Course: 4.1 弧度和度的角

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弧度和度的角

Most people are familiar with measuring angles in degrees. It is easy to picture angles like $\begin{aligned} 30^{\circ} \end{aligned}$ , $\begin{aligned} 45^{\circ} \end{aligned}$ or $\begin{aligned} 90^{\circ} \end{aligned}$ and the fact that $\begin{aligned} 360^{\circ} \end{aligned}$ makes up an entire circle. Over 2000 years ago the Babylonians used a base 60 number system and divided up a circle into 360 equal parts. This became the standard and it is how most people think of angles today.
::大多数人都熟悉度的测量角度。很容易将角度如 30 、 45 或 90 和 360 和整个圆形画出来。 2000 年, 巴比伦人使用了一个60 基数系统,将圆分为360 等分。这成了标准,也是大多数人今天对角度的思考方式。

However, there are many units with which to measure angles. For example, the gradian was invented along with the metric system and it divides a circle into 400 equal parts. The sizes of these different units are very arbitrary.
::然而,有许多单位用来测量角度。例如, 梯度是和测量系统一起发明的, 它将一个圆分为400个等分。这些不同单位的大小非常武断。

A radian is a unit of measuring angles that is based on the properties of circles. This makes it more meaningful than gradians or degrees. How many radians make up a circle?
::弧度是一个基于圆形特性的测量角度单位。这使得弧度比梯度或度更有意义。圆形由多少弧度组成?

Radians and Degrees
::弧度和度

A radian is defined to be the central angle where the subtended arc length is the same length as the radius.
::弧度的定义是子弧长度与半径相同长度的中心角。

Another way to think about radians is through the circumference of a circle. The circumference of a circle with radius $\begin{aligned} r \end{aligned}$ is $\begin{aligned} 2 π r \end{aligned}$ . Just over six radii (exactly $\begin{aligned} 2 π \end{aligned}$ radii) would stretch around any circle.
::另一种思考弧度的方法是通过圆圈环绕。圆圈半径为 r 的环绕值为 2 °r 。仅略高于 6 个弧度( 直径为 2 radi ) , 环绕任何圆圈。

To define a radian in terms of degrees, equate a circle measured in degrees to a circle measured in radians.
::要以度数来定义弧度,将以度数计量的圆与以弧度计量的圆等值。

$\begin{aligned} 360 d e g r e e s = 2 π r a d i a n s \end{aligned}$ , so $\begin{aligned} \frac{180}{π} d e g r e e s = 1 r a d i a n \end{aligned}$
::360度=2弧度,即180°=1弧度

Alternatively; $\begin{aligned} 360 d e g r e e s = 2 π r a d i a n s \end{aligned}$ , so $\begin{aligned} 1 d e g r e e = \frac{π}{180} r a d i a n s \end{aligned}$
::或; 360度=2弧度, 即 1 度= 180 弧度

The conversion factor to convert degrees to radians is: $\begin{aligned} \frac{π}{180^{\circ}} \end{aligned}$
::将度转换为弧度的转换系数为: 180

The conversion factor to convert radians to degrees is: $\begin{aligned} \frac{180^{\circ}}{π} \end{aligned}$
::将弧度转换为度的转换系数为: 180 {_____________________________________________________________________________________

If an angle has no units, it is assumed to be in radians.
::如果角度没有单位,则假定为弧度。

If you were to convert $\begin{aligned} 150^{\circ} \end{aligned}$ into radians, you would multiply $\begin{aligned} 150^{\circ} \end{aligned}$ by the correct conversion factor. You would get:
::如果您要将 150 转换成弧度, 您将会将 150 乘以正确的转换系数。您将会得到 :

$\begin{aligned} 150^{\circ} \cdot \frac{π}{180^{\circ}} = \frac{15 π}{18} = \frac{5 π}{6} r a d i a n s \end{aligned}$
::1501801518=56弧度

You can check your work by making sure the degree units appear on both the numerator and denominator.
::您可以通过确保分子和分母同时出现度单位来检查您的工作。

If you were to convert $\begin{aligned} \frac{π}{6} \end{aligned}$ radians into degrees, you would multiply $\begin{aligned} \frac{π}{6} \end{aligned}$ by the correct conversion factor. You would get $\begin{aligned} \frac{π}{6} \cdot \frac{180^{\circ}}{π} = \frac{180^{\circ}}{6} = 30^{\circ} \end{aligned}$
::如果您要将 6 弧度转换为度, 您将会以正确的换算系数乘以 6。您将会得到 61801806=30

