Section: 其他三角函数图 | 5.7 其他三角函数图

Section outline

If you already know the relationship between the equation and graph of functions then the other four functions can be found by identifying zeroes, asymptotes and key points. Are the four new functions transformations of the sine and cosine functions?
::如果您已经知道函数的方程式和图形之间的关系,那么其他四个函数可以通过识别零、零和关键点来找到。是否对正弦函数和正弦函数的四种新函数进行转换?

Graphing Other Trigonometric Functions
::绘制其他三角函数图

Secant and Cosecant
::塞坎特和塞桑特

Since secant is the reciprocal of cosine the graphs are very closely related.
::由于分离是连弦的对等关系,因此图表是密切相关的。

Notice wherever cosine is zero, secant has a vertical asymptote and where $\begin{aligned} \cos x = 1 \end{aligned}$ then $\begin{aligned} \sec x = 1 \end{aligned}$ as well. These two logical pieces allow you to graph any secant function of the form:
::备注( cosine is 0 ) , secont 具有垂直的渐变状态, 以及 Cosx=1 然后 secx=1 。这两个逻辑部分允许您绘制窗体的任何偏移函数 :

$\begin{aligned} f (x) = \pm a \cdot \sec (b (x + c)) + d \end{aligned}$
:xxx)+d)+(b(x+c)+(d)+(xx)+(xx)+

The method is to graph it as you would a cosine and then insert asymptotes and the secant curves so they touch the cosine curve at its maximum and minimum values. This technique is identical to graphing cosecant graphs. Simply use the sine graph to find the location and asymptotes.
::方法就是以正弦形式绘制图形,然后插入正弦和伸缩曲线,以便以最大值和最小值触摸余弦曲线。这种方法与正弦图形相同。只需使用正弦图形找到位置和静脉图。

Tangent and Cotangent
::切点和切点

The tangent and cotangent graphs are more difficult because they are a ratio of the sine and cosine functions.
::相切和相切图形比较困难,因为它们是正弦和余弦函数的比重。

$\begin{aligned} \tan x = \frac{\sin x}{\cos x} \end{aligned}$
::tanx=sinxcosxx

$\begin{aligned} \cot x = \frac{\cos x}{\sin x} \end{aligned}$
::COTx=cosxinxxxx

The way to think through the graph of $\begin{aligned} f (x) = \tan x \end{aligned}$ is to first determine its asymptotes. The asymptotes occur when the denominator, cosine, is zero. This happens at $\begin{aligned} \pm \frac{π}{2}, \pm \frac{3 π}{2} \dots \end{aligned}$ The next thing to plot is the zeros which occur when the numerator, sine, is zero. This happens at $\begin{aligned} 0, \pm π, \pm, 2 π \dots \end{aligned}$ From the unit circle and basic , you already know some values of $\begin{aligned} \tan x \end{aligned}$ :
::通过 f( x) =tanx 的图形来思考的方法是首先确定其小数点。当分母( cosine) 为 0 时会出现小数点。这发生在 2, 32... 下一步要绘制的是当点数( Prin) 为 0 时出现的零数。这发生在 0, , 2... 从单位圆和基本圆中, 您已经知道 tanx 的一些值 :

$\begin{aligned} \tan \frac{π}{4} = 1 \end{aligned}$
::tan $4=1 $4=1

$\begin{aligned} \tan (- \frac{π}{4}) = - 1 \end{aligned}$
::~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

By plotting all this information, you get a very good sense as to what the graph of tangent looks like and you can fill in the rest.
::通过绘制所有这些信息, 你就能很好地了解相切图的外形, 并可以填充其余的信息。

Notice that the period of tangent is $\begin{aligned} π \end{aligned}$ not $\begin{aligned} 2 π \end{aligned}$ , because it has a shorter cycle.
::请注意,相切期间不是2,因为它的周期较短。

