Course: 1.18 单圆圆内的参考角和角

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单圆的参考角和角
When you walk into math class one day, your teacher has a surprise for the class. You're going to play series of games related to the things you've been learning about in class. For the first game, your teacher hands each group of students a spinner with an "x" and "y" axis marked. The game is to see how many angles you identify correctly. However, in this game, you are supposed to give what is called the "reference angle". You spin your spinner three times. Each picture below shows one of the spins:
::当有一天你走进数学课时,你的老师会给班级一个惊喜。你会玩一系列与你在课堂上学到的东西有关的游戏。在第一个游戏中, 您的老师会把每个学生组的脊椎都用“ x” 和“y” 轴来标记。游戏是看您正确识别了多少角度。但是, 在这个游戏中, 您应该给一个叫做“ 参考角度” 的游戏。您将旋转三次。下面的每张照片都显示一个方向 :

Can you correctly identify the reference angles for these pictures?
::您能够正确识别这些图片的引用角度吗 ?

Reference Angles
::参考角度

Consider the angle $\begin{aligned} 150^{\circ} \end{aligned}$ . If we graph this angle in standard position , we see that the terminal side of this angle is a reflection of the terminal side of $\begin{aligned} 30^{\circ} \end{aligned}$ , across the $\begin{aligned} y - \end{aligned}$ axis.
::考虑角度 150 。如果我们用标准位置绘制这个角度, 我们可以看到这个角度的末端是 30 的末端的反射, 横跨 y - 轴。

Notice that $\begin{aligned} 150^{\circ} \end{aligned}$ makes a $\begin{aligned} 30^{\circ} \end{aligned}$ angle with the negative $\begin{aligned} x - \end{aligned}$ axis. Therefore we say that $\begin{aligned} 30^{\circ} \end{aligned}$ is the reference angle for $\begin{aligned} 150^{\circ} \end{aligned}$ . Formally, the reference angle of an angle in standard position is the angle formed with the closest portion of the $\begin{aligned} x - \end{aligned}$ axis. Notice that $\begin{aligned} 30^{\circ} \end{aligned}$ is the reference angle for many angles. For example, it is the reference angle for $\begin{aligned} 210^{\circ} \end{aligned}$ and for $\begin{aligned} - 30^{\circ} \end{aligned}$ .
::注意 150 以负 x - 轴为 30 角。因此, 我们表示 30 是 150 的参考角。形式上, 标准位置角的参考角是 x - 轴中最接近部分的角。注意 30 是多个角度的参考角。例如, 这是 210 和 - 30 的参考角。

In general, identifying the reference angle for an angle will help you determine the values of the trig functions of the angle.
::一般而言,确定角的参考角将有助于确定角的三角函数的值。

Identifying Reference Angles
::识别参考角度

Graph each of the following angles and identify their reference angles.
::绘制以下每个角度的图,并标明其参考角度。

a. $\begin{aligned} 140^{\circ} \end{aligned}$
::a. 140_________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

$\begin{aligned} 140^{\circ} \end{aligned}$ makes a $\begin{aligned} 40^{\circ} \end{aligned}$ angles with the negative $\begin{aligned} x - \end{aligned}$ axis. Therefore the reference angle is $\begin{aligned} 40^{\circ} \end{aligned}$ .
::140 以负 x - 轴为40 角度。因此, 参考角度为 40 。

b. $\begin{aligned} 240^{\circ} \end{aligned}$
::b. 240_________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

$\begin{aligned} 240^{\circ} \end{aligned}$ makes a $\begin{aligned} 60^{\circ} \end{aligned}$ angle with the negative $\begin{aligned} x - \end{aligned}$ axis. Therefore the reference angle is $\begin{aligned} 60^{\circ} \end{aligned}$
::240

c. $\begin{aligned} 380^{\circ} \end{aligned}$
::c. 380__________________________________________________________________________________________________________________________________________________________________________________________________________________

$\begin{aligned} 380^{\circ} \end{aligned}$ is a full rotation of $\begin{aligned} 360^{\circ} \end{aligned}$ , plus an additional $\begin{aligned} 20^{\circ} \end{aligned}$ . So this angle is co-terminal with $\begin{aligned} 20^{\circ} \end{aligned}$ , and $\begin{aligned} 20^{\circ} \end{aligned}$ is its reference angle.
::380琌360锣 20 ┮硂àcm20琌妓 20琌参旧à

