课程： 14.8 简化三角矩阵表达式 | The Way To Learn

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选择节简化三角数表达式

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简化三角数表达式
How could you write the trigonometric function $\begin{aligned} \cos θ + \cos θ (\tan^{2} θ) \end{aligned}$ more simply?
::如何更简单地写入三角函数 ?

Simplifying Trigonometric Expressions
::简化三角数表达式

Now that you are more familiar with trig identities, we can use them to simplify expressions. Remember, that you can use any of the following identities.
::现在您更熟悉三类身份, 我们可以使用它们来简化表达式。记住, 您可以使用以下任何一种身份。

: $\begin{aligned} \csc θ = \frac{1}{\sin θ}, \sec θ = \frac{1}{\cos θ}, \end{aligned}$ and $\begin{aligned} \cot θ = \frac{1}{\tan θ} \end{aligned}$
:csc1sin,sec1cos,和cot1tan)

Tangent and Cotangent Identities: $\begin{aligned} \tan θ = \frac{\sin θ}{\cos θ} \end{aligned}$ and $\begin{aligned} \cot θ = \frac{\cos θ}{\sin θ} \end{aligned}$
::相近和相切的特征:tansincos和cotcossin

: $\begin{aligned} \sin^{2} θ + \cos^{2} θ = 1, 1 + \tan^{2} θ = \sec^{2} θ, \end{aligned}$ and $\begin{aligned} 1 + \cot^{2} θ = \csc^{2} θ \end{aligned}$
::1,1+tan2sec2和1+cot2csc2

Cofunction Identities: $\begin{aligned} \sin (\frac{π}{2} - θ) = \cos θ, \cos (\frac{π}{2} - θ) = \sin θ, \end{aligned}$ and $\begin{aligned} \tan (\frac{π}{2} - θ) = \cot θ \end{aligned}$
::共认人:sin(2) =cos,cos(2) =sin, 和tan(2) =cot

Negative Angle Identities: $\begin{aligned} \sin (- θ) = - \sin θ, \cos (- θ) = \cos θ, \end{aligned}$ and $\begin{aligned} \tan (- θ) = - \tan θ \end{aligned}$
::负角识别度: sin sin, cos =cos, 和 tan tan

Let's simplify the following expressions.
::让我们简化以下表达式。

$\begin{aligned} \frac{\sec x}{\sec x - 1} \end{aligned}$
::秒 *%xsec*x- 1

When simplifying trigonometric expressions, one approach is to change everything into sine or cosine. First, we can change secant to cosine using the Reciprocal Identity .
::当简化三角表达式时,一种方法是将一切改变为正弦或正弦。首先,我们可以用“对等身份”来改变“分弦”。

$\begin{aligned} \frac{\sec x}{\sec x - 1} \to \frac{\frac{1}{\cos x}}{\frac{1}{\cos x} - 1} \end{aligned}$

::秒 *%ssec*x - 11cos *x1cos*x1cos*x-1

Now, combine the denominator into one fraction by multiplying 1 by $\begin{aligned} \frac{\cos x}{\cos x} \end{aligned}$ .
::现在,将分母结合成一个分数, 乘以 1 乘以 cos xxcos x 。

$\begin{aligned} \frac{\frac{1}{\cos x}}{\frac{1}{\cos x} - 1} \to \frac{\frac{1}{\cos x}}{\frac{1}{\cos x} - \frac{\cos x}{\cos x}} \to \frac{\frac{1}{\cos x}}{\frac{1 - \cos x}{\cos x}} \end{aligned}$

::1cos*x1cos*x1cos*x1cos*x1cos*x1cos*xxx1cos*xx1cos*xx

Change this problem into a division problem and simplify.
::将这个问题变成分裂问题和简化。

$\begin{aligned} \frac{\frac{1}{\cos x}}{\frac{1 - \cos x}{\cos x}} \to & \frac{1}{\cos x} \div \frac{1 - \cos x}{\cos x} \\ \frac{1}{\cos x} \cdot \frac{\cos x}{1 - \cos x} \\ \frac{1}{1 - \cos x} \end{aligned}$