The graph of cotangent can be found using identical logic as tangent. You know $\begin{aligned} \cot x = \frac{1}{\tan x} \end{aligned}$ . This means that the graph of cotangent will have zeros wherever tangent has asymptotes and asymptotes wherever tangent has zeroes. You also know that where tangent is 1, cotangent is also 1. Thus the graph of cotangent is:
::共切图可以使用与正切相同的逻辑查找。您知道 cotx=1tanx。这意味着正切图中, 只要正切图中含有无位数, 则余切图中就有零位数。您也知道, 相切图中, 1, 余切图中也有 1 。因此, 余切图中:

Examples
::实例

Example 1
::例1

Earlier, you were asked if the four new functions are transformations of sine and cosine. The four new functions are not purely transformations of the sine and cosine functions. However, secant and cosecant are transformations of each other as are tangent and cotangent.
::早些时候, 有人询问您这四个新函数是否是正弦和正弦的转换。这四个新函数并非纯正弦和正弦函数的转换。但是, 分离和共弦函数是相互的转换, 与正切和相切一样。

Example 2
::例2

Graph the function $\begin{aligned} f (x) = - 2 \cdot \csc (π (x - 1)) + 1 \end{aligned}$ .
::函数 f( x)\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\1\\\\\1)+1。

Graph the function as if it were a sine function. Then insert asymptotes wherever the sine function crosses the sinusoidal axis . Lastly add in the cosecant curves.
::绘制函数图时,它仿佛是一个正弦函数。然后在正弦函数横跨正弦轴的位置插入正弦函数。最后在余弦曲线中添加。

The is 2. The shape is negative sine. The function is shifted up one unit and to the right one unit.
::2. 形状为负正弦值,函数向上移一个单位,向右移一个单位。

Note that only the blue portion of the graph represents the given function.
::请注意,仅图中的蓝色部分代表给定函数。

Example 3
::例3

How do you write a tangent function as a cotangent function?
::您如何将正切函数写成余切函数 ?

There are two main ways to go between a tangent function and a cotangent function. The first method was discussed in Example A: $\begin{aligned} f (x) = \tan x = \frac{1}{\cot x} \end{aligned}$ .
::在正切函数和余切函数之间有两种主要方式。例A:f(x)=tanx=1cotx讨论了第一种方法。

The second approach involves two transformations. Start by reflecting across the $\begin{aligned} x \end{aligned}$ or the $\begin{aligned} y \end{aligned}$ axis. Notice that this produces an identical result. Next shift the function to the right or left by $\begin{aligned} \frac{π}{2} \end{aligned}$ . Again this produces an identical result. $\begin{aligned} f (x) = \tan x = - \cot (x - \frac{π}{2}) \end{aligned}$ .
::第二个方法涉及两个转换。从反射 x 或 y 轴开始。注意此结果产生相同的结果。下一步将函数向右或左移动。下次将函数向 2 转移。此结果再次产生相同的结果。 f( x) =tanx cot (x 2) 。

Example 4
::例4

Find the equation of the function in the following graph.
::在下图中查找函数的方程式。

If you connect the relative maximums and minimums of the function, it produces a shifted cosine curve that is easier to work with.
::如果将函数的相对最大值和最小值连接在一起,则生成一个较容易工作的转动余弦曲线。

The amplitude is 3. The vertical shift is 2 down. The period is 4 which implies that $\begin{aligned} b = \frac{π}{2} \end{aligned}$ . The shape is positive cosine and if you choose to start at $\begin{aligned} x = 0 \end{aligned}$ there is no phase shift .
::振幅为 3 。垂直移动向下为 2 。周期为 4 。这意味着 b2 。形状为正余弦, 如果您选择在 x=0 开始, 则没有相移。

$\begin{aligned} f (x) = 3 \cdot \csc (\frac{π}{2} x) - 2 \end{aligned}$
:xx) = 3csc ( 2x)-2

Example 5
::例5

Where are the asymptotes for tangent and why do they occur?
::相切值的微粒在哪里? 为什么会发生?