Determining the Value of Trigonometric Functions
::确定三角函数的数值

1. Find the ordered pair for $\begin{aligned} 240^{\circ} \end{aligned}$ and use it to find the value of $\begin{aligned} \sin 240^{\circ} \end{aligned}$ .
::1. 找到240的定购一对,并用它来寻找sin240的价值。

$\begin{aligned} \sin 240^{\circ} = \frac{- \sqrt{3}}{2} \end{aligned}$
::性别性别性别性别性别性别性别性别性别性别性别性别性别性别性别性别性别性别性别性别性别性别性别性别性别性别性别性别性别性别性别性别性别性别性别性别性别性别性别性别性别性别性别性别性别性别性别性别性别性别性别性别性别性别性别性别性别性别性别性别性别性别性别性别性别性别性别性别性别性别性别性别性别性别性别性别性别

As we found in part b under the question above, the reference angle for $\begin{aligned} 240^{\circ} \end{aligned}$ is $\begin{aligned} 60^{\circ} \end{aligned}$ . The figure below shows $\begin{aligned} 60^{\circ} \end{aligned}$ and the three other angles in the unit circle that have $\begin{aligned} 60^{\circ} \end{aligned}$ as a reference angle.
::如上文问题b部分所示,240的参考角为60。下图显示单位圆的60和以60为参考角的其他三个角度。

The terminal side of the angle $\begin{aligned} 240^{\circ} \end{aligned}$ represents a reflection of the terminal side of $\begin{aligned} 60^{\circ} \end{aligned}$ over both axes. So the coordinates of the point are $\begin{aligned} (- \frac{1}{2}, - \frac{\sqrt{3}}{2}) \end{aligned}$ . The $\begin{aligned} y - \end{aligned}$ coordinate is the sine value, so $\begin{aligned} \sin 240^{\circ} = - \frac{\sqrt{3}}{2} \end{aligned}$ .
::角度 240 的终端面代表两个轴上60 的终端面的反射。因此点的坐标是 (- 12, -32)。 y - 坐标是正弦值, 所以是sinQ 240 。 32 。

Just as the figure above shows $\begin{aligned} 60^{\circ} \end{aligned}$ and three related angles, we can make similar graphs for $\begin{aligned} 30^{\circ} \end{aligned}$ and $\begin{aligned} 45^{\circ} \end{aligned}$ .
::上面的图显示60个角度和三个相关角度, 我们可以为 30个角度和45个角度制作类似的图表。

Knowing these ordered pairs will help you find the value of any of the trig functions for these angles.
::了解这些定做的配对有助于您找到这些角度的任何三角函数的价值。

2. Find the value of $\begin{aligned} \cot 300^{\circ} \end{aligned}$
::2. 查找 cot300的值

$\begin{aligned} \cot 300^{\circ} = - \frac{1}{\sqrt{3}} \end{aligned}$
::30013

Using the graph above, you will find that the ordered pair is $\begin{aligned} (\frac{1}{2}, - \frac{\sqrt{3}}{2}) \end{aligned}$ . Therefore the cotangent value is $\begin{aligned} \cot 300^{\circ} = \frac{x}{y} = \frac{\frac{1}{2}}{- \frac{\sqrt{3}}{2}} = \frac{1}{2} \times - \frac{2}{\sqrt{3}} = - \frac{1}{\sqrt{3}} \end{aligned}$
::使用上面的图表,您会发现订购的一对是( 12, - 32) 。因此, 余切值是 cot300\\\ xy= 12- 32= 12\\\ 23\\\ 13

We can also use the concept of a reference angle and the ordered pairs we have identified to determine the values of the trig functions for other angles.
::我们还可以利用参考角度的概念和我们确定的定购配对的概念来确定其他角度的三角函数的值。

Examples
::实例

Example 1
::例1

Earlier, you were asked if you can correctly identify the reference angles in the pictures.
::早些时候,有人询问您能否正确识别图片中的引用角度。

Since you know how to measure reference angles now, upon examination of the spinners, you know that the first angle is $\begin{aligned} 30^{\circ} \end{aligned}$ , the second angle is $\begin{aligned} 45^{\circ} \end{aligned}$ , and the third angle is $\begin{aligned} 60^{\circ} \end{aligned}$ .
::既然你现在知道如何测量参考角度了, 在检查脊柱时, 你知道第一个角度是30, 第二个角度是45, 第三个角度是60。

Example 2
::例2

Graph $\begin{aligned} 210^{\circ} \end{aligned}$ and identify its reference angle.
::图210 并标明其参考角。

