Course: 13.6 单元圆圈和弧度措施介绍

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单元圆环和弧度措施介绍
An oddly-shaped house in Asia is built at a $\begin{aligned} 135^{\circ} \end{aligned}$ angle. How many radians is this angle equal to?
::亚洲的一座奇形房子是以 135°Q 角度建造的。这个角度有多少弧度等于 ?

Unit Circle and Radian Measure
::单位圆圆和弧度测量

The unit circle is the circle centered at the origin with radius equal to one unit. This means that the distance from the origin to any point on the circle is equal to one unit.
::单位圆是指以原圆为中心,半径等于一个单位的圆。这意味着从原圆到圆上任何点的距离等于一个单位。

Using the unit circle, we can define another unit of measure for angles, radians. is based upon the circumference of the unit circle. The circumference of the unit circle is $\begin{aligned} 2 π \end{aligned}$ ( $\begin{aligned} 2 π r \end{aligned}$ , where $\begin{aligned} r = 1 \end{aligned}$ ). So a full revolution, or $\begin{aligned} 360^{\circ} \end{aligned}$ , is equal to $\begin{aligned} 2 π \end{aligned}$ radians. Half a rotation, or $\begin{aligned} 180^{\circ} \end{aligned}$ is equal to $\begin{aligned} π \end{aligned}$ radians.
::使用单位圆, 我们可以根据单位圆的环绕来定义另一个角度的单位单位。单位圆的环绕是 2 × 2 × (r= 1) 。因此, 完全的革命, 或 360 × 等于 2 × 弧度。半旋转, 或 180 × 等于 × 弧度。

One radian is equal to the measure of $\begin{aligned} θ \end{aligned}$ , the rotation required for the arc length intercepted by the angle to be equal to the radius of the circle. In other words the arc length is 1 unit for $\begin{aligned} θ = 1 \end{aligned}$ radian.
::一个弧度等于 ° 的度量, 以角截取的弧长度需要旋转以等于圆半径。换句话说, 弧长度是 1 的 1 单位。

We can use the equality, $\begin{aligned} π = 180^{\circ} \end{aligned}$ to convert from degrees to radians and vice versa.
::我们可以利用平等, 180 来从度转换成弧度,反之亦然。

To convert from degrees to radians, multiply by $\begin{aligned} \frac{π}{180^{\circ}} \end{aligned}$ .
::从度向弧度转换为弧度,乘以180。

To convert from radians to degrees, multiply by $\begin{aligned} \frac{180^{\circ}}{π} \end{aligned}$ .
::从弧度转换为度,乘以180 。

Let's convert the following units of measure.
::让我们转换以下计量单位。

Convert $\begin{aligned} 250^{\circ} \end{aligned}$ to radians.
::将 250 转换为弧度。

To convert from degrees to radians, multiply by $\begin{aligned} \frac{π}{180^{\circ}} \end{aligned}$ . So, $\begin{aligned} \frac{250 π}{180} = \frac{25 π}{18} \end{aligned}$ .
::从度向弧度转换为弧度,乘以 180 。所以, 250 180=25 18。

Convert $\begin{aligned} 3 π \end{aligned}$ to degrees.
::将3转换为度。

To convert from radians to degrees, multiply by $\begin{aligned} \frac{180^{\circ}}{π} \end{aligned}$ . So, $\begin{aligned} 3 π \times \frac{180^{\circ}}{π} = 3 \times 180^{\circ} = 540^{\circ} \end{aligned}$ .
::从弧度转换为度,乘以180。所以,乘以3180}3×180×540。

Now, let's find two angles, one positive and one negative, coterminal to $\begin{aligned} \frac{5 π}{3} \end{aligned}$ and find its reference angle , in radians.
::现在,让我们找到两个角度,一个是正的,另一个是负的, 共同终点到5°3, 找到它的参考角,以弧度。

Since we are working in radians now we will add/subtract multiple of $\begin{aligned} 2 π \end{aligned}$ instead of $\begin{aligned} 360^{\circ} \end{aligned}$ . Before we can add, we must get a common denominator of 3 as shown below.
::既然我们现在在用弧度工作,我们将增加/减号2,而不是360。在增加之前,我们必须取得以下3个共同分母。

$\begin{aligned} \frac{5 π}{3} + 2 π = \frac{5 π}{3} + \frac{6 π}{3} = \frac{11 π}{3} a n d \frac{5 π}{3} - 2 π = \frac{5 π}{3} - \frac{6 π}{3} = - \frac{π}{3} \end{aligned}$

