课程： 1.19 负角的三角函数

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选择节负角的三角函数

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负角的三角函数
While practicing for the track team, you regularly stop to consider the values of trig functions for the angle you've covered as you run around the circular track at your school. Today, however, is different. To keep things more interesting, your coach has decided to have you and your teammates run the opposite of the usual direction on the track. From your studies at school, you know that this is the equivalent of a "negative angle".
::在为田径队练习时, 您定期停下来来考虑您在学校绕环轨运行时所覆盖的角度的三角函数值。但是, 今天不同了。为了让事情更加有趣, 您的教练决定让你和你的队友运行与通常的轨道方向相反。您在学校学习时知道这相当于“ 负角度 ” 。

You have run $\begin{aligned} - 45^{\circ} \end{aligned}$ around the track, and want to fine the value of the cosine function for this angle. Is it still possible to find the values of trig functions for these new types of angles?
::您环绕轨道运行了- 45, 并想要为此角的余弦函数值进行微调。仍然可以找到这些新类型的角的三角函数值吗 ?

Trigonometric Functions of Negative Angles
::负角的三角函数

Recall that graphing a negative angle means rotating clockwise. The graph below shows $\begin{aligned} - 30^{\circ} \end{aligned}$ .
::回顾负角度的图形表示顺时针旋转。下图显示 -30\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\N\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\

Notice that this angle is coterminal with $\begin{aligned} 330^{\circ} \end{aligned}$ . So the ordered pair is $\begin{aligned} (\frac{\sqrt{3}}{2}, - \frac{1}{2}) \end{aligned}$ . We can use this ordered pair to find the values of any of the trig functions of $\begin{aligned} - 30^{\circ} \end{aligned}$ . For example, $\begin{aligned} \cos (- 30^{\circ}) = x = \frac{\sqrt{3}}{2} \end{aligned}$ .
::注意此角度与 330 {/ { 330 } 相交。因此, 定购的对是 (32, -12) 。我们可以使用这对定购的对来查找 - 30 { { 330 { =x= 32 的任何三角函数的值。例如, cos { (- 30 }) =x= 32 。

In general, if a negative angle has a reference angle of $\begin{aligned} 30^{\circ} \end{aligned}$ , $\begin{aligned} 45^{\circ} \end{aligned}$ , or $\begin{aligned} 60^{\circ} \end{aligned}$ , or if it is a quadrantal angle , we can find its ordered pair, and so we can determine the values of any of the trig functions of the angle.
::一般来说,如果一个负角的参考角为 30,45,或60, 或者如果它是四方角, 我们可以找到它的定对, 这样我们就可以确定角的任何三角函数的值。

Finding the Value of Trigonometric Expressions
::查找三角数表达式的值

Find the value of the following expressions:
::查找下列表达式的值:

1. $\begin{aligned} \sin (- 45^{\circ}) \end{aligned}$
::1. 罪(-45)

$\begin{aligned} \sin (- 45^{\circ}) = - \frac{\sqrt{2}}{2} \end{aligned}$
:-45)

$\begin{aligned} - 45^{\circ} \end{aligned}$ is in the $\begin{aligned} 4^{t h} \end{aligned}$ quadrant, and has a reference angle of $\begin{aligned} 45^{\circ} \end{aligned}$ . That is, this angle is coterminal with $\begin{aligned} 315^{\circ} \end{aligned}$ . Therefore the ordered pair is $\begin{aligned} (\frac{\sqrt{2}}{2}, - \frac{\sqrt{2}}{2}) \end{aligned}$ and the sine value is $\begin{aligned} - \frac{\sqrt{2}}{2} \end{aligned}$ .
::~ 45位于第四象限, 参考角为 45。也就是说, 这个角与 315 相交。因此, 定购的对是 (22,- 22) , 正弦值是- 22 。

2. $\begin{aligned} \sec (- 300^{\circ}) \end{aligned}$
::2. 秒(-300)

$\begin{aligned} \sec (- 300^{\circ}) = 2 \end{aligned}$
::秒(- 300)=2

The angle $\begin{aligned} - 300^{\circ} \end{aligned}$ is in the $\begin{aligned} 1^{s t} \end{aligned}$ quadrant and has a reference angle of $\begin{aligned} 60^{\circ} \end{aligned}$ . That is, this angle is coterminal with $\begin{aligned} 60^{\circ} \end{aligned}$ . Therefore the ordered pair is $\begin{aligned} (\frac{1}{2}, \frac{\sqrt{3}}{2}) \end{aligned}$ and the secant value is $\begin{aligned} \frac{1}{x} = \frac{1}{\frac{1}{2}} = 2 \end{aligned}$ .
::角度 - 300位于第1象限中, 参考角为 60。也就是说, 这个角度是60的交界点。因此, 订购的一对是 (12, 32) , 分离值是 1x= 112= 2 。

3. $\begin{aligned} \cos (- 90^{\circ}) \end{aligned}$
::3.cos(-90)

$\begin{aligned} \cos (- 90^{\circ}) = 0 \end{aligned}$
::COs(-90)=0

The angle $\begin{aligned} - 90^{\circ} \end{aligned}$ is coterminal with $\begin{aligned} 270^{\circ} \end{aligned}$ . Therefore the ordered pair is (0, -1) and the cosine value is 0.
::角度 - 90 与 270 相交。因此, 订购的对是 (0, - 1) , 余弦值是 0 。

