课程： 1.23 引号特征 | The Way To Learn

章节大纲

选择节常规

折叠展开
常规

全部折叠展开全部
- 选择活动新闻通告
  
  新闻通告讨论区
选择节引号

折叠展开
引号
You are working in math class one day when your friend leans over and asks you what you got for the of a particular angle.
::有一天你在数学课上工作当你朋友靠过来问你从某个角度得到了什么

"I got $\begin{aligned} \frac{1}{2} \end{aligned}$ for the sine, and $\begin{aligned} \frac{\sqrt{3}}{2} \end{aligned}$ for the cosine. Why?" you ask.
::"我得到了12分的正弦,32分的连弦。为什么?"你问。

"It looks like I'm supposed to calculate the tangent function for the same angle you just did, but I can't remember the relationship for tangent. What should I do?" he says.
::“看来我应该用你刚才的同一角度来计算正切函数, 但我记不清正切关系。我该怎么办?”他说。

Do you know how you can help your friend find the answer, even if both you and he don't remember the relationship for tangent?
::你知道你怎样才能帮你的朋友找到答案即使你和他都不记得对方的关系吗?

Quotient Identities
::引号

The definitions of the trig functions led us to the , which can be seen in the Concept about that topic. They also lead us to another set of identities, the quotient identities.
::三角函数的定义引导我们找到这个概念,从关于这个主题的概念中可以看到这个概念,它们也引导我们找到另一套身份,即商数特性。

Consider first the sine, cosine, and tangent functions. For angles of rotation (not necessarily in the unit circle) these functions are defined as follows:
::首先考虑正弦、连弦和正切函数。对于旋转角度(不一定在单位圆内),这些函数的定义如下:

$\begin{aligned} \sin θ & = \frac{y}{r} \\ \cos θ & = \frac{x}{r} \\ \tan θ & = \frac{y}{x} \end{aligned}$

::来来来去去去去去去去去去去去去去去去去去去去去去去去去去去去去去去去去去去去去去去去去去去去去

Given these definitions, we can show that $\begin{aligned} \tan θ = \frac{\sin θ}{\cos θ} \end{aligned}$ , as long as $\begin{aligned} \cos θ \neq 0 \end{aligned}$ :
::鉴于这些定义,我们可以证明,只要以0计:

$\begin{aligned} \frac{\sin θ}{\cos θ} = \frac{\frac{y}{r}}{\frac{x}{r}} = \frac{y}{r} \times \frac{r}{x} = \frac{y}{x} = \tan θ . \end{aligned}$

::yrx=yrx=yx=tan。

The equation $\begin{aligned} \tan θ = \frac{\sin θ}{\cos θ} \end{aligned}$ is therefore an identity that we can use to find the value of the tangent function, given the value of the sine and cosine.
::等式 tansincos因此是一个身份, 我们可以用它来找到正弦函数的值, 以正弦函数和正弦函数的值。

Let's take a look at some problems involving quotient identities.
::让我们来研究一下与商数身份有关的一些问题。

1. Find the value of $\begin{aligned} \tan θ \end{aligned}$ ?
::1. 找到TAN的值吗?

If $\begin{aligned} \cos θ = \frac{5}{13} \end{aligned}$ and $\begin{aligned} \sin θ = \frac{12}{13} \end{aligned}$ , what is the value of $\begin{aligned} \tan θ \end{aligned}$ ?
::如果cos513和sin1213, 是什么值 tan?

$\begin{aligned} \tan θ = \frac{12}{5} \end{aligned}$
::丹尼 125

$\begin{aligned} \tan θ = \frac{\sin θ}{\cos θ} = \frac{\frac{12}{13}}{\frac{5}{13}} = \frac{12}{13} \times \frac{13}{5} = \frac{12}{5} \end{aligned}$

::1213513=1213×135=125

2. Show that $\begin{aligned} \cot θ = \frac{\cos θ}{\sin θ} \end{aligned}$
::2. 显示这些小鸡鸡

$\begin{aligned} \frac{\cos θ}{\sin θ} = \frac{\frac{x}{r}}{\frac{y}{r}} = \frac{x}{r} \times \frac{r}{y} = \frac{x}{y} = \cot θ \end{aligned}$

::cossinxryr=xrxrxry=xy=cot

3. What is the value of $\begin{aligned} \cot θ \end{aligned}$ ?
::3. 食堂的价值是什么?

If $\begin{aligned} \cos θ = \frac{7}{25} \end{aligned}$ and $\begin{aligned} \sin θ = \frac{24}{25} \end{aligned}$ , what is the value of $\begin{aligned} \cot θ \end{aligned}$ ?
::如果Cos725和sin2425,那么Cot值是多少?

