Course: 13.8 对等三角函数

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对等三角函数
A ladder propped up against a house forms an angle of $\begin{align*}30^\circ\end{align*}$ with the ground. What is the secant of this angle?
::顶着一栋房子的梯子形成30°C与地面的角角。这个角的偏移是什么?

Reciprocal Trigonometric Functions
::对等三角函数

Each of the trigonometric ratios has a reciprocal function associated with it as shown below.
::如下所示,每项三角比率都具有对应的功能。

The reciprocal of sine is cosecant : $\begin{align*}\frac{1}{\sin \theta}=\csc \theta\end{align*}$ , so $\begin{align*}\csc \theta=\frac{H}{O}\end{align*}$ (hypotenuse over opposite)
::共生是共生: 1sincsc, 所以cscHO (对面的套用)

The reciprocal of cosine is secant: $\begin{align*}\frac{1}{\cos \theta}=\sec \theta\end{align*}$ , so $\begin{align*}\sec \theta=\frac{H}{A}\end{align*}$ (hypotenuse over adjacent)
::的对等性是松散的:1cosçsec,所以cecHA(相邻的套用)

The reciprocal of tangent is cotangent : $\begin{align*}\frac{1}{\tan \theta}=\cot \theta\end{align*}$ , so $\begin{align*}\cot \theta=\frac{A}{O}\end{align*}$ (adjacent over opposite)
::相切的对等是相切的: 1tancot, 所以cotAO(相邻)

Let's use a calculator to evaluate $\begin{align*}\sec \frac{2 \pi}{5}\end{align*}$ .
::让我们用计算器来评估秒=2 5 。

First, be sure that your calculator is in radian mode. To check/change the mode, press the MODE button and make sure RADIAN is highlighted. If it is not, use the arrow keys to move the cursor to RADIANS and press enter to select RADIAN as the mode. Now we are ready to use the calculator to evaluate the reciprocal trig function . Since the calculator does not have a button for secant, however, we must utilize the reciprocal relationship between cosine and secant:
::首先, 请确定您的计算器是以弧度模式进行的。要检查/ 更改模式, 请按 MODE 按钮, 并确保 RADIAN 被突出显示。如果没有, 则使用箭头键将光标移动到 RADIANS , 按输入选择 RADIAN 作为模式。现在我们准备使用计算器来评估对等三角函数。但是, 由于计算器没有用于 seant 的按钮, 我们必须使用 cosine 和 selant 之间的对等关系 :

$\begin{aligned} Since \sec θ = \frac{1}{\cos θ}, \sec \frac{2 π}{5} = \frac{1}{\cos \frac{2 π}{5}} = 3.2361. \end{aligned}$
$\begin{align*}\text{Since}\ \sec \theta=\frac{1}{\cos \theta}, \sec \frac{2 \pi}{5}=\frac{1}{\cos \frac{2 \pi}{5}}=3.2361.\end{align*}$
::从秒1秒1秒2秒5=1秒2秒5=3.2361秒2秒5=3.2361秒

Now, let's use a calculator to evaluate $\begin{align*}\cot 100^\circ\end{align*}$ .
::现在,让我们用计算器来评估Cot100。

This time we will need to be in degree mode. After the mode has been changed we can use the reciprocal of cotangent, which is tangent, to evaluate as shown:
::这一次我们需要在度模式中。模式改变后, 我们可以使用正切的对等余切来评估 :

$\begin{aligned} Since \cot θ = \frac{1}{\tan θ}, \cot 100^{\circ} = \frac{1}{\tan 100^{\circ}} \approx - 0.1763. \end{aligned}$
$\begin{align*}\text{Since}\ \cot \theta=\frac{1}{\tan \theta}, \cot 100^\circ=\frac{1}{\tan 100^\circ} \approx -0.1763.\end{align*}$
::从1吨到1吨,1吨,1吨,1吨,1吨,1吨,1吨,1吨,1吨,1吨,1吨,1吨,1吨,1吨,1吨,1吨,1吨,1吨,1吨,1吨,1吨,1吨,1吨,1吨,1吨,1吨,1吨,1吨,1吨,1吨,1吨,1吨,1吨,1吨,1吨,1吨,1吨,1吨,1吨,1吨,1吨,1吨,1吨,1吨,1吨,1吨,1吨,1吨,1吨,1吨,1吨,1吨,1吨,1吨,1吨,1吨,1吨,1吨,1吨,1吨。

Finally, let's find the exact value of $\begin{align*}\csc \frac{5 \pi}{3}\end{align*}$ without using a calculator and give our answer in exact form.
::最后,让我们在不使用计算器的情况下找到csc53的准确价值并以准确的形式给出答案

The reciprocal of cosecant is sine so we will first find $\begin{align*}\sin \frac{5 \pi}{3}\end{align*}$ Using either the unit circle or the alternative method, we can determine that $\begin{align*}\sin \frac{5 \pi}{3}\end{align*}$ is $\begin{align*}-\frac{\sqrt{3}}{2}\end{align*}$ using a $\begin{align*}60^\circ\end{align*}$ reference angle in the fourth quadrant. Now, find its reciprocal: $\begin{align*}\frac{1}{-\frac{\sqrt{3}}{2}}=-\frac{2}{\sqrt{3}}=-\frac{2 \sqrt{3}}{3}\end{align*}$ .
::共生体的对等是必不可缺的,所以我们首先会发现 sin53 使用单位圆或替代方法,我们可以确定 sin53 是-32 使用第四象限的60 参考角度。现在,我们发现它的对等性:1-3223233。

