1.15 标准位置的旋转角度

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  标准位置的旋转角度

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  While playing a game with friends, you are using a spinner. You know that the best number to land on is $\begin{array}{r}7\end{array}$ . The spinner looks like this:
  ::在和朋友玩游戏时,你正在使用一个旋转轴。你知道着陆的最佳数字是7。 旋转轴看起来像这样:

  

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  Can you determine how to represent the angle of the spinner if it lands on the $\begin{array}{r}7\end{array}$ ?
  ::如果旋转器降落在7号上,你能确定如何代表旋转器的角度吗?

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  Angles of Rotation in Standard Positions
  ::标准位置的旋转角度

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  Consider our game that is played with a spinner. When you spin the spinner, how far has it gone? You can answer this question in several ways. You could say something like “the spinner spun around 3 times.” This means that the spinner made 3 complete rotations, and then landed back where it started.
  ::想想我们与一个旋转器玩过的游戏。 当你旋转旋转时, 它走了多远? 您可以以几种方式回答这个问题。 您可以说“ 旋转器旋转了3次 ” 。 这意味着旋转器旋转了3次, 然后又回到了它开始的位置。

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  We can also measure the rotation in degrees. In the previous lesson we worked with angles in triangles, measured in degrees. You may recall from geometry that a full rotation is 360 degrees, usually written as $\begin{array}{r}{360}^{\circ }\end{array}$ . Half a rotation is then $\begin{array}{r}{180}^{\circ }\end{array}$ and a quarter rotation is $\begin{array}{r}{90}^{\circ }\end{array}$ . Each of these measurements will be important in this Concept. We can use our knowledge of graphing to represent any angle. The figure below shows an angle in what is called standard position .
  ::我们也可以用度来测量旋转。 在上一课中, 我们用三角形的角度, 以度计。 您可以从几何中记得, 完全旋转是360度, 通常写成360度。 半个旋转是180度, 四分之一旋转是90度。 每一个测量都对这个概念很重要 。 我们可以使用我们的图形知识来代表任何角度 。 下图显示一个角度, 即所谓的标准位置 。

  

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  The initial side of an angle in standard position is always on the positive $\begin{array}{r}x-\end{array}$ axis. The terminal side always meets the initial side at the origin. Notice that the rotation goes in a counterclockwise direction. This means that if we rotate clockwise , we will generate a negative angle . Below are several examples of angles in standard position.
  ::标准位置角度的初始侧面总是在正 X - 轴上。 终端侧面总是在源头的初始侧面相匹配。 注意旋转在逆时针方向进行。 这意味着如果我们按时针旋转, 我们就会产生负角度。 下面是标准位置角度的几个例子 。

  

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  The 90 degree angle is one of four quadrantal angles. A quadrantal angle is one whose terminal side lies on an axis. Along with $\begin{array}{r}{90}^{\circ },0^{\circ },{180}^{\circ }\end{array}$ and $\begin{array}{r}{270}^{\circ }\end{array}$ are quadrantal angles.
  ::90 度角是 4 度角之一。 4 度角是 轴的终点边。 与 90 度、 0 度、 180 度 和 270 度 一样, 也是 4 度角 。

  

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  These angles are referred to as quadrantal because each angle defines a quadrant. Notice that without the arrow indicating the rotation, $\begin{array}{r}{270}^{\circ }\end{array}$ looks as if it is a $\begin{array}{r}-{90}^{\circ }\end{array}$ , defining the fourth quadrant. Notice also that $\begin{array}{r}{360}^{\circ }\end{array}$ would look just like $\begin{array}{r}{0}^{\circ }\end{array}$ .
  ::这些角度被称为四方形, 因为每个角度都定义了方形 。 请注意, 没有箭头指示旋转, 270 看起来像是 - 90 , 定义了第四个方形 。 通知 360 也看起来像 0 。

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  Finding the Angle of Rotation
  ::寻找旋转的角

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  Identify what the angle is in this graph:
  ::识别此图中的角度 :

  

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  The angle drawn out is $\begin{array}{r}{135}^{\circ }\end{array}$ .
  ::引出的角度是135。

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  Identifying Angles 
  ::识别角度

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  Identifying the angles within the following graphs. 
  ::在下图中辨别角度。

  1. 

