}$ is a coterminal angle.
  ::共同终点角是一个与 - 90 {} 相同但有不同价值的终端角。 在这种情况下, 270 { 是一个共同终点角 。

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  Example 4
  ::例4

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  Find two coterminal angles to $\begin{array}{r}{70}^{\circ }\end{array}$ by rotating in the positive direction around the circle.
  ::通过旋转圆圈周围的积极方向,找到两个共同终点角到 70\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\"\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\

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  Rotating once around the circle gives a coterminal angle of $\begin{array}{r}{430}^{\circ }\end{array}$ . Rotating again around the circle gives a coterminal angle of $\begin{array}{r}{790}^{\circ }\end{array}$ .
  ::环绕圆圈旋转一次, 共断角为 430 。 环绕圆圈旋转一次, 共断角为 790 。

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  Review
  ::回顾

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  1. Is $\begin{array}{r}{315}^{\circ }\end{array}$ co-terminal with $\begin{array}{r}-{45}^{\circ }\end{array}$ ?
     ::315和45是共同终结的吗?
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  2. Is $\begin{array}{r}{90}^{\circ }\end{array}$ co-terminal with $\begin{array}{r}-{90}^{\circ }\end{array}$ ?
     ::90和90是共同终结的吗?
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  3. Is $\begin{array}{r}{350}^{\circ }\end{array}$ co-terminal with $\begin{array}{r}-{370}^{\circ }\end{array}$ ?
     ::350 和370是共同终结的吗?
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  4. Is $\begin{array}{r}{15}^{\circ }\end{array}$ co-terminal with $\begin{array}{r}{1095}^{\circ }\end{array}$ ?
     ::15和1095是共同终结的吗?
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  5. Is $\begin{array}{r}{85}^{\circ }\end{array}$ co-terminal with $\begin{array}{r}{1880}^{\circ }\end{array}$ ?
     ::85和1880年是共同终结的吗?

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  For each diagram, name the angle in 3 ways. At least one way should use negative degrees.
  ::每个图表用三种方式命名角度。 至少一种方式应该使用负度。

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  11. Name the angle of the 8 on a standard clock two different ways.
      ::在标准时钟上以两种不同的方式命名 8 的角度 。
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  12. Name the angle of the 11 on a standard clock two different ways.
      ::在标准时钟上以两种不同的方式命名 11 的角度 。
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  13. Name the angle of the 4 on a standard clock two different ways.
      ::在标准时钟上用两种不同的方式命名 4 的角度 。
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  14. Explain how to determine whether or not two angles are co-terminal.
      ::解释如何确定两个角度是否为共同终点。
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  15. How many rotations is $\begin{array}{r}{4680}^{\circ }\end{array}$ ?
      ::4680年轮调了多少次?

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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 。