Course: 3.1 偶数和奇数特征

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偶数和奇数
You and your friend are in math class together. You enjoy talking a lot outside of class about all of the interesting topics you cover in class. Lately you've been covering trig functions and the unit circle . As it turns out, trig functions of certain angles are pretty easy to remember. However, you and your friend are wishing there was an easy way to ‘‘shortcut’’ calculations so that if you knew a trig function for an angle you could relate it to the trig function for another angle; in effect giving you more reward for knowing the first trig function.
::你和你的朋友一起上数学课。您喜欢在课外谈论很多您在课内覆盖的所有有趣的话题。最近您一直在覆盖三角函数和单位圆。事实证明, 某些角度的三角函数很容易被记住。但是, 您和你的朋友希望有一个简单的方法来进行“ shortcut” 计算, 这样如果您知道一个角度的三角函数, 您可以将其与三角函数联系起来, 换个角度; 事实上, 您知道第一个三角函数会得到更多的奖励。

You're examining some notes and starting writing down trig functions at random. You eventually write down:
::您正在检查一些笔记, 并开始随机写下 trig 函数。您最终会写下 :

$\begin{aligned} \cos (\frac{π}{18}) \end{aligned}$
::COs ( 18)

Is there any way that if you knew how to compute this, you'd automatically know the answer for a different angle?
::有没有办法,如果你知道如何计算这个, 你会自动知道答案不同的角度?

Even and Odd Identities
::偶数和奇数

An even function is a function where the value of the function acting on an argument is the same as the value of the function when acting on the negative of the argument. Or, in short:
::even 函数是一个函数,根据某一参数行事的函数的价值与根据该参数的否定作用的函数的价值相同。或简而言之:

$\begin{aligned} f (x) = f (- x) \end{aligned}$
:xx)=f(x)

So, for example, if f(x) is some function that is even, then f(2) has the same answer as f(-2). f(5) has the same answer as f(-5), and so on.
::因此,例如,如果f(x)是某种偶数函数,那么f(2)的回答与f(-2).f(5)的回答与f(-5)等相同。

In contrast, an odd function is a function where the negative of the function's answer is the same as the function acting on the negative argument. In math terms, this is:
::反之,奇数函数是指函数回答的负数与以负参数作用的函数相同。

$\begin{aligned} - f (x) = f (- x) \end{aligned}$
::-f(x)=f(x)

If a function were negative, then f(-2) = -f(2), f(-5) = -f(5), and so on.
::如果函数为负,则f(-2)=-f(2)、f(-5)=-f(5),等等。

Functions are even or odd depending on how the end behavior of the graphical representation looks. For example, $\begin{aligned} y = x^{2} \end{aligned}$ is considered an even function because the ends of the parabola both point in the same direction and the parabola is symmetric about the $\begin{aligned} y - \end{aligned}$ axis. $\begin{aligned} y = x^{3} \end{aligned}$ is considered an odd function for the opposite reason. The ends of a cubic function point in opposite directions and therefore the parabola is not symmetric about the $\begin{aligned} y - \end{aligned}$ axis. What about the trig functions? They do not have exponents to give us the even or odd clue (when the degree is even, a function is even, when the degree is odd, a function is odd).
::函数是偶数或奇数, 取决于图形表达式的结束行为。例如, y=x2 被视为偶数函数, 因为 parbola 的结尾都指向同一方向, 而 parbola 则对准 y- 轴。 y=x3 则因相反的原因被视为奇数函数。立方函数点对向相反方向的结尾, 因此parbola 对准 y- 轴。有关 y- 轴的函数是什么 ? 它们没有向我们提供偶数或奇数线索的引号( 当程度相等时, 函数是偶数, 当程度奇数时, 函数是奇数 ) 。

$\begin{aligned} \underline{Even Function} & \underline{Odd Function} \\ y = (- x)^{2} = x^{2} & y = (- x)^{3} = - x^{3} \end{aligned}$