  

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  The angle drawn out is $\begin{array}{r}{0}^{\circ }\end{array}$ .
  ::引出的角度是 0 。

  2. 

  

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  The angle drawn out is $\begin{array}{r}{30}^{\circ }\end{array}$
  ::引出的角度是30

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  Examples
  ::实例

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  Example 1
  ::例1

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  Earlier, you were asked to determine how to represent the angle of the spinner if it lands on the 7. 
  ::早些时候,你被要求确定 如何代表旋转器的角度 如果它降落在7。

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  Since you know that the angle between the horizontal and vertical directions is $\begin{array}{r}{90}^{\circ }\end{array}$ , each number on the spinner takes up $\begin{array}{r}{30}^{\circ }\end{array}$ . Therefore, since you are on the $\begin{array}{r}7\end{array}$ , you know that you are $\begin{array}{r}\frac{2}{3}\end{array}$ of the way to the vertical. Therefore, the angle of the spinner when it lands on $\begin{array}{r}7\end{array}$ is $\begin{array}{r}{60}^{\circ }\end{array}$ .
  ::既然你知道水平方向和垂直方向之间的角是 90 , 旋转器上每个数字会增加30 。 因此, 既然你在7 上, 你就会知道自己是 23 的垂直方向。 因此, 旋转器在7 上降落时的角是 60 。

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  Example 2
  ::例2

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  Identify what the angle is in this graph, using negative angles:
  ::使用负角度识别此图中的角度 :

  

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  The angle drawn out is $\begin{array}{r}-{135}^{\circ }\end{array}$ .
  ::引出的角度是-135。

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  Example 3
  ::例3

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  Identify what the angle is in this graph, using negative angles:
  ::使用负角度识别此图中的角度 :

  

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  The angle drawn out is $\begin{array}{r}-{180}^{\circ }\end{array}$ .
  ::引出的角度是-180。

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  Example 4
  ::例4

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  Identify what the angle is in this graph, using negative angles:
  ::使用负角度识别此图中的角度 :

  

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   The angle drawn out is $\begin{array}{r}-{225}^{\circ }\end{array}$ .
  ::引出的角度是-225。

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  Review
  ::回顾

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  1. Draw an angle of $\begin{array}{r}{90}^{\circ }\end{array}$ .
     ::绘制 90 的角 。
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  2. Draw an angle of $\begin{array}{r}{45}^{\circ }\end{array}$ .
     ::绘制角度为 45。
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  3. Draw an angle of $\begin{array}{r}-{135}^{\circ }\end{array}$ .
     ::绘制- 135\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
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  4. Draw an angle of $\begin{array}{r}-{45}^{\circ }\end{array}$ .
     ::绘制到- 45\\\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ - 45\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ - \ \ \ \ \ \ \ \ \ \ \
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  5. Draw an angle of $\begin{array}{r}-{270}^{\circ }\end{array}$ .
     ::绘制- 270\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
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  6. Draw an angle of $\begin{array}{r}{315}^{\circ }\end{array}$ .
     ::绘制 315 角度 。

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  For each diagram, identify the angle. Write the angle using positive degrees.
  ::每个图表都标明角度。 使用正度写入角度 。

  7. 

  8. 

  9. 

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  For each diagram, identify the angle. Write the angle using negative degrees.
  ::每个图表都标明角度。 使用负度写入角度 。

  10. 

  11. 

  12. 

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  13. Explain how to convert between angles that use positive degrees and angles that use negative degrees.
      ::解释如何在使用正度的角与使用负度的角之间转换。
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  14. At what angle is the 7 on a standard 12-hour clock? Use positive degrees.
      ::标准 12 小时钟的 7 角度是什么 ? 使用正度 。
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  15. At what angle is the 2 on a standard 12-hour clock? Use positive degrees.
      ::标准 12 小时钟上的 2 角度是什么 ? 使用正度 。

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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 。