Section: 简单度量等式表 | 3.3 简化的三角等量形式

Section outline

Sometimes things are simpler than they look. For example, trigonometric identities can sometimes be reduced to simpler forms by applying other rules. For example, can you find a way to simplify $\begin{aligned} \cos^{3} θ = \frac{3 \cos θ + \cos 3 θ}{4} \end{aligned}$ ?
::有时事情比看上去简单。例如,三角特征有时可以通过应用其他规则被简化为更简单的形式。例如,你能找到一种简化 cos3 {3cos}}{3}{3}4}的方法吗?

Trigonometric Equations
::三角数等量

By this time in your school career you have probably seen trigonometric functions represented in many ways: ratios between the side lengths of right triangles, as functions of coordinates as one travels along the unit circle and as abstract functions with graphs. Now it is time to make use of the properties of the trigonometric functions to gain knowledge of the connections between the functions themselves. The patterns of these connections can be applied to simplify trigonometric expressions and to solve trigonometric equations.
::在你的学校生涯中,你可能已经看到过以多种方式表现的三角函数:右三角侧长之间的比例,作为坐标的函数,在单圆上行走,以及作为带有图形的抽象函数。现在是利用三角函数的属性来了解函数本身之间的联系的时候了。这些连接的模式可以用于简化三角表达式和解析三角方程式。

In order to do this, look for parts of the complex trigonometric expression that might be reduced to fewer trig functions if one of the identities you already know is applied to the expression. As you apply identities, some complex trig expressions have parts that can be cancelled out, others can be reduced to fewer trig functions. Observe how this is accomplished in the examples below.
::要做到这一点, 请查找复杂的三角表达式中如果您已经知道的身份之一被应用到该表达式中, 可能会缩小为更少的三角函数的部分。在应用身份时, 一些复杂的三角表达式中有可以取消的部分, 其它的则可以缩小为更少的三角函数。请看看下面的例子中是如何做到这一点的。

Simplifying Expressions
::简化表达式

1. Simplify the following expression using the basic trigonometric identities: $\begin{aligned} \frac{1 + \tan^{2} x}{\csc^{2} x} \end{aligned}$
::1. 使用基本三角特性简化以下表达式: 1+tan2xcsc2x

$\begin{aligned} \frac{1 + \tan^{2} x}{\csc^{2} x} & \dots (1 + \tan^{2} x = \sec^{2} x) Pythagorean Identity \\ \frac{\sec^{2} x}{\csc^{2} x} & \dots (\sec^{2} x = \frac{1}{\cos^{2} x} and \csc^{2} x = \frac{1}{\sin^{2} x}) Reciprocal Identity \\ \frac{\frac{1}{\cos^{2} x}}{\frac{1}{\sin^{2} x}} & = (\frac{1}{\cos^{2} x}) \div (\frac{1}{\sin^{2} x}) \\ (\frac{1}{\cos^{2} x}) \cdot (\frac{\sin^{2} x}{1}) & = \frac{\sin^{2} x}{\cos^{2} x} \\ = \tan^{2} x \to Quotient Identity \end{aligned}$

::1+tan2xcsc2x...( 1+tan2x=sec2x) Pythagoran 身份2xc2xcsc2xxx...( sec2xx=1cos2x和csc2x=1sin2xxx)R对等身份1cos2x1x2xxx=1cos2xxxx}[1+tan2}x=1cos2x1xxxxxx=1c2xxxxx1) 身份( 1cos2xx) (1cos2x}( sin2xxx1) =sin2xxxx

2. Simplify the following expression using the basic trigonometric identities: $\begin{aligned} \frac{\sin^{2} x + \tan^{2} x + \cos^{2} x}{\sec x} \end{aligned}$
::2. 使用基本三角特性简化以下表达式: sin2x+tan2x+cos2xsecx

$\begin{aligned} \frac{\sin^{2} x + \tan^{2} x + \cos^{2} x}{\sec x} & \dots (\sin^{2} x + \cos^{2} x = 1) Pythagorean Identity \\ \frac{1 + \tan^{2} x}{\sec x} & \dots (1 + \tan^{2} x = \sec^{2} x) Pythagorean Identity \\ \frac{\sec^{2} x}{\sec x} & = \sec x \end{aligned}$

::sin2x+cos2xseecx... (辛2x+cos2x=1) Pythagoren 身份1+tan2xsecx... (1+tan2x=sec2x) Pythagoren 身份2xx=sec*x

3. Simplify the following expression using the basic trigonometric identities: $\begin{aligned} \cos x - \cos^{3} x \end{aligned}$
::3. 使用基本三角特性简化以下表达式:cosx-cos3x

$\begin{aligned} \cos x - \cos^{3} x \\ \cos x (1 - \cos^{2} x) \dots Factor out \cos x and \sin^{2} x = 1 - \cos^{2} x \\ \cos x (\sin^{2} x) \end{aligned}$

::cosx- coos3xxcos%x( 1- cos2x)... folor out cosx 和 sin2x=1- cos2xxx( sin2x)

Examples
::实例

Example 1
::例1

Earlier, you were asked to simplify $\begin{aligned} \cos^{3} θ = \frac{3 \cos θ + \cos 3 θ}{4} \end{aligned}$ .
::早些时候,你被要求简化C33cosCos3CosCos34。

The easiest way to start is to recognize the triple angle identity :
::最容易开始的方法就是认清三重角度特征:

$\begin{aligned} \cos 3 θ = \cos^{3} θ - 3 \sin^{2} θ \cos θ \end{aligned}$
::3333332222222222222233333333333333333333332222222CO222222CO222222222222222222222222222222222222222

