Course: 4.4 三角测量 | The Way To Learn

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三角形

Trigonometry is the study of triangles. If you know the angles of a triangle and one side length, you can use the properties of similar triangles and proportions to completely solve for the missing sides.
::三角形是三角形的研究。如果您知道三角形的角度和侧边长度,您可以使用相似三角形的属性和比例来完全解析缺失的边。

Imagine trying to measure the height of a flag pole. It would be very difficult to measure vertically because it could be several stories tall. Instead walk 10 feet away and notice that the flag pole makes a 65 degree angle with your feet. Using this information, what is the height of the flag pole?
::想象一下如何测量旗杆的高度。垂直测量将非常困难, 因为它可能具有几层高。相反, 它会走10英尺远, 并且注意到旗杆用你的脚来测出65度角。使用这个信息, 旗杆的高度是多少?

Trigonometric Functions
::三角函数

The six trigonometric functions are sine, cosine, tangent, cotangent , secant and cosecant . Opp stands for the side opposite of the angle $\begin{aligned} θ \end{aligned}$ , hyp stands for hypotenuse and adj stands for side adjacent to the angle $\begin{aligned} θ \end{aligned}$ .
::6个三角函数是正弦、正弦、正弦、正切、正切、断裂和正弦。 Opp 代表角的对面, Hyp 代表低压, adj 代表角的对面。 Opp 代表角的对面, hyp 代表低压, adj 代表角的对面。

$\begin{aligned} \sin θ = \frac{o p p}{h y p} \end{aligned}$
::

$\begin{aligned} \cos θ = \frac{a d j}{h y p} \end{aligned}$
::

$\begin{aligned} \tan θ = \frac{o p p}{a d j} \end{aligned}$
::

$\begin{aligned} \cot θ = \frac{a d j}{o p p} \end{aligned}$
::cotadjopp 连接

$\begin{aligned} \sec θ = \frac{h y p}{a d j} \end{aligned}$
::

$\begin{aligned} \csc θ = \frac{h y p}{o p p} \end{aligned}$
::csc

The reason why these trigonometric functions exist is because two triangles with the same interior angles will have side lengths that are always proportional. Trigonometric functions are used by identifying two known pieces of information on a triangle and one unknown, setting up and solving for the unknown. Calculators are important because the operations of , and are already programmed in. The other three (cot, sec and csc) are not usually in calculators because there is a reciprocal relationship between them and tan, cos and sec.
::这些三角函数之所以存在,是因为两个具有相同内角的三角形的侧长总是成比例的。三角函数用于识别一个三角形和一个未知的三角形上已知的两块信息,为未知的三角形设置和解决。计算器很重要, 因为它的操作, 并且已经被编程在其中。其它三个三角形( cot、 sec 和 csc) 通常不是在计算器中, 因为它们与棕色、 cos 和 sec 之间存在对等关系。

$\begin{aligned} \sin θ = \frac{o p p}{h y p} = \frac{1}{\csc θ} \end{aligned}$
::

$\begin{aligned} \cos θ = \frac{a d j}{h y p} = \frac{1}{\sec θ} \end{aligned}$
::

$\begin{aligned} \tan θ = \frac{o p p}{a d j} = \frac{1}{\cot θ} \end{aligned}$
::@tanoppadj=1cot

Keep in mind that your calculator can be in degree mode or radian mode. Be sure you can toggle back and forth so that you are always in the appropriate units for each problem.
::记住您的计算器可以是度模式或弧度模式。请确定您可以前后切换, 以便您总是在适合每个问题的单位中。

Note that the images throughout this concept are not drawn to scale. If you were given the following triangle and asked to solve for side $\begin{aligned} b \end{aligned}$ , you would use sine to find $\begin{aligned} b \end{aligned}$ .
::请注意, 整个概念中的图像不会被绘制为缩放大小。如果您被给下一个三角形, 并被要求解析 b 侧, 您将使用正弦查找 b 。

$\begin{aligned} \sin (\frac{2 π}{7}) & = \frac{b}{14} \\ b & = 14 \cdot \sin (\frac{2 π}{7}) \approx 10.9 i n \end{aligned}$