::Even 函数_Odd 函数_y=(- x)2=x2y=(- x)3x3x3

Let’s consider sine. Start with $\begin{aligned} \sin (- x) \end{aligned}$ . Will it equal $\begin{aligned} \sin x \end{aligned}$ or $\begin{aligned} - \sin x \end{aligned}$ ? Plug in a couple of values to see.
::让我们考虑正弦。从sin(-x)开始。它是否等于sinx 或-sinx? 插在几个价值观中。

$\begin{aligned} \sin (- 30^{\circ}) & = \sin 330^{\circ} = - \frac{1}{2} = - \sin 30^{\circ} \\ \sin (- 135^{\circ}) & = \sin 225^{\circ} = - \frac{\sqrt{2}}{2} = - \sin 135^{\circ} \end{aligned}$

::=============================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================

From this we see that sine is odd . Therefore, $\begin{aligned} \sin (- x) = - \sin x \end{aligned}$ , for any value of $\begin{aligned} x \end{aligned}$ . For cosine, we will plug in a couple of values to determine if it’s even or odd.
::从这里我们可以看到正弦是奇数。因此, sin( ~x) sinx, 任何值为 x。对于 cosine, 我们将插入几个值, 以确定它是偶数还是奇数。

$\begin{aligned} \cos (- 30^{\circ}) & = \cos 330^{\circ} = \frac{\sqrt{3}}{2} = \cos 30^{\circ} \\ \cos (- 135^{\circ}) & = \cos 225^{\circ} = - \frac{\sqrt{2}}{2} = \cos 135^{\circ} \end{aligned}$

::cos(-30)=cos33032=cos30cos(-135)=cos22522=cos135

This tells us that the cosine is even . Therefore, $\begin{aligned} \cos (- x) = \cos x \end{aligned}$ , for any value of $\begin{aligned} x \end{aligned}$ . The other four trigonometric functions are as follows:
::这告诉我们余弦是均衡的。因此, cos (- x) =cosx, 任何值为 x。其他四个三角函数如下:

$\begin{aligned} \tan (- x) & = - \tan x \\ \csc (- x) & = - \csc x \\ \sec (- x) & = \sec x \\ \cot (- x) & = - \cot x \end{aligned}$

::~ ~ (x) ~ (x) ~ (x) ~ (x) ~ (x) ~ (x) ~ (x) ~ (x) ~ (x) ~ (x) ~ (x) ~ (x) ~ (x) ~ (x) ~ (x) ~ (x) ~ (x) ~ (x)

Notice that cosecant is odd like sine and secant is even like cosine.
::注意酸是奇特的像正弦和松脱甚至是焦素。

Finding Even and Odd Identities
::寻找偶数和奇数

1. Find sinx
::1. 找出罪过

If $\begin{aligned} \cos (- x) = \frac{3}{4} \end{aligned}$ and $\begin{aligned} \tan (- x) = - \frac{\sqrt{7}}{3} \end{aligned}$ , find $\begin{aligned} \sin x \end{aligned}$ .
::如果cos(-x) =34 和 tan(-x) 73, 找到 sinx 。

We know that sine is odd. Cosine is even, so $\begin{aligned} \cos x = \frac{3}{4} \end{aligned}$ . Tangent is odd, so $\begin{aligned} \tan x = \frac{\sqrt{7}}{3} \end{aligned}$ . Therefore, sine is positive and $\begin{aligned} \sin x = \frac{\sqrt{7}}{4} \end{aligned}$ .
::我们知道正弦是奇数。正弦是偶数, 所以cosx=34。唐恩是奇数, 所以tanx=73。因此, 正弦是正弦, sinux=74。

2. Find sin(-x)
::2. 查明罪(-x)

If $\begin{aligned} \sin (x) = .25 \end{aligned}$ , find $\begin{aligned} \sin (- x) \end{aligned}$
::如果 sin *( x) =.25, 发现 sin *( - x)

Since sine is an odd function, $\begin{aligned} \sin (- θ) = - \sin (θ) \end{aligned}$ .
::因为正弦是一个奇特的函数, sin sin 。