::1cos*x1 -cos*xx*x*x*x*1x*x*1 -cos*x*x*x*x1x*x*x*x*x*_cos*x*x*x*1_cos*x11_cos*x*x

$\begin{aligned} \frac{\sin^{4} x - \cos^{4} x}{\sin^{2} x - \cos^{2} x} \end{aligned}$
::x-cos4xsin2x-cos2xxxx2xxxxxxxxx

With this problem, we need to factor the numerator and denominator and see if anything cancels. The rules of factoring a quadratic and the special quadratic formulas can be used in this scenario.
::在此问题上, 我们需要计数数和分母, 看看是否有东西取消。在此情况下可以使用二次方程和特殊二次方程的计算规则。

$\begin{aligned} \frac{\sin^{4} x - \cos^{4} x}{\sin^{2} x - \cos^{2} x} \to \frac{(\sin^{2} x - \cos^{2} x) (\sin^{2} x + \cos^{2} x)}{(\sin^{2} x - \cos^{2} x)} \to \sin^{2} x + \cos^{2} x \to 1 \end{aligned}$

:sin2)\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\

In the last step, we simplified to the left hand side of the Pythagorean Identity. Therefore, this expression simplifies to 1.
::最后一步,我们简化为毕达哥里人身份的左手侧。因此,这一表达式简化为1。

$\begin{aligned} \sec θ \tan^{2} θ + \sec θ \end{aligned}$
::

First, pull out the GCF.
::首先,拿出绿色气候基金。

$\begin{aligned} \sec θ \tan^{2} θ + \sec θ \to \sec θ (\tan^{2} θ + 1) \end{aligned}$

: )

Now, $\begin{aligned} \tan^{2} θ + 1 = \sec^{2} θ \end{aligned}$ from the Pythagorean Identities, so simplify further.
::现在,来自毕达哥利人身份的Tan2=1=sec2=2, 进一步简化。

$\begin{aligned} \sec θ (\tan^{2} θ + 1) \to \sec θ \cdot \sec^{2} θ \to \sec^{3} θ \end{aligned}$

::seecsec2sec3

Examples
::实例

Example 1
::例1

Earlier, you were asked to simplify the trigonometric function $\begin{aligned} \cos θ + \cos θ (\tan^{2} θ) \end{aligned}$ .
::先前曾要求您简化三角函数 cosççççççç(tan2) 。

Notice that the terms in the expression $\begin{aligned} \cos θ + \cos θ (\tan^{2} θ) \end{aligned}$ have a common factor of $\begin{aligned} \cos θ \end{aligned}$ , so start by factoring this common term out.
::请注意,Coscos(tan2)表达式中的术语有一个共同的 Cos系数,因此首先考虑这一共同术语。

$\begin{aligned} \cos θ + \cos θ (\tan^{2} θ) \\ \cos θ (1 + t a n^{2} θ) \end{aligned}$

::coscos(tan2)cos(1+tan2)

Now, use the trigonometric identity $\begin{aligned} 1 + \tan^{2} θ = \sec^{2} θ \end{aligned}$ , substitute, and simplify.
::现在,使用三角特征 1+tan2sec2, 替代, 简化。

$\begin{aligned} \cos θ (1 + t a n^{2} θ) \\ = \cos θ (s e c^{2} θ) \\ = \cos θ (\frac{1}{c o s^{2} θ)} \\ = \frac{1}{c o s θ} \\ = \sec θ \end{aligned}$

::cos( 1+tan2) =cos( sec2) =cos( 1cos2) =1cossec

Simplify the following trigonometric expressions.
::简化以下三角表达式。

Example 2
::例2

$\begin{aligned} \cos (\frac{π}{2} - x) \cot x \end{aligned}$
::COs(%2-x)cotx

Use the Cotangent Identity and the Cofunction Identity $\begin{aligned} \cos (\frac{π}{2} - θ) = \sin θ \end{aligned}$ .
::使用余切身份和共控身份cos(2)=sin。