27) =b14b=14sin( 27) 10.9 英寸

Examples
::实例

Example 1
::例1

Earlier, you were asked about the height of a flagpole that you are 10 feet away from. You notice that the flag pole makes a $\begin{aligned} 65^{\circ} \end{aligned}$ angle with your feet.
::早些时候,有人问起你身处距离你10英尺远处的旗杆的高度。你注意到旗杆用你的脚用65角度。

If you are 10 feet from the base of a flagpole and assume that the flagpole makes a $\begin{aligned} 90^{\circ} \end{aligned}$ angle with the ground, you can use the following triangle to model the situation.
::如果您离旗杆底部10英尺,并假设旗杆与地面的角为90°Q,您可以用以下三角形来模拟情况。

\begin{aligned} \tan 65^{\circ} & = \frac{x}{10} \\ x & = 10 \tan 65^{\circ} \approx 21.4 f t \end{aligned}

::{\fn黑体\fs22\bord1\shad0\3aHBE\4aH00\fscx67\fscy66\2cHFFFFFF\3cH808080}为什么? {\fn黑体\fs22\bord1\shad0\3aHBE\4aH00\fscx67\fscy66\2cHFFFFFF\3cH808080}为什么?

Example 2
::例2

Solve for angle $\begin{aligned} A \end{aligned}$ .
::解决A角度的问题。

This problem can be solved using sin, cos or tan because the opposite, adjacent and hypotenuse lengths are all given.
::这个问题可以用罪、罪或晒黑来解决,因为反面、相邻和低温长度都给定了。

The argument, or input, of a sin function is always an angle. The arcsin, or $\begin{aligned} \sin^{- 1} θ \end{aligned}$ , function on the calculator has an argument that is a ratio of the triangle sides.
::罪函数的参数或输入始终是一个角度。计算器上的 Arcsin 函数或 sin-1 函数有一个三角边之比的参数。

\begin{aligned} \sin A & = \frac{5}{13} \\ \sin^{- 1} (\sin A) & = \sin^{- 1} (\frac{5}{13}) \\ A & = \sin^{- 1} (\frac{5}{13}) \approx 0.39 r a d i a n \approx {22.6}^{\circ} \end{aligned}

::A=513sin-1(sina)=sin-1(513)A=sin-1(513)A=sin-1(513) 0.39弧度22.6

Example 3
::例3

Given a right triangle with $\begin{aligned} a = 12 i n \end{aligned}$ , $\begin{aligned} m ∠ B = 20^{\circ}, \end{aligned}$ and $\begin{aligned} m ∠ C = 90^{\circ} \end{aligned}$ , find the length of the hypotenuse.
::根据右三角形的 a=12, mB=20和 mC=90, 找到下限长度。

It is helpful to draw a diagram to represent the data given in a question.
::绘制一个图表来代表一个问题中提供的数据很有帮助。

\begin{aligned} \cos 20^{\circ} & = \frac{12}{c} \\ c & = \frac{12}{\cos 20^{\circ}} \approx 12.77 i n \end{aligned}

::12ccc=12cos 20\\\12.77英寸

Example 4
::例4

Given $\begin{aligned} △ A B C \end{aligned}$ where $\begin{aligned} B \end{aligned}$ is a right angle, $\begin{aligned} m ∠ C = 18^{\circ} \end{aligned}$ , and $\begin{aligned} c = 12 \end{aligned}$ . What is $\begin{aligned} a \end{aligned}$ ?
::考虑到ABC B 是右角, mC=18 和 c=12。什么是?

Drawing out this triangle, it looks like:
::画出这个三角形, 它看起来像:

\begin{aligned} \tan 18^{\circ} & = \frac{12}{a} \\ a & = \frac{12}{\tan 18^{\circ}} \approx 36.9 \end{aligned}

::{\fn黑体\fs22\bord1\shad0\3aHBE\4aH00\fscx67\fscy66\2cHFFFFFF\3cH808080}为什么? {\fn黑体\fs22\bord1\shad0\3aHBE\4aH00\fscx67\fscy66\2cHFFFFFF\3cH808080}... {\fn黑体\fs22\bord1\shad0\3aHBE\4aH00\fscx67\fscy66\2cHFFFFFF\3cH808080}...