We can also use our knowledge of reference angles and ordered pairs to find the values of trig functions of angles with measure greater than 360 degrees.
::我们还可以利用我们对参考角度和定购配对的知识,找到测量度大于360度的角的三角函数的值。

Examples
::实例

Example 1
::例1

Earlier, you were asked if it is still possible to find the values of trig functions for the new type of angles.
::早些时候,有人询问您是否仍然可以找到新类型角度的三角函数值。

What you want to find is the value of the expression: $\begin{aligned} \cos (- 45^{\circ}) \end{aligned}$
::您想要找到的是表达式的值 : cos(- 45)

$\begin{aligned} \cos (- 45^{\circ}) = \frac{\sqrt{2}}{2} \end{aligned}$
::COs(- 45)=22

$\begin{aligned} - 45^{\circ} \end{aligned}$ is in the $\begin{aligned} 4^{t h} \end{aligned}$ quadrant, and has a reference angle of $\begin{aligned} 45^{\circ} \end{aligned}$ . That is, this angle is coterminal with $\begin{aligned} 315^{\circ} \end{aligned}$ . Therefore the ordered pair is $\begin{aligned} (\frac{\sqrt{2}}{2}, - \frac{\sqrt{2}}{2}) \end{aligned}$ and the cosine value is $\begin{aligned} \frac{\sqrt{2}}{2} \end{aligned}$ .
::~ 45位于第四象限, 参考角为 45。也就是说, 这个角与 315 相交。因此, 订购的对是 (22, - 22) , 余弦值是 22 。

Example 2
::例2

Find the value of the expression: $\begin{aligned} \cos - 180^{\circ} \end{aligned}$
::查找表达式的值: cos- 180 {} @ label

The angle $\begin{aligned} - 180^{\circ} \end{aligned}$ is coterminal with $\begin{aligned} 180^{\circ} \end{aligned}$ . Therefore the ordered pair of points is (-1, 0). The cosine is the "x" coordinate, so here it is -1.
::角度 - 180 与 180 相交。因此, 定点对数是 (-1, 0) 。余弦是“ x” 坐标, 所以这里是 - 1 。

Example 3
::例3

Find the value of the expression: $\begin{aligned} \sin - 90^{\circ} \end{aligned}$
::查找表达式的值: sin- 90 {} @ label

The angle $\begin{aligned} - 90^{\circ} \end{aligned}$ is coterminal with $\begin{aligned} 270^{\circ} \end{aligned}$ . Therefore the ordered pair of points is (0, -1). The sine is the "y" coordinte, so here it is -1.
::角度- 90 与 270 相交。因此, 定点对点是 (0, - 1) 。正弦是“ y ” coordide, 所以这里是 - 1 。

Example 4
::例4

Find the value of the expression: $\begin{aligned} \tan - 270^{\circ} \end{aligned}$
::查找表达式的值: tan- 270 {____________________________________________________________

The angle $\begin{aligned} - 270^{\circ} \end{aligned}$ is coterminal with $\begin{aligned} 90^{\circ} \end{aligned}$ . Therefore the ordered pair of points is (0, 1). The tangent is the "y" coordinate divided by the "x" coordinate. Since the "x" coordinate is 0, the tangent is undefined.
::角度 - 270 {} 是 90 {} 的交界点。因此, 定序的一对点是 (0, 1) 。正切点是“ y” 坐标除以“ x” 坐标。由于“ x” 坐标是 0, 正切点是未定义的。

Review
::回顾

Calculate each value.
::计算每个数值。

$\begin{aligned} \sin - 120^{\circ} \end{aligned}$
::120-120 ____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

$\begin{aligned} \cos - 120^{\circ} \end{aligned}$
::COS-1120

$\begin{aligned} \tan - 120^{\circ} \end{aligned}$
::泰坦 - 120 ________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

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

$\begin{aligned} \sec - 120^{\circ} \end{aligned}$
::空间 - 120

$\begin{aligned} \cot - 120^{\circ} \end{aligned}$
::COT-120

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

$\begin{aligned} \sec - 45^{\circ} \end{aligned}$
::空间-45

$\begin{aligned} \tan - 45^{\circ} \end{aligned}$
::坦桑尼亚-45 _______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

$\begin{aligned} \cos - 135^{\circ} \end{aligned}$
::COS-135____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

$\begin{aligned} \csc - 135^{\circ} \end{aligned}$
::csc-135

$\begin{aligned} \sec - 135^{\circ} \end{aligned}$
::空间-135 *

$\begin{aligned} \tan - 210^{\circ} \end{aligned}$
::泰 - 210

$\begin{aligned} \sin - 270^{\circ} \end{aligned}$
::和/或/或/或/或/或/或/或/或/或/或/或/或/或/或/或/或/或/或/或/或/或/或/或/或/或/或/或/或/或/或/或/或/或/或/或/或

$\begin{aligned} \cot - 90^{\circ} \end{aligned}$
::COT-90

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

负角的三角函数

Trigonometric Functions of Negative Angles ::负角的三角函数

Finding the Value of Trigonometric Expressions ::查找三角数表达式的值

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)

Trigonometric Functions of Negative Angles
::负角的三角函数

Finding the Value of Trigonometric Expressions
::查找三角数表达式的值

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)