Since $\begin{aligned} \tan x = \frac{\sin x}{\cos x} \end{aligned}$ the asymptotes occur whenever $\begin{aligned} \cos x = 0 \end{aligned}$ which is $\begin{aligned} \pm \frac{π}{2}, \pm \frac{3 π}{2}, \dots \end{aligned}$
::由于 tanx = sinxcos x x x x x x x x x x x x x x x, osx=0 = 2, 32, 2, 32...

Summary

Secant and cosecant graphs are closely related to cosine and sine graphs, respectively, as they are their reciprocals.
::相干和共生图形分别与共生和正正弦图形密切相关,因为它们是相互的。

To graph tangent, first determine its asymptotes (where cosine is zero) and zeros (where sine is zero), then plot known values and fill in the rest. The period of tangent is $\begin{aligned} π, \end{aligned}$ not $\begin{aligned} 2 π . \end{aligned}$
::图形正切, 首先确定它的零点( 共弦为零) 和零点( 正弦为零) , 然后绘制已知值, 并填满其余值。相切期为 __ , 而不是 2__ 。

Cotangent graph can be found using similar logic as tangent, with zeros where tangent has asymptotes and asymptotes where tangent has zeros.
::共切图可以使用与正切相类似的逻辑来查找,在正切为零时,正切为零,正切为零时,正切为零,正切为零时,正切为零。

Review
::回顾

1. What function can you use to help you make a sketch of $\begin{aligned} f (x) = \sec x \end{aligned}$ ? Why?
::1. 您可以用什么函数来帮助您绘制 f( x) =secx 的草图? 为什么?

2. What function can you use to help you make a sketch of $\begin{aligned} g (x) = \csc x \end{aligned}$ ? Why?
::2. 您可以用什么函数来帮助您绘制 g( x) = csc* x 的草图? 为什么?

Make a sketch of each of the following from memory.
::从记忆中绘制以下各点的草图。

3. $\begin{aligned} f (x) = \sec x \end{aligned}$
::3. f(x) =secx

4. $\begin{aligned} g (x) = \csc x \end{aligned}$
::4. g(x)=cscx

5. $\begin{aligned} h (x) = \tan x \end{aligned}$
::5. h(x)=tanx

6. $\begin{aligned} k (x) = \cot x \end{aligned}$
::6. k(x) =cotx

Graph each of the following.
::分别绘制以下各图。

7. $\begin{aligned} f (x) = 2 \csc (x) + 1 \end{aligned}$
::7. f(x)=2csc(x)+1

8. $\begin{aligned} g (x) = 2 \csc (\frac{π}{2} x) + 1 \end{aligned}$
::8. g(x)=2csc ( 2x)+1

9. $\begin{aligned} h (x) = 2 \csc (\frac{π}{2} (x - 3)) + 1 \end{aligned}$
::9. h(x) = 2csc (%2(x-3))+1

10. $\begin{aligned} j (x) = \cot (\frac{π}{2} x) + 3 \end{aligned}$
::10. j(x)=cot( 2x)+3

11. $\begin{aligned} k (x) = - \sec (\frac{π}{3} (x + 1)) - 4 \end{aligned}$
::11. k( x) sec( 3( x+1))- 4

12. $\begin{aligned} m (x) = - \tan (x) + 1 \end{aligned}$
::12. (x) (x)+1

13. $\begin{aligned} p (x) = - 2 \tan (x - \frac{π}{2}) + 1 \end{aligned}$
::13. p(x) 2tan (x) 2+1

14. Find two ways to write $\begin{aligned} \sec x \end{aligned}$ in terms of other trigonometric functions.
::14. 找到两种方法,用其他三角函数来写cex。

15. Find two ways to write $\begin{aligned} \csc x \end{aligned}$ in terms of other trigonometric functions.
::15. 找到两种方法用其他三角函数来写cscx。

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

Graphing Other Trigonometric Functions ::绘制其他三角函数图

Secant and Cosecant ::塞坎特和塞桑特

Tangent and Cotangent ::切点和切点

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)