Section: 基本三角特征 | 6.1 基本三角特征

Section outline

The basic trigonometric identities are ones that can be logically deduced from the definitions and graphs of the six trigonometric functions . Previously, some of these identities have been used in a casual way, but now they will be formalized and added to the toolbox of trigonometric identities.
::基本三角特征是从六种三角函数的定义和图表中逻辑推导出来的。以前,其中一些特征是随意使用的,但现在它们将被正式化并添加到三角特征的工具箱中。

How can you use the trigonometric identities to simplify the following expression?
::您如何使用三角特征来简化以下表达式 ?

$\begin{aligned} {[\frac{\sin (\frac{π}{2} - θ)}{\sin (- θ)}]}^{- 1} \end{aligned}$
::[sin_()_()_________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

Trigonometric Identities
::三角度数特征

An identity is a mathematical sentence involving the symbol “=” that is always true for variables within the domains of the expressions on either side.
::身份是指一个带有符号“=”的数学句,对两侧表达式范围内的变量来说,始终如此。

Reciprocal Identities
::相互身份

The refer to the connections between the trigonometric functions like sine and cosecant . Sine is opposite over hypotenuse and cosecant is hypotenuse over opposite. This logic produces the following six identities.
::指的是三角函数之间的联系, 如正弦和正弦。 Sine 与下限相对, 共弦和共弦等。共弦是相反的, 共正弦是相反的。此逻辑产生以下六个特性。

$\begin{aligned} \sin θ = \frac{1}{\csc θ} \end{aligned}$
::国家国家国家国家国家国国国国国国国国国国国国国国国国国国国国国国国国国国国国国国国国国国国国国国国国国国国国国国国国国国一国国国国国国国国国国国国国国国国国国国国国国国国国国国国国国

$\begin{aligned} \cos θ = \frac{1}{\sec θ} \end{aligned}$
:cos1sec) (cos1sec) (cos1sec)

$\begin{aligned} \tan θ = \frac{1}{\cot θ} \end{aligned}$
::{\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,不,不,不,不,不,不,不,不,不,不,不,不,不,不,不,不,不,不,不,不,不,不,不,不,不,不,不,不,不,不,不,不,不,不,不,不,不,不,不,不,不,不,不,不,不,不,不,不,不,不,不,不,不,不,不,不,不,不,不,不,不,不,不,不,不,不,不

$\begin{aligned} \cot θ = \frac{1}{\tan θ} \end{aligned}$
::柯特・1吨・・・・・・・・・・・・・・・・・・・・・・・・・・・・・・・・・・・・・・・・・・・・・・・・

$\begin{aligned} \sec θ = \frac{1}{\cos θ} \end{aligned}$
::11111111111111112

$\begin{aligned} \csc θ = \frac{1}{\sin θ} \end{aligned}$
:csc1sin) (csc1sin) (csc1sin1}) (csc1sin}) (csc1sin}) (csc1sin}) (csc1sin1sin}) (csc1sin}) (csin1sin%) (c) (c) (c) (c) (c) (c) (a) (c) (c) (c) (c) (c) (c) (c) (c) (a) (c) (c) (c) (c) (c) (c) (c)) (c) (c) (c) (c) (c) (c) (c) (c) (c) (c) (c) (c) (c) (c) (c) (c) (c)

Quotient Identities
::引号

The follow from the definition of sine, cosine and tangent.
::与正弦、正弦和正弦定义的接轨。

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

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

Odd/Even Identities
::奇数/偶数

The odd-even identities follow from the fact that only cosine and its reciprocal secant are even and the rest of the trigonometric functions are odd.
::奇异的奇异身份来源于这样一个事实,即只有连弦和对等分离是均衡的,其余的三角函数是奇异的。

$\begin{aligned} \sin (- θ) = - \sin θ \end{aligned}$
:)

$\begin{aligned} \cos (- θ) = \cos θ \end{aligned}$
::cos() =cos() =cos() =cos()

$\begin{aligned} \tan (- θ) = - \tan θ \end{aligned}$
::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~

$\begin{aligned} \cot (- θ) = - \cot θ \end{aligned}$
:) () () () () () () () () () ()

$\begin{aligned} \sec (- θ) = \sec θ \end{aligned}$
::秒

$\begin{aligned} \csc (- θ) = - \csc θ \end{aligned}$
::csc()csc()csc()csc()

Cofunction Identities
::共同用途

The cofunction identities make the connection between trigonometric functions and their “co” counterparts like . Graphically, all of the cofunctions are reflections and horizontal shifts of each other.
::共同功能身份使三角函数与其“共同”对应方(如 .) 之间的联系。从图形上看,所有共同功能都是反射和横向移动。

$\begin{aligned} \cos (\frac{π}{2} - θ) = \sin θ \end{aligned}$
::=sin =sin

$\begin{aligned} \sin (\frac{π}{2} - θ) = \cos θ \end{aligned}$
::-================================================================================================================================================== ==================================================================================================================

