Course: 13.5 旋转角、交界角和参考角介绍

Section outline

Select section General

Collapse Expand
General

Collapse all Expand all
- Select activity 新闻通告
  
  新闻通告 Forum
Select section 旋转角、交界角和参考角的介绍

Collapse Expand
旋转角、交界角和参考角的介绍
In which quadrant does the terminal side of the angle $\begin{aligned} - 500^{o} \end{aligned}$ lie and what is the reference angle for this angle?
::角度 - 500o 的端端面在哪个象限是 ? 这个角的参考角是什么 ?

Angles of Rotation
::旋转角度

Angles of rotation are formed in the coordinate plane between the positive $\begin{aligned} x \end{aligned}$ -axis (initial side) and a ray (terminal side) . Positive angle measures represent a counterclockwise rotation while negative angles indicate a clockwise rotation.
::在正 X 轴( 初始面) 和射线( 终点面) 之间的坐标平面中形成旋转角度。积极的角度测量表示逆时针旋转, 而负角度则表示顺时针旋转。

Since the $\begin{aligned} x \end{aligned}$ and $\begin{aligned} y \end{aligned}$ axes are perpendicular, each axis then represents an increment of ninety degrees of rotation. The diagrams below show a variety of angles formed by rotating a ray through the quadrants of the coordinate plane.
::由于 x 和 y 轴是垂直的, 因此每个轴的递增为90 度的旋转。下面的图表显示了通过坐标平面的四角旋转射线而形成的各种角度。

An angle of rotation can be described infinitely many ways. It can be described by a positive or negative angle of rotation or by making multiple full circle rotations through $\begin{aligned} 360^{o} \end{aligned}$ . The example below illustrates this concept.
::旋转的角度可以描述得无穷无尽,可以用旋转的正或负角度来描述,也可以用旋转的正或负角度来描述,或者通过360o进行多次全圆旋转来描述。下面的例子说明了这个概念。

For the angle $\begin{aligned} 525^{o} \end{aligned}$ , an entire $\begin{aligned} 360^{o} \end{aligned}$ rotation is made and then we keep going another $\begin{aligned} 165^{o} \end{aligned}$ to $\begin{aligned} 525^{o} \end{aligned}$ . Therefore, the resulting angle is equivalent to $\begin{aligned} 525^{o} - 360^{o} \end{aligned}$ , or $\begin{aligned} 165^{o} \end{aligned}$ . In other words, the terminal side is in the same location as the terminal side for a $\begin{aligned} 165^{o} \end{aligned}$ angle. If we subtract $\begin{aligned} 360^{o} \end{aligned}$ again, we get a negative angle, $\begin{aligned} - 195^{o} \end{aligned}$ . Since they all share the same terminal side, they are called .
::角度 525o , 整个 360o 旋转, 然后我们再继续 165o 到 525o 。因此, 由此产生的角度相当于 525o - 360o , 或者 165o 。换句话说, 终端侧的位置与终端侧的位置相同 165o 角度。如果我们再减去 360o, 我们得到一个负角度 , 也就是 1- 195o 。由于它们都具有相同的终端侧面, 它们被调用。

Let's determine two coterminal angles to $\begin{aligned} 837^{o} \end{aligned}$ , one positive and one negative.
::让我们确定两个共同终点角度到837o, 一个正数和一个负数。

To find coterminal angles we simply add or subtract $\begin{aligned} 360^{o} \end{aligned}$ multiple times to get the angles we desire. $\begin{aligned} 837^{o} - 360^{o} = 477^{o} \end{aligned}$ , so we have a positive coterminal angle. Now we can subtract $\begin{aligned} 360^{o} \end{aligned}$ again to get $\begin{aligned} 477^{o} - 360^{o} = 117^{o} \end{aligned}$ .
::为了找到共同终点角度,我们只需多加或减去360o次即可获得我们想要的角。 837o- 360o=477o, 因此我们有一个正共同终点角度。现在我们可以再减去360o, 以获得477o- 360o=117o。

Reference Angle
::参考角

A reference angle is the acute angle between the terminal side of an angle and the $\begin{aligned} x \end{aligned}$ – axis. The diagram below shows the reference angles for terminal sides of angles in each of the four quadrants.
::参考角是角度末端与 x - 轴之间的急性角。下图显示了四个方位中每个角度的末端角的参考角。