::5 3+2 5 3+6 3= 11 3和5 3 - 2 5 3 - 6 3

Now, to find the reference angle, first determine in which quadrant $\begin{aligned} \frac{5 π}{3} \end{aligned}$ lies. If we think of the measures of the angles on the axes in terms of $\begin{aligned} π \end{aligned}$ and more specifically, in terms of $\begin{aligned} \frac{π}{3} \end{aligned}$ , this task becomes a little easier.
::现在,为了找到参考角度, 首先确定 5°3 的象限。如果我们从 ° 和更具体地说, 从 ° 3 的角度来考虑轴轴角的度量, 那么任务就变得更容易了。

Consider $\begin{aligned} π \end{aligned}$ is equal to $\begin{aligned} \frac{3 π}{3} \end{aligned}$ and $\begin{aligned} 2 π \end{aligned}$ is equal to $\begin{aligned} \frac{6 π}{3} \end{aligned}$ as shown in the diagram. Now we can see that the terminal side of $\begin{aligned} \frac{5 π}{3} \end{aligned}$ lies in the fourth quadrant and thus the reference angle will be:
::考虑 = 等于 3 3, 2 = 6 3, 如图所示。现在我们可以看到 5 = 3 的终点在第四个象限中, 因此参考角度将是 :

$\begin{aligned} \frac{6 π}{3} - \frac{5 π}{3} = \frac{π}{3} \end{aligned}$

Finally, let's find two angles coterminal to $\begin{aligned} \frac{7 π}{6} \end{aligned}$ , one positive and one negative, and find its reference angle, in radians.
::最后,让我们从两个角度共同到7 6,一个是正的,一个是负的, 在弧度中找到参考角度。

This time we will add multiples of $\begin{aligned} 2 π \end{aligned}$ with a common denominator of 6, or $\begin{aligned} \frac{2 π}{1} \times \frac{6}{6} = \frac{12 π}{6} \end{aligned}$ . For the positive angle, we add to get $\begin{aligned} \frac{7 π}{6} + \frac{12 π}{6} = \frac{19 π}{6} \end{aligned}$ . For the negative angle , we subtract to get $\begin{aligned} \frac{7 π}{6} - \frac{12 π}{6} = \frac{5 π}{6} \end{aligned}$ .
::这一次,我们将增加2的倍数, 公分母为 6 或 21×66= 12.6。对于正角, 我们增加76+126= 196。对于负角, 我们减为 76 - 126= 56。

In this case $\begin{aligned} π \end{aligned}$ is equal to $\begin{aligned} \frac{6 π}{6} \end{aligned}$ and $\begin{aligned} 2 π \end{aligned}$ is equal to $\begin{aligned} \frac{12 π}{6} \end{aligned}$ as shown in the diagram. Now we can see that the terminal side of $\begin{aligned} \frac{7 π}{6} \end{aligned}$ lies in the third quadrant and thus the reference angle will be:
::在此情况下 = 等于 6 6, 2 = 12 6, 如图所示。现在我们可以看到 7 = 6 的终点在第三个象限中, 因此引用角度将是 :

$\begin{aligned} \frac{7 π}{6} - \frac{6 π}{6} = \frac{π}{6} \end{aligned}$

Examples
::实例

Example 1
::例1

Earlier, you were asked to convert $\begin{aligned} 135^{\circ} \end{aligned}$ to radians.
::早些时候,你被要求将135转换成弧度。

To convert from degrees to radians, multiply by $\begin{aligned} \frac{π}{180^{\circ}} \end{aligned}$ . So, $\begin{aligned} \frac{135 π}{180} = \frac{3 π}{4} \end{aligned}$ .
::从度向弧度转换为弧度,乘以 180。所以,乘以 135180=34。

Convert the following angle measures from degrees to radians.
::将以下角度的度量从度转换为弧度。

Example 2
::例2

$\begin{aligned} - 45^{\circ} \end{aligned}$

$\begin{aligned} - 45^{\circ} \times \frac{π}{180^{\circ}} = - \frac{π}{4} \end{aligned}$

Example 3
::例3

$\begin{aligned} 120^{\circ} \end{aligned}$

$\begin{aligned} 120^{\circ} \times \frac{π}{180^{\circ}} = \frac{2 π}{3} \end{aligned}$

Example 4
::例4

$\begin{aligned} 330^{\circ} \end{aligned}$

$\begin{aligned} 330^{\circ} \times \frac{π}{180^{\circ}} = \frac{11 π}{6} \end{aligned}$