Therefore, $\begin{aligned} \sin (- x) = - \sin (x) = - .25 \end{aligned}$
::因此,sin(-x)sin(x)25。

3. Find cos(-x)
::3. 查找(x)cs(x)

If $\begin{aligned} \cos (x) = .75 \end{aligned}$ , find $\begin{aligned} \cos (- x) \end{aligned}$
::如果 cos(x) =. 75, 找到 cos(-x)

Since cosine is an even function, $\begin{aligned} \cos (x) = \cos (- x) \end{aligned}$ .
::由于 Cosine 是一个偶数函数, cos (x) = cos (-x) 。

Therefore, $\begin{aligned} \cos (- x) = .75 \end{aligned}$
::因此,cos(-x)=.75

Examples
::实例

Example 1
::例1

Earlier, you were asked to compute $\begin{aligned} \cos (\frac{π}{18}) \end{aligned}$ .
::早些时候,你被要求计算Cos( 18) 。

Since you now know that cosine is an even function, you get to know the cosine of the negative of an angle automatically if you know the cosine of the positive of the angle.
::既然你现在知道连弦是一个偶数函数, 你就会知道一个角度的负余度是自动的, 如果你知道角度的正余度的话。

Therefore, since $\begin{aligned} \cos (\frac{π}{18}) = .9848 \end{aligned}$ , you automatically know that $\begin{aligned} \cos (- \frac{π}{18}) = \cos (\frac{17 π}{18}) = .9848 \end{aligned}$ .
::因此,既然Cos(18)=9848, 你自然会知道Cos(18)=cos(1718)=9848。

Example 2
::例2

What two angles have a value for cosine of $\begin{aligned} \frac{\sqrt{3}}{2} \end{aligned}$ ?
::哪些两个角度的余弦值为 32 ?

On the unit circle, the angles $\begin{aligned} 30^{\circ} \end{aligned}$ and $\begin{aligned} 330^{\circ} \end{aligned}$ both have $\begin{aligned} \frac{\sqrt{3}}{2} \end{aligned}$ as their value for cosine. $\begin{aligned} 330^{\circ} \end{aligned}$ can be rewritten as $\begin{aligned} - 30^{\circ} \end{aligned}$
::在单位圆圆上,角30°和角330°都具有32的余弦值。 330°可以改写为-30°C。

Example 3
::例3

If $\begin{aligned} \cos θ = \frac{\sqrt{3}}{2} \end{aligned}$ , find $\begin{aligned} \sec (- θ) \end{aligned}$
::如果cos32,请找到 sec()

There are 2 ways to think about this problem. Since $\begin{aligned} \cos θ = \cos - θ \end{aligned}$ , you could say $\begin{aligned} \sec (- θ) = \frac{1}{\cos (- θ)} = \frac{1}{\cos (θ)} \end{aligned}$ Or you could leave the cosine function the way it is and say that $\begin{aligned} \sec (- θ) = \sec (θ) = \frac{1}{\cos θ} \end{aligned}$ . But either way, the answer is $\begin{aligned} \frac{2}{\sqrt{3}} \end{aligned}$
::有两种方法来思考这个问题。因为 coscos, 您可以说 sec = 1cos = 1cos 或者你可以离开 cos , 并说 sec =sec = 1cos。但无论哪种方式, 答案都是 23 。

Example 4
::例4

If $\begin{aligned} \cot θ = - \sqrt{3} \end{aligned}$ find $\begin{aligned} \cot - θ \end{aligned}$
::如果 Cot% 3 找到 cot% 3 。

Since $\begin{aligned} \cot (- θ) = - \cot (θ) \end{aligned}$ , if $\begin{aligned} \cot θ = - \sqrt{3} \end{aligned}$ then $\begin{aligned} - \cot (- θ) = - \sqrt{3} \end{aligned}$ . Therefore, $\begin{aligned} \cot (- θ) = \sqrt{3} \end{aligned}$ .
::从科特尔()到科特尔(),如果科特尔()3,那么-科特尔() ()3。因此,科特尔()=3。