Note: A reference angle is never determined by the angle between the terminal side and the $\begin{aligned} y \end{aligned}$ – axis. This is a common error for students, especially when the terminal side appears to be closer to the $\begin{aligned} y \end{aligned}$ – axis than the $\begin{aligned} x \end{aligned}$ – axis.
::注:一个参考角度从不由终端面与 y 轴之间的角来决定。对于学生来说,这是一个常见错误,特别是当终端面似乎接近 y 轴而不是 x 轴时。

Now, let's determine the quadrant in which $\begin{aligned} - 745^{o} \end{aligned}$ lies and hence determine the reference angle.
::现在,让我们来决定 745o 所在的象限, 从而决定参考角度。

Since our angle is more than one rotation, we need to add $\begin{aligned} 360^{o} \end{aligned}$ until we get an angle whose absolute value is less than $\begin{aligned} 360^{o} \end{aligned}$ : $\begin{aligned} - 745^{o} + 360^{o} = - 385^{o} \end{aligned}$ , again $\begin{aligned} - 385^{o} + 360^{o} = - 25^{o} \end{aligned}$ .
::由于我们的角度不只一个旋转,我们需要增加360o, 直到我们得到一个绝对值低于360o的角: - 745o+360o___385o, 也就是 - 385o+360o_25o。

Now we can plot the angle and determine the reference angle:
::现在我们可以绘制角度,确定参考角度:

Note that the reference angle is positive $\begin{aligned} 25^{o} \end{aligned}$ . All reference angles will be positive as they are acute angles (between $\begin{aligned} 0^{o} \end{aligned}$ and $\begin{aligned} 90^{o} \end{aligned}$ ).
::请注意,参考角为正25o。所有参考角均为正25o。所有参考角均为正25o,因为它们是急性角(0o至90o之间)。

Finally, let's give two coterminal angles to $\begin{aligned} 595^{\circ} \end{aligned}$ , one positive and one negative, and find the reference angle.
::最后,让我们给595两个共同终点角度, 一个是正的,一个是负的, 并找到参考角度。

To find the coterminal angles we can add/subtract $\begin{aligned} 360^{\circ} \end{aligned}$ . In this case, our angle is greater than $\begin{aligned} 360^{\circ} \end{aligned}$ so it makes sense to subtract $\begin{aligned} 360^{\circ} \end{aligned}$ to get a positive coterminal angle: $\begin{aligned} 595^{\circ} - 360^{\circ} = 235^{\circ} \end{aligned}$ . Now subtract again to get a negative angle: $\begin{aligned} 235^{\circ} - 360^{\circ} = - 125^{\circ} \end{aligned}$ .
::要找到共同终点角, 我们可以添加/ 减号 360 。在此情况下, 我们的角大于 360 。因此, 减去 360 以获得正共终点角是有道理的 : 595 360 235 。现在再次减号, 以获得负角 : 235 360 125 。

By plotting any of these angles we can see that the terminal side lies in the third quadrant as shown.
::通过绘制这些角度中的任何角度,我们可以看到,终端侧面位于显示的第三个象限。

Since the terminal side lies in the third quadrant, we need to find the angle between $\begin{aligned} 180^{\circ} \end{aligned}$ and $\begin{aligned} 235^{\circ} \end{aligned}$ , so $\begin{aligned} 235^{\circ} - 180^{\circ} = 55^{\circ} \end{aligned}$ .
::因为终点在第三个象限, 我们需要找到180和235之间的角, 所以是235和180。

Examples
::实例

Example 1
::例1

Earlier, you were asked to find the reference angle of $\begin{aligned} - 500^{o} \end{aligned}$ and find the quadrant in which the terminal side lies.
::早些时候,有人要求你找到- 500o 的参考角,并找到终点侧所在的象限。

Since our angle is more than one rotation, we need to add $\begin{aligned} 360^{o} \end{aligned}$ until we get an angle whose absolute value is less than $\begin{aligned} 360^{o} \end{aligned}$ : $\begin{aligned} - 500^{o} + 360^{o} = - 200^{o} \end{aligned}$ .
::由于我们的角度不只一个旋转,我们需要增加360o, 直到我们得到一个绝对值低于360o的角: - 500o+360o200o。