$\begin{aligned} \tan (\frac{π}{2} - θ) = \cot θ \end{aligned}$
::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~

$\begin{aligned} \cot (\frac{π}{2} - θ) = \tan θ \end{aligned}$
:

$\begin{aligned} \sec (\frac{π}{2} - θ) = \csc θ \end{aligned}$
::

$\begin{aligned} \csc (\frac{π}{2} - θ) = \sec θ \end{aligned}$
::csc = sec = sec = csc = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =

Examples
::实例

Example 1
::例1

Earlier, you were asked how you could simplify the trigonometric expression:
::早些时候,有人问你如何简化三角表达式:

It can be simplified to be equivalent to negative tangent as shown below:
::可以简化为相当于负正切值,如下所示:

\begin{aligned} {[\frac{\sin (\frac{π}{2} - θ)}{\sin (- θ)}]}^{- 1} & = \frac{\sin (- θ)}{\sin (\frac{π}{2} - θ)} \\ = \frac{- \sin θ}{\cos θ} \\ = - \tan θ \end{aligned}

::[sin()()()()()()()()()()()()()()()()()()()()()()()。

Example 2
::例2

If $\begin{aligned} \sin θ = 0.87 \end{aligned}$ , find $\begin{aligned} \cos (θ - \frac{π}{2}) \end{aligned}$ .
::如果罪过是0.87, 找到cos(2) 。

While it is possible to use a calculator to find $\begin{aligned} θ \end{aligned}$ , using identities works very well too.
::虽然可以使用计算器找到 ,但使用身份也很有效。

First you should factor out the negative from the argument. Next you should note that cosine is even and apply the odd-even identity to discard the negative in the argument. Lastly recognize the cofunction identity .
::首先,您应该从参数中计出负值。接下来您应该注意, cosine 是偶数, 并且应用奇数身份来丢弃参数中的负值。最后, 您应该识别共用身份。

$\begin{aligned} \cos (θ - \frac{π}{2}) = \cos (- (\frac{π}{2} - θ)) = \cos (\frac{π}{2} - θ) = \sin θ = 0.87 \end{aligned}$
:)=cos()=cos()=cos()=sin0.87

Example 3
::例3

If $\begin{aligned} \cos (θ - \frac{π}{2}) = 0.68 \end{aligned}$ then determine $\begin{aligned} \csc (- θ) \end{aligned}$ .
::如果cos(2) =0.68 那么确定 csc() 。

You need to show that $\begin{aligned} \cos (θ - \frac{π}{2}) = \cos (\frac{π}{2} - θ) \end{aligned}$ .
::您需要显示 cos( 2) =cos( 2) 。

\begin{aligned} 0.68 & = \cos (θ - \frac{π}{2}) \\ = \cos (\frac{π}{2} - θ) \\ = \sin (θ) \end{aligned}

::0.68=cos() =cos() =sin()

Then, $\begin{aligned} \csc (- θ) = - \csc θ \end{aligned}$
::然后,csc()csc()csc()

\begin{aligned} = - \frac{1}{\sin θ} \\ = - (0.68)^{- 1} \\ \approx - 1.47 \end{aligned}

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

Example 4
::例4

Use identities to prove the following: $\begin{aligned} \cot (- β) \cot (\frac{π}{2} - β) \sin (- β) = \cos (β - \frac{π}{2}) \end{aligned}$ .
::使用身份来证明 : comt cot( 2) sin =cos( 2)。

When doing trigonometric proofs, it is vital that you start on one side and only work with that side until you derive what is on the other side. Sometimes it may be helpful to work from both sides and find where the two sides meet, but this work is not considered a proof. You will have to rewrite your steps so they follow from only one side. In this case, work with the left side and keep rewriting it until you have $\begin{aligned} \cos (β - \frac{π}{2}) \end{aligned}$ .
::在进行三角验证时,至关重要的是,你必须从一方开始,只与另一方合作,直到你得出另一方的答案。有时,从双方开展工作并找到双方相会的地点也许是有益的,但这项工作不被视为证据。你必须改写你的步骤,以便它们只从一方走。在这种情况下,与左方合作,并不断重写,直到你找到答案为止。

\begin{aligned} \cot (- β) \cot (\frac{π}{2} - β) \sin (- β) & = - \cot β \tan β \cdot - \sin β \\ = - 1 \cdot - \sin β \\ = \sin β \\ = \cos (\frac{π}{2} - β) \\ = \cos (- (β - \frac{π}{2})) \\ = \cos (β - \frac{π}{2}) \end{aligned}

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Example 5
::例5

Prove the following trigonometric identity by working with only one side.
::通过只与一方合作来证明以下三角特征。