$\begin{aligned} \cot θ = \frac{7}{24} \end{aligned}$
::科特724

$\begin{aligned} \cot θ = \frac{\cos θ}{\sin θ} = \frac{\frac{7}{25}}{\frac{24}{25}} = \frac{7}{25} \times \frac{25}{24} = \frac{7}{24} \end{aligned}$

::7252425=725×2524=724

Examples
::实例

Example 1
::例1

Earlier, you were asked if you can help your friend find the answer.
::之前有人问过你能否帮你朋友找到答案

Since you now know that:
::既然你现在知道:

$\begin{aligned} \tan θ = \frac{\sin θ}{\cos θ} \end{aligned}$
::

you can use this knowledge to help your friend with the sine and cosine values you measured for yourself earlier:
::您可以使用此知识帮助您的朋友使用您早些时候为自己测量的正弦和余弦值 :

$\begin{aligned} \tan θ = \frac{\sin θ}{\cos θ} = \frac{\frac{1}{2}}{\frac{\sqrt{3}}{2}} = \frac{1}{\sqrt{3}} \end{aligned}$
::1232=13=13

Example 2
::例2

If $\begin{aligned} \cos θ = \frac{17}{145} \end{aligned}$ and $\begin{aligned} \sin θ = \frac{144}{145} \end{aligned}$ , what is the value of $\begin{aligned} \tan θ \end{aligned}$ ?
::如果Cos17145 和sin144145, 是什么价值 tan?

$\begin{aligned} \tan θ = \frac{144}{17} \end{aligned}$ . We can see this from the relationship for the tangent function:
::tan14417. 我们从相切函数的关系中可以看到这一点:

$\begin{aligned} \tan θ = \frac{\sin θ}{\cos θ} = \frac{\frac{144}{145}}{\frac{17}{145}} = \frac{144}{145} \times \frac{145}{17} = \frac{144}{17} \end{aligned}$

::1414517145=144145×14517=14417=14417

Example 3
::例3

If $\begin{aligned} \sin θ = \frac{63}{65} \end{aligned}$ and $\begin{aligned} \cos θ = \frac{16}{65} \end{aligned}$ , what is the value of $\begin{aligned} \tan θ \end{aligned}$ ?
::如果sin6365和cos1665, 是什么价值 tan?

$\begin{aligned} \tan θ = \frac{63}{16} \end{aligned}$ . We can see this from the relationship for the tangent function:
::tan6316。我们从相切函数的关系中可以看到这一点:

$\begin{aligned} \tan θ = \frac{\sin θ}{\cos θ} = \frac{\frac{63}{65}}{\frac{16}{65}} = \frac{63}{65} \times \frac{65}{16} = \frac{63}{16} \end{aligned}$

::{\fn黑体\fs22\bord1\shad0\3aHBE\4aH00\fscx67\fscy66\2cHFFFFFF\3cH808080}=63651665=6365×6516=6316 {\fn黑体\fs22\bord1\shad0\3aHBE\4aH00\fscx67\fscy66\2cHFFFFFF\3cH808080}=6316=6316

Example 4
::例4

If $\begin{aligned} \tan θ = \frac{40}{9} \end{aligned}$ and $\begin{aligned} \cos θ = \frac{9}{41} \end{aligned}$ , what is the value of $\begin{aligned} \sin θ \end{aligned}$ ?
::如果Tan409和cos941, 什么是罪恶的价值?

$\begin{aligned} \sin θ = \frac{40}{41} \end{aligned}$ . We can see this from the relationship for the tangent function:
::4041. 我们从相切函数的关系中可以看到这一点:

$\begin{aligned} \tan θ = \frac{\sin θ}{\cos θ} \\ \sin θ = (\tan θ) (\cos θ) \\ \sin θ = \frac{40}{9} \times \frac{9}{41} \\ \sin θ = \frac{40}{41} \end{aligned}$

::{\fn黑体\fs22\bord1\shad0\3aHBE\4aH00\fscx67\fscy66\2cHFFFFFF\3cH808080}不... {\fn黑体\fs22\bord1\shad0\3aHBE\4aH00\fscx67\fscy66\2cHFFFFFF\3cH808080}不... {\fn黑体\fs22\bord1\shad0\3aHBE\4aH00\fscx67\fscy66\2cHFFFFFF\3cH808080}不 {\fn黑体\fs22\bord1\shad0\3aHBE\4aH00\fscx67\fscy66\2cHFFFFFF\3cH808080}不... {\fn黑体\fs22\bord1\shad0\3aHBE\4aH00\fscx67\fscy66\2cHFFFFFF\3cH808080}不... {\fn黑体\fs22\bord1\shad0\3aHBE\4aH00\fscx67\fscy66\2cHFFFFFF\3cH808080}不...

