课程： 2.2 度与弧度之间的转换

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度与弧度之间的转换
You are hard at work in the school science lab when your teacher asks you to turn a knob on a detector you are using $\begin{aligned} 75^{\circ} \end{aligned}$ degrees. Unfortunately, you have been working in radians for a while, and so you're having trouble remembering how far to turn the knob. Is there a way to translate the instructions in degrees to radians?
::您在学校科学实验室工作很辛苦, 当您的老师要求您在使用 75 ° 度 75 ° 度的检测器上打开一个把手。不幸的是, 您在弧度中工作了一段时间, 所以您记不清要翻转点有多远。您有办法将指示的度数转换为弧度吗 ?

Conversion between Degrees and Radians
::度与弧度之间的转换

Since degrees and radians are different ways of measuring the distance moved around the circumference of a circle, it is reasonable to suppose that there is a conversion formula between these two units. This formula works for all degrees and radians. Remember that: $\begin{aligned} π radians = 180^{\circ} \end{aligned}$ . If you divide both sides of this equation by $\begin{aligned} π \end{aligned}$ , you will have the conversion formula:
::由于度和弧度是测量环圆周围移动的距离的不同方式,因此有理由假定这两个单位之间有一个转换公式。这个公式适用于所有度和弧度。要记住 : 弧度=180 。如果将此方程的两边除以 ,您将拥有转换公式:

$\begin{aligned} radians \times \frac{180}{π} = degrees \end{aligned}$

::弧度x180 度

If we have a degree measure and wish to convert it to radians, then manipulating the equation above gives:
::如果我们有一个度量度,并希望将其转换成弧度,那么操纵上面的方程式就会给出:

$\begin{aligned} degrees \times \frac{π}{180} = radians \end{aligned}$

::°180=弧度

Let's look at a few problems where we convert between degrees and radians.
::让我们来研究几个问题, 我们如何在度和弧度之间转换。

1. Convert $\begin{aligned} \frac{11 π}{3} \end{aligned}$ to degree measure.
::1. 将113转换为度量。

From the last section, you should recognize that this angle is a multiple of $\begin{aligned} \frac{π}{3} \end{aligned}$ (or 60 degrees), so there are 11, $\begin{aligned} \frac{π}{3} \end{aligned}$ 's in this angle, $\begin{aligned} \frac{π}{3} \times 11 = 60^{\circ} \times 11 = 660^{\circ} \end{aligned}$ .
::从上一节中,您应该认识到这个角度是% 3 (或60度) 的倍数, 所以这个角度有 11 ,% 3 3 = 60 11= 660 = 。

Here is what it would look like using the formula:
::使用公式看起来是这样的:

$\begin{aligned} radians \times \frac{180}{π} = degrees \end{aligned}$
::弧度x180 度

2. Convert $\begin{aligned} - 120^{\circ} \end{aligned}$ to . Leave the answer in terms of $\begin{aligned} π \end{aligned}$ .
::2. 将-120改为.将答案保留为-。

$\begin{aligned} degrees \times \frac{π}{180} & = radians \\ - 120^{\circ} \times \frac{π}{180} & = \frac{- 120^{\circ} π}{180} \end{aligned}$

::180=弧度-120 180 120 120 180

and reducing to lowest terms gives us $\begin{aligned} - \frac{2 π}{3} \end{aligned}$
::减到最低给我们 -2 3

You could also have noticed that 120 is $\begin{aligned} 2 \times 60 \end{aligned}$ . Since $\begin{aligned} 60^{\circ} \end{aligned}$ is $\begin{aligned} \frac{π}{3} \end{aligned}$ radians, then 120 is 2, $\begin{aligned} \frac{π}{3} \end{aligned}$ ’s, or $\begin{aligned} \frac{2 π}{3} \end{aligned}$ . Make it negative and you have the answer, $\begin{aligned} - \frac{2 π}{3} \end{aligned}$ .
::您也可以注意到 120 是 2x60 。既然 60 = 3 弧度, 那么 120 是 2 = 3 弧度, 或者 2 = 3, 或 2 = 3 。使其为负值, 您就会得到答案 : - 2 = 3 。

3. Express $\begin{aligned} \frac{11 π}{12} \end{aligned}$ radians terms of degrees.
::3. 速度为11°12的弧度值。

$\begin{aligned} radians \times \frac{180}{π} = degrees \end{aligned}$
::弧度x180 度

Note: Sometimes students have trouble remembering if it is $\begin{aligned} \frac{180}{π} \end{aligned}$ or $\begin{aligned} \frac{π}{180} \end{aligned}$ . It might be helpful to remember that radian measure is almost always expressed in terms of $\begin{aligned} π \end{aligned}$ . If you want to convert from radians to degrees, you want the $\begin{aligned} π \end{aligned}$ to cancel out when you multiply, so it must be in the denominator.
::注意: 有时学生难以记住是180 或 180 。提醒一下, 弧度测量几乎总是以表示。如果您想要从弧度转换成度, 您需要在乘法时取消 , 所以它必须是分母。

