标准位置的旋转角度

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Introduction
::导言

 

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To do this, calculations for the area of the fields to be irrigated have to be converted so that the length of the irrigation pipe and the angle that it is rotated are used. Until now, the grower only used area measurements that considered linear dimensions, such as length and width. To calculate the area given the length of the irrigation pipe, the grower first has to understand angles and how they are measured.
::为此,必须转换灌溉田面积的计算方法,以便使用灌溉管道的长度和旋转角。到目前为止,种植者只使用考虑线性尺寸(如长度和宽度)的区域测量方法。要计算灌溉管道长度的面积,种植者首先必须了解角度和如何测量角度。

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

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Rotation angles can be measured in degrees. You may recall from geometry that one full rotation is 360 degrees, usually written as $\begin{array}{r}{360}^{\circ }\end{array}$ . Half a rotation is  $\begin{array}{r}{180}^{\circ },\end{array}$ and a quarter rotation is $\begin{array}{r}{90}^{\circ }\end{array}$ . As linear quantities  $\begin{array}{r}x\end{array}$  and  $\begin{array}{r}y\end{array}$  are represented in a standard way graphically, a standard is needed for angular measure. The figure below shows an angle in what is called standard position .
::旋转角度可以以度度来测量。 您可以从几何中记得一个完全旋转是360度, 通常写为360°。 半旋转是180°, 四分之一旋转是90°。 由于直线数量 x 和 y 以图形方式以标准方式表示, 角度测量需要一个标准。 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 the angle measures a counterclockwise rotation, the  angle has a positive measure. A clockwise rotation corresponds to a negative measure. 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- 轴值上。 终端侧面总是在源头与初始侧面相匹配。 注意旋转以逆时针方向进行。 这意味着, 如果角度用反时针旋转, 角度会有一个正度的度量。 顺时针旋转与负度相对。 下面是标准位置角度的几个例子 :

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Learn, Play, and Explore with Angles of Rotation: 
::学习、玩耍和探索与旋转角度:

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Coterminal Angles
::交界形角

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In the example below, the angle whose measure is  $\begin{array}{r}{30}^{\circ }\end{array}$  could also be expressed as  $\begin{array}{r}{390}^{\circ }\end{array}$  or even  $\begin{array}{r}-{330}^{\circ }\end{array}$ . These three measures have the same initial side and the same terminal side, so they represent the same angle. They are called .
::在下面的例子中,测量量为30°C的角也可以表示为390°C,甚至330°C。这三项测量量的初始面和终端面相同,因此代表了同一角度。它们被称为 。

 

 

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Quadrantal Angles
::四方角

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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. Quadrantal angles are  $\begin{array}{r}{0}^{\circ },{90}^{\circ },{180}^{\circ },\end{array}$ and $\begin{array}{r}{270}^{\circ }\end{array}$ .
::90 度角是 4 个 方 角 。 方 方 角 是 轴 的 末端 。 方 角 是 0 、 90 、 180 和 270 。

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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 4th 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, 没有箭头显示旋转, 270( 270) 看起来像是- 90( - 90) , 定义了第四方形 。 同时通知 360( 360) 看起来像 0( 0) 。

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Examples
::实例

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Example 1
::例1

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What are the angles in standard position represented by these graphs?
::这些图表代表的标准位置的角度是什么 ?

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Solution :  
::解决方案 :

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The angles represented are  $\begin{array}{r}{0}^{\circ }\end{array}$ ,  $\begin{array}{r}{30}^{\circ }\end{array}$   and  $\begin{array}{r}{135}^{\circ }\end{array}$ .
::表示的角度是 0, 30和135。

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Example 2
::例2

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Identify the measure of the angle represented below. Use a negative measure.
::确定以下角度的测量值。使用负计量值。

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Solution :  
::解决方案 :

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The angle's measure is   $\begin{array}{r}-{135}^{\circ }\end{array}$ .
::角度的量度是 -135

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Example 3
::例3

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Identify the measure of the angle represented below. Use a positive and a negative measure.
::确定以下角度的度量。使用正和负的度量。

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Solution :  
::解决方案 :

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The angle is $\begin{array}{r}{180}^{\circ }\end{array}$ or  $\begin{array}{r}-{180}^{\circ }\end{array}$  .
::角度是180或-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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Solution :  
::解决方案 :

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The angle's measure is $\begin{array}{r}-{225}^{\circ }\end{array}$ .
::角度的量度是-225。

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Example 5
::例5

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Which of the following angles are coterminal with $\begin{array}{r}{45}^{\circ }\end{array}$ ?
::以下哪个角度与 45 Q 相交 ?

-45°, 405°, -315°, 135°

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Solution:
::解决方案 :

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405° and -315° are coterminal with 45°.
::405°和-315°是45°的共交点。

 

 

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Summary
::摘要

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• Angles in standard position have their initial side on the positive $\begin{array}{r}x-\end{array}$ axis . The positive direction for the angle is counterclockwise. The angle ends at its terminal side.
  ::标准位置的角在正 X - 轴上有其初始边。角的正方向是逆时针。角在终端边端结束 。
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• Coterminal angles are angles that can be represented in many ways but are equivalent angles, because they have the same initial side and terminal side. 
  ::直角是可以用多种方式表示的角,但具有同等角度,因为它们有相同的初始侧面和末端面。
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• Quadrantal angles have their terminal side on one of the axes.
  ::四方角度的终端侧面 位于轴上

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Review
::回顾

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1. Draw an angle with measure  $\begin{array}{r}{90}^{\circ }\end{array}$ .
   ::用度量 90\\\ 绘制角度 。
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2. Draw an angle with measure  $\begin{array}{r}{45}^{\circ }\end{array}$ .
   ::绘制带有度量 45 的角 。
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3. Draw an angle with measure  $\begin{array}{r}-{135}^{\circ }\end{array}$ .
   ::绘制角度, 度量为- 135\\\\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ - \ - \ \ \ - \ - \ - \ - \ - \ - \ - \ - \ - \ - \ - \ - \ - \ - \ - \ - \ - \
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4. Draw an angle with measure  $\begin{array}{r}-{45}^{\circ }\end{array}$ .
   ::绘制一个角度, 度量为- 45\\\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ - \ \ \
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5. Draw an angle with measure  $\begin{array}{r}-{270}^{\circ }\end{array}$ .
   ::用度量 - 270\\\\ 来绘制角度 。
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6. Draw an angle with measure  $\begin{array}{r}{315}^{\circ }\end{array}$ .
   ::用度量为 315\\\\\ 绘制角度 。

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For each diagram, identify the angle's positive measure:
::对于每一图表,请标明角度的积极度量:

7.

8.

9.

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For each diagram below, identify the angle. Write the angle using negative degrees.
::下面的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.  Which standard positive angle has a terminal angle that includes 7 on a standard 12-hour clock? Use positive degrees.
::13. 哪个标准正角具有终端角,包括标准12小时钟的7个角?使用正度。

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14.  Which standard positive angle has a terminal angle that includes 2 on a standard 12-hour clock? Use positive degrees.
::14. 哪个标准正角具有终端角,包括标准12小时钟2个角?使用正度。

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Review (Answers) 
::回顾(答复)

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Please see the Appendix.
::请参看附录。