Examples
::实例

Example 1
::例1

Earlier, you were asked to find the secant of the angle of the ladder propped up against the house.
::早些时候,有人要求你找到梯子的角角的偏移点爬上的房子。

The secant is the reciprocal of the cosine. So to find $\begin{align*}\sec 30^\circ\end{align*}$ , use the cosine.
::分离是连弦的对等。所以要找到 sec30,使用连弦。

$\begin{align*}\cos 30^\circ = \frac{\sqrt{3}}{2}\end{align*}$ .
::3032 3032

Therefore, $\begin{align*}\sec 30^\circ = \frac{2}{\sqrt{3}} = \frac{2\sqrt{3}}{3}\end{align*}$ .
::因此,cec3023=233。

Use your calculator to evaluate the following reciprocal trigonometric functions .
::使用您的计算器来评估以下对等三角函数。

Example 2
::例2

$\begin{align*}\csc \frac{7 \pi}{8}\end{align*}$
:c) 7_______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

$\begin{align*}\csc \frac{7 \pi}{8}=\frac{1}{\sin \frac{7 \pi}{8}}=2.6131\end{align*}$
::csc78=1sin78=2.6131

Example 3
::例3

$\begin{align*}\cot 85^\circ\end{align*}$
::柯特85

$\begin{align*}\cot 85^\circ=\frac{1}{\tan 85^\circ}=0.0875\end{align*}$
::851tan850.0875

Evaluate the following without using a calculator. Give all answers in exact form.
::在不使用计算器的情况下评估以下内容。以准确的形式给出所有答案。

Example 4
::例4

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

$\begin{align*}\sec 225^\circ\end{align*}$ is the reciprocal of $\begin{align*}\cot 225^\circ\end{align*}$ , a $\begin{align*}45^\circ\end{align*}$ reference angle in quadrant three where cosine is negative. Because $\begin{align*}\cos 45^\circ=\frac{1}{\sqrt{2}}, \cos 225^\circ=-\frac{1}{\sqrt{2}}\end{align*}$ , and $\begin{align*}\sec 225^\circ=-\sqrt{2}\end{align*}$ .
::QQQQQQQ225是cotQQQQQ225的对等关系, 共弦度为负的三象限中45参考角。因为 cosQQ4512, cos22512, 和 sec2252。

Example 5
::例5

$\begin{align*}\csc \frac{5 \pi}{6}\end{align*}$
:c) 56

$\begin{align*}\csc \frac{5 \pi}{6}\end{align*}$ is the reciprocal of $\begin{align*}\sin \frac{5 \pi}{6}\end{align*}$ , a $\begin{align*}\frac{\pi}{6}\end{align*}$ or $\begin{align*}30^\circ\end{align*}$ reference angle in the second quadrant where sine is positive. Because $\begin{align*}\sin \frac{\pi}{6}=\frac{1}{2}, \sin \frac{5 \pi}{6}=\frac{1}{2},\end{align*}$ and $\begin{align*}\csc \frac{5 \pi}{6}=2\end{align*}$ .
::csc=56是罪的对等 56, 在正弦正弦的第二个象限中, 一个6 或 30 的参考角。因为 sin= 6= 12, sin= 56=12, csc= 56=2 。

Review
::回顾

Use your calculator to evaluate the reciprocal trigonometric functions. Round your answers to four decimal places.
::使用您的计算器来评价对等三角函数。将您的答案四舍五入到小数点后四位。

$\begin{align*}\csc 95^\circ\end{align*}$
:csc95) (csc95)

$\begin{align*}\cot 278^\circ\end{align*}$
::288

$\begin{align*}\sec \frac{14 \pi}{5}\end{align*}$
::秒14=5

$\begin{align*}\cot(-245^\circ)\end{align*}$
:- 245)

$\begin{align*}\sec \frac{6 \pi}{7}\end{align*}$
::6+7 秒

$\begin{align*}\csc \frac{23 \pi}{13}\end{align*}$
:c) 章次页次

$\begin{align*}\cot 333^\circ\end{align*}$
::comt333

$\begin{align*}\csc \frac{9 \pi}{5}\end{align*}$
::csc95

Evaluate the following trigonometric functions without using a calculator. Give your answers exactly.
::在不使用计算器的情况下评估以下三角函数。请准确回答。

$\begin{align*}\sec \frac{5 \pi}{6}\end{align*}$
::秒=56

$\begin{align*}\csc \left(-\frac{3 \pi}{2}\right)\end{align*}$
::csc(-32)

$\begin{align*}\cot 225^\circ\end{align*}$
::柯特225

$\begin{align*}\sec \frac{11 \pi}{3}\end{align*}$
::秒=11%3

$\begin{align*}\csc \frac{7 \pi}{6}\end{align*}$
::cc=76 (csc=76)

$\begin{align*}\sec 270^\circ\end{align*}$
::270

$\begin{align*}\cot \frac{5 \pi}{3}\end{align*}$
::COT=53

$\begin{align*}\csc 315^\circ\end{align*}$
::==================================================================================

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

对等三角函数

Reciprocal Trigonometric Functions ::对等三角函数

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)

Reciprocal Trigonometric Functions
::对等三角函数

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)