Example 5
::例5

Given $\begin{aligned} △ M N O \end{aligned}$ where $\begin{aligned} O \end{aligned}$ is a right angle, $\begin{aligned} m = 12 \end{aligned}$ , and $\begin{aligned} n = 14 \end{aligned}$ . What is the measure of angle $\begin{aligned} M \end{aligned}$ ?
Drawing out the triangle, it looks like:
::鉴于 MNO 右角 O 的位置, m= 12 和 n= 14。角 M 的量度是多少? 绘制三角形, 它看起来像 :

\begin{aligned} \tan M & = \frac{12}{14} \\ M & = \tan^{- 1} (\frac{12}{14}) \approx 0.7 r a d i a n \approx {40.6}^{\circ} \end{aligned}

::tan m=1214M=tan-1(1214) 0.7 弧度 40.6

Summary

The six trigonometric functions are sine, cosine, tangent, cotangent, secant and cosecant.

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

::六个三角函数是正弦、正弦、正弦、正切、正切、断裂和正弦。 sinopphyp cschypoppcosadjhyp se hypadjtanoppadj cotadjopp

Review
::回顾

For 1-15, information about the sides and/or angles of right triangle $\begin{aligned} A B C \end{aligned}$ is given. Completely solve the triangle (find all missing sides and angles) to 1 decimal place.
::1-15 给出关于右三角 ABC 的边和/或角的信息。完全解析三角( 找到所有缺失的边和角) 到小数点后 1 位。

Problem Number ::问题编号	$\begin{aligned} A \end{aligned}$ ::A A A	$\begin{aligned} B \end{aligned}$ ::BB ,B	$\begin{aligned} C \end{aligned}$	$\begin{aligned} a \end{aligned}$ ::a a/	$\begin{aligned} b \end{aligned}$ ::b b b	$\begin{aligned} c \end{aligned}$ ::c , c , c , c
1.	$\begin{aligned} 90^{\circ} \end{aligned}$				4	7
2.	$\begin{aligned} 90^{\circ} \end{aligned}$		$\begin{aligned} 37^{\circ} \end{aligned}$	18
3.		$\begin{aligned} 90^{\circ} \end{aligned}$	$\begin{aligned} 15^{\circ} \end{aligned}$		32
4.			$\begin{aligned} 90^{\circ} \end{aligned}$	6		11
5.	$\begin{aligned} 90^{\circ} \end{aligned}$	$\begin{aligned} 12^{\circ} \end{aligned}$		19
6.		$\begin{aligned} 90^{\circ} \end{aligned}$			17	10
7.	$\begin{aligned} 90^{\circ} \end{aligned}$	$\begin{aligned} 10^{\circ} \end{aligned}$			2
8.	$\begin{aligned} 4^{\circ} \end{aligned}$	$\begin{aligned} 90^{\circ} \end{aligned}$		0.3
9.	$\begin{aligned} \frac{π}{2} \end{aligned}$ radian ::2 弧度		1 radian ::1 弧度			15
10.		$\begin{aligned} \frac{π}{2} \end{aligned}$ radian ::2 弧度		12	15
11.			$\begin{aligned} \frac{π}{2} \end{aligned}$ radian ::2 弧度		9	14
12.	$\begin{aligned} \frac{π}{4} \end{aligned}$ radian ::4 弧度	$\begin{aligned} \frac{π}{4} \end{aligned}$ radian ::4 弧度			5
13.	$\begin{aligned} \frac{π}{2} \end{aligned}$ radian ::2 弧度			26	13
14.		$\begin{aligned} \frac{π}{2} \end{aligned}$ radian ::2 弧度			19	16
15.			$\begin{aligned} \frac{π}{2} \end{aligned}$ radian ::2 弧度	10		$\begin{aligned} 10 \sqrt{2} \end{aligned}$

Review (Answers)
::回顾(答复)

To see the answer key for this book, go to the and click on the Answer Key under the ' ' option.

Section outline

General

三角形

Trigonometric Functions ::三角函数

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)