Substituting this into the original equation gives:
::将其替换为原始方程时给出的:

$\begin{aligned} \cos^{3} θ = \frac{3 \cos θ + (\cos^{3} θ - 3 \sin^{2} θ \cos θ)}{4} \end{aligned}$
:cos33322224)4

Notice that you can then multiply by four and subtract a $\begin{aligned} \cos^{3} θ \end{aligned}$ term:
::注意您可以乘以 4 并减去一个 cos3 {} 条件 :

$\begin{aligned} 3 \cos^{3} θ = 3 \cos θ - 3 \sin^{2} θ \cos θ \end{aligned}$
::3cs33csin22223cscos

And finally pulling out a three and dividing:
::最后拉出三分两裂:

$\begin{aligned} \cos^{3} θ = \cos θ - \sin^{2} θ \cos θ \end{aligned}$
::

Then pulling out a $\begin{aligned} \cos θ \end{aligned}$ and dividing:
::然后拉出一个球杆并分裂:

$\begin{aligned} \cos^{2} θ = 1 - \sin^{2} θ \end{aligned}$
::CO2%1 - sin2 *% 1 - sin2 *% 1 - sin2

Example 2
::例2

Simplify $\begin{aligned} \tan^{3} (x) \csc^{3} (x) \end{aligned}$ .
::简化 tan3 (x) csc3 (x) 。

$\begin{aligned} \tan^{3} (x) \csc^{3} (x) \\ = \frac{\sin^{3} (x)}{\cos^{3} (x)} \times \frac{1}{\sin^{3} (x)} \\ = \frac{1}{\cos^{3} (x)} \\ = \sec^{3} (x) \end{aligned}$

::tan3}(x)csc3}(x)=sin3}(x)csc3}(x)=sin3}(x)cos3}(x)x1sin3}(x)=1cos3}(x)=sec3}(x)=sec3}(x)

Example 3
::例3

Show that $\begin{aligned} \cot^{2} (x) + 1 = \csc^{2} (x) \end{aligned}$ .
::显示 comt2 ( x)+1=csc2}( x) 。

Start with $\begin{aligned} \sin^{2} (x) + \cos^{2} (x) = 1 \end{aligned}$ , and divide everything through by $\begin{aligned} \sin^{2} (x) \end{aligned}$ :
::以 sin2 ( x) +cos2 ( x) = 1 开始, 并用 sin2 ( x) 来分隔一切 :

$\begin{aligned} \sin^{2} (x) + \cos^{2} (x) = 1 \\ = \frac{\sin^{2} (x)}{\sin^{2} (x)} + \frac{\cos^{2} (x)}{\sin^{2} (x)} = \frac{1}{\sin^{2} (x)} \\ = 1 + \cot^{2} (x) = \csc^{2} (x) \end{aligned}$

:x)=1\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\

Example 4
::例4

Simplify $\begin{aligned} \frac{\csc^{2} (x) - 1}{\csc^{2} (x)} \end{aligned}$ .
::简化 csc2( x) - 1csc2( x) 。

$\begin{aligned} \frac{\csc^{2} (x) - 1}{\csc^{2} (x)} \end{aligned}$

::csc2(x)-1csc2(x)

Using $\begin{aligned} \cot^{2} (x) + 1 = \csc^{2} (x) \end{aligned}$ that was proven in #2, you can find the relationship: $\begin{aligned} \cot^{2} (x) = \csc^{2} (x) - 1 \end{aligned}$ , you can substitute into the above expression to get:
::使用 # 2 中证明的 comt2 {( x) +1= csc2} {( x) , 您可以找到关系: comt2} {( x) = csc2} {( x) - 1, 您可以替换为上述表达式 :

$\begin{aligned} \frac{\cot^{2} (x)}{\csc^{2} (x)} \\ = \frac{\frac{\cos^{2} (x)}{\sin^{2} (x)}}{\frac{1}{\sin^{2} (x)}} \\ = \cos^{2} (x) \end{aligned}$

::comt2(x)csc2(x)=cos2(x)sin2(x)sin2(x)1sin2(x)sin2(x)=cos2(x)

Review
::回顾

Simplify each trigonometric expression as much as possible.
::尽可能简化每个三角表达式。

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

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

$\begin{aligned} \frac{1 + \tan (x)}{1 + \cot (x)} \end{aligned}$
::1+tan(x)1+cot(x)

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

$\begin{aligned} \frac{\sin^{2} (x)}{1 + \cos (x)} \end{aligned}$
::sin2 ( x) 1+cos ( x)

$\begin{aligned} (1 + \tan^{2} (x)) (\sec^{2} (x)) \end{aligned}$
:1+tan2(x)(sec2(x)))

$\begin{aligned} \sin (x) (\tan (x) + \cot (x)) \end{aligned}$
:x(tan)(x)+Cot(x))

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

$\begin{aligned} \frac{\sin (x)}{\cot^{2} (x)} - \frac{\sin (x)}{\cos^{2} (x)} \end{aligned}$
:x)cot2(x)-sin(x)cos2(x)

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

$\begin{aligned} \frac{\sin^{2} (x) - \sin^{4} (x)}{\cos^{2} (x)} \end{aligned}$
:x) - 辛4\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\

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

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

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

$\begin{aligned} (\sec^{2} (x) + \csc^{2} (x)) - (\tan^{2} (x) + \cot^{2} (x)) \end{aligned}$
:sec2(x)+csc2(x))-(tan2(x)+cot2(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

Trigonometric Equations ::三角数等量

Simplifying Expressions ::简化表达式

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)

Trigonometric Equations
::三角数等量

Simplifying Expressions
::简化表达式

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)