Review
::回顾

Fill in each blank with a trigonometric function.
::用三角函数填充每个空白。

$\begin{aligned} \tan θ = \frac{\sin θ}{?} \end{aligned}$
::{\fn黑体\fs20\shad2\2aH82\3aH20\4aH33\fscx95\3cH592001\be1}塔纳辛?

$\begin{aligned} \cos θ = \frac{\sin θ}{?} \end{aligned}$
::?

$\begin{aligned} \cot θ = \frac{?}{\sin θ} \end{aligned}$
::{\fn黑体\fs22\bord1\shad0\3aHBE\4aH00\fscx67\fscy66\2cHFFFFFF\3cH808080}这... {\fn黑体\fs22\bord1\shad0\3aHBE\4aH00\fscx67\fscy66\2cHFFFFFF\3cH808080}这...

$\begin{aligned} \cos θ = (\cot θ) \cdot (?) \end{aligned}$
:cot)(?)(?)(?)

If $\begin{aligned} \cos θ = \frac{5}{13} \end{aligned}$ and $\begin{aligned} \sin θ = \frac{1}{13} \end{aligned}$ , what is the value of $\begin{aligned} \tan θ \end{aligned}$ ?
::如果Cos513和sin113, 塔纳值是多少?

If $\begin{aligned} \sin θ = \frac{3}{5} \end{aligned}$ and $\begin{aligned} \cos θ = \frac{4}{5} \end{aligned}$ , what is the value of $\begin{aligned} \tan θ \end{aligned}$ ?
::如果是罪35和罪45, 是什么价值 tan?

If $\begin{aligned} \cos (θ) = \frac{7}{25} \end{aligned}$ and $\begin{aligned} \sin (θ) = \frac{24}{25}, \end{aligned}$ what is the value of $\begin{aligned} \cot (θ) ? \end{aligned}$
::如果cos()=725和sin()=2425,那么cot()值是多少?

If $\begin{aligned} \csc (θ) = \frac{37}{12} \end{aligned}$ and $\begin{aligned} \sec (θ) = \frac{37}{35}, \end{aligned}$ what is the value of $\begin{aligned} \tan (θ) ? \end{aligned}$
::如果 csc = 3712 和 se = 3735, tan 值是多少 ?

If $\begin{aligned} \cos (θ) = \frac{20}{29} \end{aligned}$ and $\begin{aligned} \tan (θ) = \frac{21}{20}, \end{aligned}$ what is the value of $\begin{aligned} \sin (θ) ? \end{aligned}$
::如果cos()=2029 和 tan()=220,那么sin()值是多少?

If $\begin{aligned} \sin (θ) = \frac{39}{89} \end{aligned}$ and $\begin{aligned} \tan (θ) = \frac{39}{80}, \end{aligned}$ what is the value of $\begin{aligned} \sec (θ) ? \end{aligned}$
::如果 sin = 3989 和 tan = 3980, 秒值是多少 ?

If $\begin{aligned} \cos θ = \frac{48}{73} \end{aligned}$ and $\begin{aligned} \sin θ = \frac{55}{73} \end{aligned}$ , what is the value of $\begin{aligned} \tan θ \end{aligned}$ ?
::如果Cos4873 和sin5573, 是什么价值 tan?

If $\begin{aligned} \csc (θ) = \frac{97}{65} \end{aligned}$ and $\begin{aligned} \cot (θ) = \frac{72}{65}, \end{aligned}$ what is the value of $\begin{aligned} \cos (θ) ? \end{aligned}$
::如果csc()=9765和cot()=7265,cos()值是多少?

If $\begin{aligned} \cos θ = \frac{1}{2} \end{aligned}$ and $\begin{aligned} \cot θ = \frac{\sqrt{3}}{3} \end{aligned}$ , what is the value of $\begin{aligned} \sin θ \end{aligned}$ ?
::如果cos12和cot33, 什么是罪恶的价值?

If $\begin{aligned} \tan θ = 0 \end{aligned}$ and $\begin{aligned} \cos θ = - 1 \end{aligned}$ , what is the value of $\begin{aligned} \sin θ \end{aligned}$ ?
::如果Tan0和cos1, 什么是罪恶的价值?

If $\begin{aligned} \cot θ = - 1 \end{aligned}$ and $\begin{aligned} \sin θ = - \frac{\sqrt{2}}{2} \end{aligned}$ , what is the value of $\begin{aligned} \cos θ \end{aligned}$ ?
::如果Cot1和sin22, Cos值是多少?

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

引号

Quotient Identities ::引号

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)

Quotient Identities
::引号

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)