Convert the following angle measures from radians to degrees.
::将以下角度的度量从弧度转换为度。

Example 5
::例5

$\begin{aligned} \frac{5 π}{6} \end{aligned}$

$\begin{aligned} \frac{5 π}{6} \times \frac{180^{\circ}}{π} = 150^{\circ} \end{aligned}$

Example 6
::例6

$\begin{aligned} \frac{13 π}{4} \end{aligned}$

$\begin{aligned} \frac{13 π}{4} \times \frac{180^{\circ}}{π} = 585^{\circ} \end{aligned}$

Example 7
::例7

$\begin{aligned} - \frac{5 π}{2} \end{aligned}$

$\begin{aligned} - \frac{5 π}{2} \times \frac{180^{\circ}}{π} = - 450^{\circ} \end{aligned}$

Example 8
::例8

Find two to $\begin{aligned} \frac{11 π}{4} \end{aligned}$ , one positive and one negative, and its reference angle.
::发现2到11 4, 1正1和1负1, 及其参考角度。

There are many possible coterminal angles, here are some possibilities:
::有许多可能的共同终点角度,

positive coterminal angle: $\begin{aligned} \frac{11 π}{4} + \frac{8 π}{4} = \frac{19 π}{4} \end{aligned}$ or $\begin{aligned} \frac{11 π}{4} - \frac{8 π}{4} = \frac{3 π}{4} \end{aligned}$ ,
::阳性共界角:114+84=194或114-84=34;

negative coterminal angle: $\begin{aligned} \frac{11 π}{4} - \frac{16 π}{4} = - \frac{5 π}{4} \end{aligned}$ or $\begin{aligned} \frac{11 π}{4} - \frac{24 π}{4} = - \frac{13 π}{4} \end{aligned}$
::负共同终点角: 114 - 164 - 54 或 114 - 244 - 134

Using the coterminal angle, $\begin{aligned} \frac{3 π}{4} \end{aligned}$ , which is $\begin{aligned} \frac{π}{4} \end{aligned}$ from $\begin{aligned} \frac{4 π}{4} \end{aligned}$ . So the terminal side lies in the second quadrant and the reference angle is $\begin{aligned} \frac{π}{4} \end{aligned}$ .
::使用 3+4 的共界角度, 3+4 , 也就是 4+4 的 4。因此终端侧面位于第二个象限, 参考角度是 ++4 。

Review
::回顾

For problems 1-5, convert the angle from degrees to radians. Leave answers in terms of $\begin{aligned} π \end{aligned}$ .
::对于问题1-5, 将角度从度转换为弧度。允许回答。

$\begin{aligned} 135^{\circ} \end{aligned}$

$\begin{aligned} 240^{\circ} \end{aligned}$

$\begin{aligned} - 330^{\circ} \end{aligned}$

$\begin{aligned} 450^{\circ} \end{aligned}$

$\begin{aligned} - 315^{\circ} \end{aligned}$

For problems 6-10, convert the angle measure from radians to degrees.
::对于问题 6-10, 将角度量度从弧度转换为度。

$\begin{aligned} \frac{7 π}{3} \end{aligned}$

$\begin{aligned} - \frac{13 π}{6} \end{aligned}$

$\begin{aligned} \frac{9 π}{2} \end{aligned}$

$\begin{aligned} - \frac{3 π}{4} \end{aligned}$

$\begin{aligned} \frac{5 π}{6} \end{aligned}$

For problems 11-15, find two coterminal angles (one positive, one negative) and the reference angle for each angle in radians.
::对于问题11-15, 发现两个共同终点角(一个是正角,一个是负角)和每个角的弧度参考角。

$\begin{aligned} \frac{8 π}{3} \end{aligned}$

$\begin{aligned} \frac{11 π}{4} \end{aligned}$

$\begin{aligned} - \frac{π}{6} \end{aligned}$

$\begin{aligned} \frac{4 π}{3} \end{aligned}$

$\begin{aligned} - \frac{17 π}{6} \end{aligned}$

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

单元圆环和弧度措施介绍

Unit Circle and Radian Measure ::单位圆圆和弧度测量

Examples ::实例

Example 1 ::例1

Example 2 ::例2

Example 3 ::例3

Example 4 ::例4

Example 5 ::例5

Example 6 ::例6

Example 7 ::例7

Example 8 ::例8

Review ::回顾

Review (Answers) ::回顾(答复)

Unit Circle and Radian Measure
::单位圆圆和弧度测量

Examples
::实例

Example 1
::例1

Example 2
::例2

Example 3
::例3

Example 4
::例4

Example 5
::例5

Example 6
::例6

Example 7
::例7

Example 8
::例8

Review
::回顾

Review (Answers)
::回顾(答复)