$\begin{aligned} \cos x \sin x \tan x \cot x \sec x \csc x = 1 \end{aligned}$
::COSxsinxtanxcotxsecxcxcscx=1

\begin{aligned} \cos x \sin x \tan x \cot x \sec x \csc x & = \cos x \sin x \tan x \cdot \frac{1}{\tan x} \cdot \frac{1}{\cos x} \cdot \frac{1}{\sin x} \\ = 1 \end{aligned}

::xxsinxxtanxxxxxxxxsecxcscxx=1 xxxxxxxxxxxxxin}xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx=1

Summary

An identity is a mathematical sentence involving the symbol “=” that is always true for variables within the domains of the expressions on either side.
::身份是指一个带有符号“=”的数学句,对两侧表达式范围内的变量来说,始终如此。

Reciprocal Identities:

$\begin{aligned} \sin θ = \frac{1}{\csc θ} & \csc θ = \frac{1}{\sin θ} \\ \cos θ = \frac{1}{\sec θ} & \sec θ = \frac{1}{\cos θ} \\ \tan θ = \frac{1}{\cot θ} & \cot θ = \frac{1}{\tan θ} \end{aligned}$

::对等身份: sin_1csc_csc_1sin_1sin_cos_1sec_2_1c_1c_1c_1c_1c_1c_

Quotient Identities:

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

::引言名称: tansincoscoscotcossin

Odd/Even Identities:

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

::偶数/奇数:sin()()()()()()()()()()()()()()()()()()()()()()()()()()()()()()。

Cofunction Identities:

$\begin{aligned} \sin θ = \cos (\frac{π}{2} - θ) & \csc θ = \sec (\frac{π}{2} - θ) \\ \cos θ = \sin (\frac{π}{2} - θ) & \sec θ = \csc (\frac{π}{2} - θ) \\ \tan θ = \cot (\frac{π}{2} - θ) & \cot θ = \tan (\frac{π}{2} - θ) \end{aligned}$

:) () () () () () () () () () () () () () () () () () () () () () () () () () () () () () () () () () ()

Review
::回顾

1. Prove the quotient identity for cotangent using sine and cosine.
::1. 使用正弦和余弦来证明共切物的商数特性。

2. Explain why $\begin{aligned} \cos (\frac{π}{2} - θ) = \sin θ \end{aligned}$ using graphs and transformations.
::2. 解释为何使用图表和变换的cos(2)=sin。

3. Explain why $\begin{aligned} \sec θ = \frac{1}{\cos θ} \end{aligned}$ .
::3. 解释一下为什么秒1秒1秒。

4. Prove that $\begin{aligned} \tan θ \cdot \cot θ = 1 \end{aligned}$ .
::4. 证明塔那尼科特1号

5. Prove that $\begin{aligned} \sin θ \cdot \csc θ = 1 \end{aligned}$ .
::5. 证明这一罪行是有罪的。

6. Prove that $\begin{aligned} \sin θ \cdot \sec θ = \tan θ \end{aligned}$ .
::6. 证明这一点。

7. Prove that $\begin{aligned} \cos θ \cdot \csc θ = \cot θ \end{aligned}$ .
::7. 证明这一点。

8. If $\begin{aligned} \sin θ = 0.81 \end{aligned}$ , what is $\begin{aligned} \sin (- θ) \end{aligned}$ ?
::8. 如果罪为0.81,什么是罪?

9. If $\begin{aligned} \cos θ = 0.5 \end{aligned}$ , what is $\begin{aligned} \cos (- θ) \end{aligned}$ ?
::9. 如果Cos0.5,什么是Cos()?

10. If $\begin{aligned} \cos θ = 0.25 \end{aligned}$ , what is $\begin{aligned} \sec (- θ) \end{aligned}$ ?
::10. 如果cos0.25,什么是se()?

11. If $\begin{aligned} \csc θ = 0.7 \end{aligned}$ , what is $\begin{aligned} \sin (- θ) \end{aligned}$ ?
::11. 如果csc+0.7,什么是罪?

12. How can you tell from a graph if a function is even or odd?
::12. 你如何从图表中看出函数是偶数还是奇数?

13. Prove $\begin{aligned} \frac{\tan x \cdot \sec x}{\csc x} \cdot \cot x = \tan x \end{aligned}$ .
::13. 探矿技术:

14. Prove $\begin{aligned} \frac{\sin^{2} x \cdot \sec x}{\tan x} \cdot \csc x = 1 \end{aligned}$ .
::14. 证明 sin2x secxtan x x cscx=1。

15. Prove $\begin{aligned} \cos x \cdot \tan x = \sin x \end{aligned}$ .
::15. 证明cosxx=sinx。

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

Trigonometric Identities ::三角度数特征

Reciprocal Identities ::相互身份

Quotient Identities ::引号

Odd/Even Identities ::奇数/偶数

Cofunction Identities ::共同用途

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)