Notice $\begin{aligned} π \end{aligned}$ appears in both the numerator and denominator and $\begin{aligned} \frac{π}{π} = 1 \end{aligned}$ .
::通知在分子、分母和% 1 中同时出现。

Examples
::实例

Example 1
::例1

Earlier, you were asked how many radians make up a circle. Exactly $\begin{aligned} 2 π \end{aligned}$ radians describe a circular arc. This is because $\begin{aligned} 2 π \end{aligned}$ radii wrap around the circumference of any circle.
::早些时候,有人问您有多少弧度组成一个圆圈。确切的弧度为 2 表示圆弧。这是因为 2 的弧度环绕着任何圆圈环绕。

Example 2
::例2

Convert $\begin{aligned} (6 π)^{\circ} \end{aligned}$ into radians.
::将 (6) 转换成弧度。

Don’t be fooled just because this has $\begin{aligned} π \end{aligned}$ . This number is about $\begin{aligned} 19^{\circ} \end{aligned}$
::不要因为这个数字就被愚弄。这个数字大约是19个。

$\begin{aligned} (6 π)^{\circ} \cdot \frac{π}{180^{\circ}} = \frac{6 π^{2}}{180} = \frac{π^{2}}{3} \end{aligned}$

It is very unusual to ever have a $\begin{aligned} π^{2} \end{aligned}$ term, but it can happen.
::有"二"这个词是非常罕见的,但有可能发生。

Example 3
::例3

Convert $\begin{aligned} \frac{5 π}{6} \end{aligned}$ into degrees.
::将56转换为度。

$\begin{aligned} \frac{5 π}{6} \cdot \frac{180^{\circ}}{π} = \frac{5 \cdot 30^{\circ}}{1} = 150^{\circ} \end{aligned}$

Example 4
::例4

Convert $\begin{aligned} 210^{\circ} \end{aligned}$ into radians.
::将210转换成弧度

$\begin{aligned} 210^{\circ} \cdot \frac{π}{180^{\circ}} = \frac{7 \cdot 30 \cdot π}{6 \cdot 30} = \frac{7 π}{6} \end{aligned}$

Example 5
::例5

Draw a $\begin{aligned} \frac{π}{2} \end{aligned}$ angle by first drawing a $\begin{aligned} 2 π \end{aligned}$ angle, halving it and halving the result. Recall that $\begin{aligned} \frac{π}{2} = 90^{\circ} \end{aligned}$ .
::绘制 2 角度, 首先绘制 2 角度, 将其减半, 并将结果减半。回顾 2 = 90 。

Summary

A radian is the central angle where the subtended arc length is the same length as the radius.
::弧度是子弧长度与半径相同长度的中心角。

A circle with radius $\begin{aligned} r \end{aligned}$ has a circumference of $\begin{aligned} 2 π r, \end{aligned}$ and exactly $\begin{aligned} 2 π \end{aligned}$ radii would stretch around any circle.
::一个半径为r的圆圆环为2°r,确切的说,2半径将环绕任何圆圈。

The conversion factor to convert degrees to radians is: $\begin{aligned} \frac{π}{180 °} \end{aligned}$
::将度转换成弧度的转换系数为: 180°

If an angle has no units, it is assumed to be in radians.
::如果角度没有单位,则假定为弧度。

Review
::回顾

Find the radian measure of each angle.
::查找每个角度的弧度量。

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

2. $\begin{aligned} 300^{\circ} \end{aligned}$

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

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

5. $\begin{aligned} 270^{\circ} \end{aligned}$

6. $\begin{aligned} 45^{\circ} \end{aligned}$

7. $\begin{aligned} (5 π)^{\circ} \end{aligned}$

Find the degree measure of each angle.
::查找每个角度的度量。

8. $\begin{aligned} \frac{7 π}{6} \end{aligned}$

9. $\begin{aligned} \frac{5 π}{4} \end{aligned}$

10. $\begin{aligned} \frac{3 π}{2} \end{aligned}$

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

12. $\begin{aligned} π \end{aligned}$

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

14. 3

15. Explain why if you are given an angle in degrees and you multiply it by $\begin{aligned} \frac{π}{180} \end{aligned}$ you will get the same angle in radians.
::15. 解释如果给您一个角度以度数乘以 180, 为何在弧度上得到相同的角度。

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

弧度和度的角

Radians and Degrees ::弧度和度

Examples ::实例

Example 1 ::例1

Example 2 ::例2

Example 3 ::例3

Example 4 ::例4

Example 5 ::例5

Review ::回顾

Review (Answers) ::回顾(答复)