Examples
::实例

Example 1
::例1

Earlier, you were asked is there a way to translate the instructions in degrees to radians.
::早些时候,有人问过你有没有办法将指示的度值转换成弧度。

Since you now know that the conversion for a measurement in degrees to radians is
::既然你现在知道以度为度的测量转换成弧度是

$\begin{aligned} degrees \times \frac{π}{180} = radians \end{aligned}$

::°180=弧度

you can find the solution to convert $\begin{aligned} 75^{\circ} \end{aligned}$ to radians:
::您可以找到将 75 转换为弧度的解决方案 :

$\begin{aligned} 75^{\circ} \times \frac{π}{180} = \frac{75 π}{180} = \frac{5 π}{12} \end{aligned}$

Example 2
::例2

Convert the following degree measures to radians. All answers should be in terms of $\begin{aligned} π \end{aligned}$ .
::将以下度量转换为弧度。所有答案应使用 __ 。

$\begin{aligned} 240^{\circ} \end{aligned}$ , $\begin{aligned} 270^{\circ} \end{aligned}$ , $\begin{aligned} 315^{\circ} \end{aligned}$ , $\begin{aligned} - 210^{\circ} \end{aligned}$ , $\begin{aligned} 120^{\circ} \end{aligned}$

$\begin{aligned} \frac{4 π}{3} \end{aligned}$ , $\begin{aligned} \frac{3 π}{2} \end{aligned}$ , $\begin{aligned} \frac{7 π}{4} \end{aligned}$ , $\begin{aligned} - \frac{7 π}{6} \end{aligned}$ , $\begin{aligned} \frac{2 π}{3} \end{aligned}$

Example 3
::例3

Convert the following degree measures to radians. All answers should be in terms of $\begin{aligned} π \end{aligned}$ .
::将以下度量转换为弧度。所有答案应使用 __ 。

$\begin{aligned} 15^{\circ} \end{aligned}$ , $\begin{aligned} - 450^{\circ} \end{aligned}$ , $\begin{aligned} 72^{\circ} \end{aligned}$ , $\begin{aligned} 720^{\circ} \end{aligned}$ , $\begin{aligned} 330^{\circ} \end{aligned}$

$\begin{aligned} \frac{π}{12} \end{aligned}$ , $\begin{aligned} - \frac{5 π}{2} \end{aligned}$ , $\begin{aligned} \frac{π}{5} \end{aligned}$ , $\begin{aligned} 4 π \end{aligned}$ , $\begin{aligned} \frac{11 π}{6} \end{aligned}$

Example 4
::例4

Convert the following radian measures to degrees
::将以下弧度量转换为度

$\begin{aligned} \frac{π}{2} \end{aligned}$ , $\begin{aligned} \frac{11 π}{5} \end{aligned}$ , $\begin{aligned} \frac{2 π}{3} \end{aligned}$ , $\begin{aligned} 5 π \end{aligned}$ , $\begin{aligned} \frac{7 π}{2} \end{aligned}$

$\begin{aligned} 90^{\circ} \end{aligned}$ , $\begin{aligned} 396^{\circ} \end{aligned}$ , $\begin{aligned} 120^{\circ} \end{aligned}$ , $\begin{aligned} 540^{\circ} \end{aligned}$ , $\begin{aligned} 630^{\circ} \end{aligned}$

Review
::回顾

Convert the following degree measures to radians. All answers should be in terms of $\begin{aligned} π \end{aligned}$ .
::将以下度量转换为弧度。所有答案应使用 __ 。

$\begin{aligned} 90^{\circ} \end{aligned}$

$\begin{aligned} 360^{\circ} \end{aligned}$

$\begin{aligned} 50^{\circ} \end{aligned}$

$\begin{aligned} 110^{\circ} \end{aligned}$

$\begin{aligned} 495^{\circ} \end{aligned}$

$\begin{aligned} - 85^{\circ} \end{aligned}$

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

Convert the following radian measures to degrees.
::将以下弧度量转换为度。

$\begin{aligned} \frac{5 π}{12} \end{aligned}$

$\begin{aligned} \frac{3 π}{5} \end{aligned}$

$\begin{aligned} \frac{8 π}{15} \end{aligned}$

$\begin{aligned} \frac{7 π}{10} \end{aligned}$

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

$\begin{aligned} 3 π \end{aligned}$

$\begin{aligned} \frac{7 π}{2} \end{aligned}$

Why do you think there are two different ways to measure angles? When do you think it might be more convenient to use radians than degrees?
::为什么你认为有两种不同的方式来测量角度?你觉得什么时候使用弧度比度更方便?

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。

章节大纲

常规

度与弧度之间的转换

Conversion between Degrees and Radians ::度与弧度之间的转换

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)

Conversion between Degrees and Radians
::度与弧度之间的转换

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)