$\begin{aligned} \cos (\frac{π}{2} - x) \cot x \to \sin x \cdot \frac{\cos x}{\sin x} \to \cos x \end{aligned}$
::coms(%2-x) cotxsinxcosxsinxxxcosxxxxxxxx

Example 3
::例3

$\begin{aligned} \frac{\sin (- x) \cos x}{\tan x} \end{aligned}$
:-x)cosxtanxxx

Use the Negative Angle Identity and the Tangent Identity.
::使用负角身份和切换身份。

$\begin{aligned} \frac{\sin (- x) \cos x}{\tan x} \to \frac{- \sin x \cos x}{\frac{\sin x}{\cos x}} \to - \sin x \cos x \cdot \frac{\cos x}{\sin x} \to - \cos^{2} x \end{aligned}$
::xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxs2xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

Example 4
::例4

$\begin{aligned} \frac{\cot x \cos x}{\tan (- x) \sin (\frac{π}{2} - x)} \end{aligned}$
::xxcosxtan(- x)sin( 2- x)

In this problem, you will use several identities.
::在这个问题上,你将使用几个身份。

$\begin{aligned} \frac{\cot x \cos x}{\tan (- x) \sin (\frac{π}{2} - x)} \to \frac{\frac{\cos x}{\sin x} \cdot \cos x}{- \frac{\sin x}{\cos x} \cdot \cos x} \to \frac{\frac{\cos^{2} x}{\sin x}}{- \sin x} \to \frac{\cos^{2} x}{\sin x} \cdot - \frac{1}{\sin x} \to - \frac{\cos^{2} x}{\sin^{2} x} \to - \cot^{2} x \end{aligned}$
::xcosxtan(xxxxxinxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx_xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

Review
::回顾

Simplify the following expressions.
::简化下列表达式。

$\begin{aligned} \cot x \sin x \end{aligned}$
::comtxsinxxxx

$\begin{aligned} \cos^{2} x \tan (- x) \end{aligned}$
::COs2xtan(- x)

$\begin{aligned} \frac{\cos (- x)}{\sin (- x)} \end{aligned}$
::COs(- x) sin(- x)

$\begin{aligned} \sec x \cos (- x) - \sin^{2} x \end{aligned}$
::sin2 xxxxxxxxx

$\begin{aligned} \sin x (1 + \cot^{2} x) \end{aligned}$
::exinx( 1+cot2+x)

$\begin{aligned} 1 - \sin^{2} (\frac{π}{2} - x) \end{aligned}$
::1- 辛2( 2- x)

$\begin{aligned} 1 - \cos^{2} (\frac{π}{2} - x) \end{aligned}$
::1 - COs2(%2-x)

$\begin{aligned} \frac{\tan (\frac{π}{2} - x) \sec x}{1 - \csc^{2} x} \end{aligned}$
::~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

$\begin{aligned} \frac{\cos^{2} x \tan^{2} x - 1}{\cos^{2} x} \end{aligned}$
::CO2 xtan2x- 1cos2x

$\begin{aligned} \cot^{2} x + \sin^{2} x + \cos^{2} (- x) \end{aligned}$
::comt2%x+sin2x+cos2(-x)

$\begin{aligned} \frac{\sec x \sin x + \cos (\frac{π}{2} - x)}{1 + \cos x} \end{aligned}$
::=============================================================================================================================================================================================================

$\begin{aligned} \frac{\cos (- x)}{1 + \sin (- x)} \end{aligned}$
::COs(- x)1+sin(- x)

$\begin{aligned} \frac{\sin^{2} (- x)}{\tan^{2} x} \end{aligned}$
:- x)tan2 x

$\begin{aligned} \tan (\frac{π}{2} - x) \cot x - \csc^{2} x \end{aligned}$
::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~

$\begin{aligned} \frac{\csc x (1 - \cos^{2} x)}{\sin x \cos x} \end{aligned}$
::cscx( 1) - cos2x) sin *xcos*xxx

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。

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