If we plot this angle we see that it is $\begin{aligned} - 200^{o} \end{aligned}$ clockwise from the origin or $\begin{aligned} 160^{o} \end{aligned}$ counterclockwise. $\begin{aligned} 160^{o} \end{aligned}$ lies in the second quadrant.
::如果我们绘制这个角度, 我们就会看到它从源头到正时速为- 200o, 或者从正时速为 160o。 160o 位于第二个象限。

Now determine the reference angle: $\begin{aligned} 180^{o} - 160^{o} = 20^{o} \end{aligned}$ .
::现在确定参考角: 180o- 160o=20o。

Example 2
::例2

Find two coterminal angles to $\begin{aligned} 138^{o} \end{aligned}$ , one positive and one negative.
::发现两个共同终点角到138o,一个是正,另一个是负。

$\begin{aligned} 138^{o} + 360^{o} = 498^{o} \end{aligned}$ and $\begin{aligned} 138^{o} - 360^{o} = - 222^{o} \end{aligned}$
::138+360o=498o和138-360o222o

Example 3
::例3

Find the reference angle for $\begin{aligned} 895^{o} \end{aligned}$ .
::查找895o的引用角度。

$\begin{aligned} 895^{o} - 360^{o} = 535^{o}, 535^{o} - 360^{o} = 175^{o} \end{aligned}$ . The terminal side lies in the second quadrant, so we need to determine the angle between $\begin{aligned} 175^{o} \end{aligned}$ and $\begin{aligned} 180^{o} \end{aligned}$ , which is $\begin{aligned} 5^{o} \end{aligned}$ .
::895o-360o=535o,535o-360o=175o。终端侧面位于第二个象限,所以我们需要确定175o至180o之间的角,即5o。

Example 4
::例4

Find the reference angle for $\begin{aligned} 343^{o} \end{aligned}$ .
::查找343o 的引用角度。

$\begin{aligned} 343^{o} \end{aligned}$ is in the fourth quadrant so we need to find the angle between $\begin{aligned} 343^{o} \end{aligned}$ and $\begin{aligned} 360^{o} \end{aligned}$ which is $\begin{aligned} 17^{o} \end{aligned}$ .
::343度是第四象限所以我们需要找到343度到360度之间的角也就是17度

Review
::回顾

Find two coterminal angles to each angle measure, one positive and one negative.
::在每个角度测量中找到两个共同终点角,一个是正角,另一个是负角。

$\begin{aligned} - 98^{o} \end{aligned}$
::- 98度

$\begin{aligned} 475^{o} \end{aligned}$
::475o

$\begin{aligned} - 210^{o} \end{aligned}$
::- 210o

$\begin{aligned} 47^{o} \end{aligned}$
::47o) 47o

$\begin{aligned} - 1022^{o} \end{aligned}$
::- 1022o

$\begin{aligned} 354^{o} \end{aligned}$
::354o

$\begin{aligned} - 7^{o} \end{aligned}$
::-7度

Determine the quadrant in which the terminal side lies and find the reference angle for each of the following angles.
::确定终端侧面所在的象限,并找到以下每个角度的参考角度。

$\begin{aligned} 102^{o} \end{aligned}$
::102o

$\begin{aligned} - 400^{o} \end{aligned}$
::- 4000o

$\begin{aligned} 1307^{o} \end{aligned}$
::1307o

$\begin{aligned} - 820^{o} \end{aligned}$
::-820o

$\begin{aligned} 304^{o} \end{aligned}$
::304o

$\begin{aligned} 251^{o} \end{aligned}$
::251o

$\begin{aligned} - 348^{o} \end{aligned}$
::- 348o

Explain why the reference angle for an angle between $\begin{aligned} 0^{o} \end{aligned}$ and $\begin{aligned} 90^{o} \end{aligned}$ is equal to itself.
::解释0o至90o之间角的参考角度为何等于本身。

Review (Answers)
::回顾(答复)

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。

Section outline

General

旋转角、交界角和参考角的介绍

Angles of Rotation ::旋转角度

Reference Angle ::参考角

Examples ::实例

Example 1 ::例1

Example 2 ::例2

Example 3 ::例3

Example 4 ::例4

Review ::回顾

Review (Answers) ::回顾(答复)

Angles of Rotation
::旋转角度

Reference Angle
::参考角

Examples
::实例

Example 1
::例1

Example 2
::例2

Example 3
::例3

Example 4
::例4

Review
::回顾

Review (Answers)
::回顾(答复)