Review
::回顾

Identify whether each function is even or odd.
::确定每个函数是偶数还是奇数。

$\begin{aligned} y = \sin (x) \end{aligned}$
::y=sin(x)

$\begin{aligned} y = \cos (x) \end{aligned}$
::y=cos(x)

$\begin{aligned} y = \cot (x) \end{aligned}$
::y=cot(x)

$\begin{aligned} y = x^{4} \end{aligned}$
::y=x4 y=x4

$\begin{aligned} y = x \end{aligned}$
::y=x y=x

If $\begin{aligned} \sin (x) = .3 \end{aligned}$ , what is $\begin{aligned} \sin (- x) \end{aligned}$ ?
::如果sin *(x)=3, 什么是sin *(-x)?

If $\begin{aligned} \cos (x) = .5 \end{aligned}$ , what is $\begin{aligned} \cos (- x) \end{aligned}$ ?
::如果cos(x)=5, 什么是cos(-x)?

If $\begin{aligned} \tan (x) = .1 \end{aligned}$ , what is $\begin{aligned} \tan (- x) \end{aligned}$ ?
::如果 tan(x) = 1, 什么是 tan(-x) ?

If $\begin{aligned} \cot (x) = .3 \end{aligned}$ , what is $\begin{aligned} \cot (- x) \end{aligned}$ ?
::如果cot(x)=3, 什么是cot(-x)?

If $\begin{aligned} \csc (x) = .3 \end{aligned}$ , what is $\begin{aligned} \csc (- x) \end{aligned}$ ?
::如果csc(x)=3, 什么是csc(-x)?

If $\begin{aligned} \sec (x) = 2 \end{aligned}$ , what is $\begin{aligned} \sec (- x) \end{aligned}$ ?
::如果秒(x)=2, 什么是秒(-x)?

If $\begin{aligned} \sin (x) = - .2 \end{aligned}$ , what is $\begin{aligned} \sin (- x) \end{aligned}$ ?
::如果sin(x)2, 什么是sin(-x)?

If $\begin{aligned} \cos (x) = - .25 \end{aligned}$ , what is $\begin{aligned} \sec (- x) \end{aligned}$ ?
::如果cos(x)/25,什么是se(-x)?

If $\begin{aligned} \csc (x) = 4 \end{aligned}$ , what is $\begin{aligned} \sin (- x) \end{aligned}$ ?
::如果csc(x)=4, 什么是罪? (-x)?

If $\begin{aligned} \tan (x) = - .2 \end{aligned}$ , what is $\begin{aligned} \cot (- x) \end{aligned}$ ?
::如果tan(x)2, 什么是cot(-x)?

If $\begin{aligned} \sin (x) = - .5 \end{aligned}$ and $\begin{aligned} \cos (x) = - \frac{\sqrt{3}}{2} \end{aligned}$ , what is $\begin{aligned} \cot (- x) \end{aligned}$ ?
::如果sin(x)5和cos(x)32, 什么是cot(-x)?

If $\begin{aligned} \cos (x) = - .5 \end{aligned}$ and $\begin{aligned} \sin (x) = \frac{\sqrt{3}}{2} \end{aligned}$ , what is $\begin{aligned} \tan (- x) \end{aligned}$ ?
::如果cos(x)5 和sin(x)=32, 什么是tan(x)?

If $\begin{aligned} \cos (x) = - \frac{\sqrt{2}}{2} \end{aligned}$ and $\begin{aligned} \tan (x) = - 1 \end{aligned}$ , what is $\begin{aligned} \sin (- x) \end{aligned}$ ?
::如果cos(x)(22) 和 tan(x)(1), 什么是罪(-x)?

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

偶数和奇数

Even and Odd Identities ::偶数和奇数

Finding Even and Odd Identities ::寻找偶数和奇数

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)

Even and Odd Identities
::偶数和奇数

Finding Even and Odd Identities
::寻找偶数和奇数

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)