The graph of $\begin{aligned} 210^{\circ} \end{aligned}$ looks like this:
::210的图象是这样的:

and since the angle makes a $\begin{aligned} 30^{\circ} \end{aligned}$ angle with the negative "x" axis, the reference angle is $\begin{aligned} 30^{\circ} \end{aligned}$ .
::由于角度为负“x”轴的30角, 参考角为 30。

Example 3
::例3

Graph $\begin{aligned} 315^{\circ} \end{aligned}$ and identify its reference angle.
::图315 并标明其参考角。

The graph of $\begin{aligned} 315^{\circ} \end{aligned}$ looks like this:
::315的图象是这样的:

and since the angle makes a $\begin{aligned} 45^{\circ} \end{aligned}$ angle with the positive "x" axis, the reference angle is $\begin{aligned} 45^{\circ} \end{aligned}$ .
::由于角度为45角,正“x”轴为45角,参考角为45角。

Example 4
::例4

Find the ordered pair for $\begin{aligned} 150^{\circ} \end{aligned}$ and use it to find the value of $\begin{aligned} 150^{\circ} \end{aligned}$ .
::找到150的定购一对,然后用它来找到150的价值。

Since the reference angle is $\begin{aligned} 30^{\circ} \end{aligned}$ , we know that the coordinates for the point on the unit circle are $\begin{aligned} (- \frac{\sqrt{3}}{2}, \frac{1}{2}) \end{aligned}$ . This is the same as the value for $\begin{aligned} 30^{\circ} \end{aligned}$ , except the "x" coordinate is negative instead of positive. Knowing this,
::由于参考角度是 30 , 我们知道单位圆上点的坐标是 (- 32, 12) , 这与 30 的值相同, 但“ x” 坐标是负的而不是正的。知道这一点,

$\begin{aligned} \cos 150^{\circ} = \frac{a d j a c e n t}{h y p o t e n u s e} = \frac{- \frac{\sqrt{3}}{2}}{1} = - \frac{\sqrt{3}}{2} \end{aligned}$

::150 3213232

Review
::回顾

Graph $\begin{aligned} 100^{\circ} \end{aligned}$ and identify its reference angle.
::图100 并标明其参考角度。

Graph $\begin{aligned} 200^{\circ} \end{aligned}$ and identify its reference angle.
::图200 并标明其参考角度。

Graph $\begin{aligned} 290^{\circ} \end{aligned}$ and identify its reference angle.
::图290 并标明其参考角。

Calculate each value using the unit circle and special right triangles.
::使用单位圆和特殊右三角形计算每个数值。

$\begin{aligned} \sin 120^{\circ} \end{aligned}$
::问题120 120 _________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

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

$\begin{aligned} \csc 120^{\circ} \end{aligned}$
::csc120__________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

$\begin{aligned} \cos 135^{\circ} \end{aligned}$
:cos135) (cos135) (cos135)

$\begin{aligned} \sin 135^{\circ} \end{aligned}$
::135 ____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

$\begin{aligned} \tan 135^{\circ} \end{aligned}$
::{\fn黑体\fs22\bord1\shad0\3aHBE\4aH00\fscx67\fscy66\2cHFFFFFF\3cH808080}不,不,不,不,不,不,不,不,不,不,不,不,不,不,不,不,不,不,不,不,不,不,不,不,不,不,不,不,不,不,不,不,不,不,不,不,不,不,不,不,不,不,不,不,不,不,不,不,不,不,不,不,不,不,不,不,不,不,不,不,不,不,不,不,不,不,不

$\begin{aligned} \sin 210^{\circ} \end{aligned}$
::-21021012222222222222222222222222222222222222222222222222222222222222222222222222222222

$\begin{aligned} \cos 210^{\circ} \end{aligned}$
::COS210_______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

$\begin{aligned} \cot 210^{\circ} \end{aligned}$
::科特210_____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

$\begin{aligned} \sin 225^{\circ} \end{aligned}$
::确实如此... . 225 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \

$\begin{aligned} \cos 225^{\circ} \end{aligned}$
::2252222222222222222222222222222222222222222222222222222222222222222222222222222222222

$\begin{aligned} \sec 225^{\circ} \end{aligned}$
::秒% 225

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

单圆的参考角和角

Reference Angles ::参考角度

Identifying Reference Angles ::识别参考角度

Determining the Value of Trigonometric Functions ::确定三角函数的数值

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)

Reference Angles
::参考角度

Identifying Reference Angles
::识别参考角度

Determining the Value of Trigonometric Functions
::确定三角函数的数值

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)