Course: 4.8 基本三角测量应用

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基本三角测量应用

Deciding when to use SOH, CAH, TOA, or the is not always obvious. Sometimes more than one approach will work and sometimes correct computations can still lead to incorrect results. This is because correct interpretation is still essential.
::选择何时使用 SOH、CAH、TOA 或并非总是显而易见的。有时,不止一种方法会起作用,有时纠正计算仍然可能导致错误的结果。这是因为正确的解释仍然至关重要。

If you use both the Law of Cosines and the Law of Sines on a triangle with sides 4, 7, 10 you end up with conflicting answers. Why?
::如果你在4、7、10两边的三角形上既使用科辛斯定律也使用辛那斯定律,你最终会得到相互矛盾的答案。为什么?

Trigonometry Applications
::三角测量应用

When applying trigonometry, it is important to have a clear toolbox of mathematical techniques to use. Some of the techniques may be review like the fact that all three angles in a triangle sum to be $\begin{aligned} 180^{\circ} \end{aligned}$ , other techniques may be newer like the Law of Cosines. Take a look at all the tools you have in your toolbox to solve applications with trigonometry.
::在应用三角测量时,必须有一个清晰的数学技术工具箱来使用。有些技术可能是要审查的,比如三角形和三角形中的所有三个角度都等于180,其他技术可能比科辛斯定律更新。看看工具箱中用三角法解决应用程序的所有工具。

Toolbox
::工具箱

The three angles in a triangle sum to be $\begin{aligned} 180^{\circ} \end{aligned}$ .
::三角形和三角形中的三个角度为 180 。

There are $\begin{aligned} 360^{\circ} \end{aligned}$ in a circle and this can help us interpret negative angles as positive angles.
::圆圈里有360,这可以帮助我们将负面角度解释为积极角度。

The states that for legs $\begin{aligned} a, b \end{aligned}$ and hypotenuse $\begin{aligned} c \end{aligned}$ in a right triangle, $\begin{aligned} a^{2} + b^{2} = c^{2} \end{aligned}$ .
::规定右三角形的腿 a、b 和下角 c, a2+b2=c2。

The Triangle Inequality Theorem states that for any triangle, the sum of any two of the sides must be greater than the third side.
::三角不平等理论指出,对于任何三角,任何两边的总和必须大于第三边。

The Law of Cosines: $\begin{aligned} c^{2} = a^{2} + b^{2} - 2 a b \cos C \end{aligned}$
::C2=a2+b2-2-2abcosC

The Law of Sines: $\begin{aligned} \frac{a}{\sin A} = \frac{b}{\sin B} \end{aligned}$ or $\begin{aligned} \frac{\sin A}{a} = \frac{\sin B}{b} \end{aligned}$ (Be careful for the ambiguous case)
::Sines法:asinA=bsinB或sinAa=sinBb(注意模棱两可的案件)

SOH CAH TOA is a mnemonic device to help you remember the three original trig functions:
::SOH CHAH TOA 是一个记忆设备, 帮助您记住三个原始的三角函数 :

\begin{aligned} \sin θ = \frac{o p p}{h y p} \cos θ = \frac{a d j}{h y p} \tan θ = \frac{o p p}{a d j} \end{aligned}

30-60-90 right triangles have side ratios $\begin{aligned} x, x \sqrt{3}, 2 x \end{aligned}$
::30-60-90 右三角形有侧翼比率 x,x3,2x

45-45-90 right triangles have side ratios $\begin{aligned} x, x, x \sqrt{2} \end{aligned}$
::45 - 45 - 90 右三角形有侧比xxxxxx2

Pythagorean number triples are exceedingly common and should always be recognized in right triangle problems. Examples of triples are 3, 4, 5 and 5, 12, 13.
::Pydagorena数字的三倍非常常见,在正确的三角问题中应始终得到承认,例如,3、4、5和5、12、13是3、4、5和3。

A few definitions will also be necessary to solve applications. Angle of elevation is the angle at which you view and object above the horizon. Angle of depression is the angle at which you view and object below the horizon. Bearing is how direction is measured at sea. North is 0º , East is 9 0 º , South is 18 0 º and West is 27 0 º .
::解决应用问题也需要几个定义。高度角是您在地平线上查看和天体的角。沉积角是您在地平线下查看和天体的角。沉积角是您在地平线下查看和天体的角。沉积角是如何在海上测量方向。北纬0o, 东经90o, 南纬180o, 西经270o。

Examples
::实例

Example 1
::例1

Earlier, you were asked why you may end up with conflicting answers if you use both the Law of Sines and the Law of Cosines. Sometimes when using the Law of Sines you can get answers that do not match the Law of Cosines. Both answers can be correct computationally, but the Law of Sines may involve interpretation when the triangle is obtuse. The Law of Cosines does not require this interpretation.
::早些时候,有人问到,如果你同时使用《辛那斯定律》和《科辛定律》,为什么你最后会遇到相互矛盾的答案。有时,当使用《辛那斯定律》和《科辛定律》时,你可以得到与《科辛定律》不相符的答案。这两个答案可以计算正确,但当三角形模糊时,《辛那定律》可能涉及解释。《科辛定律》不需要这种解释。

First, use Law of Cosines to find $\begin{aligned} ∠ B \end{aligned}$ :
::首先,使用科辛斯定律来寻找#B:

\begin{aligned} 12^{2} & = 3^{2} + 14^{2} - 2 \cdot 3 \cdot 14 \cdot \cos B \\ ∠ B & = \cos^{- 1} (\frac{12^{2} - 3^{2} - 14^{2}}{- 2 \cdot 3 \cdot 14}) \approx {43.43}^{\circ} \end{aligned}

::122=32+142-231414BBB=Cos-11(122-32-142-2314)434343

Then, use Law of Sines to find $\begin{aligned} ∠ C \end{aligned}$ . Use the unrounded value for $\begin{aligned} B \end{aligned}$ even though a rounded value is shown.
::然后,使用Sines Law 查找 QC。即使显示四舍五入值, 也使用 B 的未四舍五入值。

\begin{aligned} \frac{\sin {43.43}^{\circ}}{12} & = \frac{\sin C}{14} \\ \frac{14 \sin {43.43}^{\circ}}{12} & = \sin C \\ ∠ C & = \sin^{- 1} (\frac{14 \sin {43.43}^{\circ}}{12}) \approx {53.3}^{\circ} \end{aligned}

14sin43.4312) (53.33)

Use the Law of Cosines to double check $\begin{aligned} ∠ C \end{aligned}$ .
::使用科辛斯定律双倍检查"C"

\begin{aligned} 14^{2} & = 3^{2} + 12^{2} - 2 \cdot 3 \cdot 12 \cdot \cos C \\ C & = \cos^{- 1} (\frac{14^{2} - 3^{2} - 12^{2}}{- 2 \cdot 3 \cdot 12}) \approx {126.7}^{\circ} \end{aligned}

::142=32+122-2312cCC=cos-1(142-32-122-2312)126.7

Notice that the last two answers do not match, but they are supplementary. This is because this triangle is obtuse and the $\begin{aligned} \sin^{- 1} (\frac{o p p}{h y p}) \end{aligned}$ function is restricted to only producing acute angles.
::请注意后两个答案不匹配, 但它们是补充的。这是因为这个三角形是模糊的, 而 sin - 1\\\\\\\ (opphyp) 函数仅限于生成急性角度。

Example 2
::例2

A surveying crew is given the job of verifying the height of a cliff. From point $\begin{aligned} A \end{aligned}$ , they measure an angle of elevation to the top of the cliff to be $\begin{aligned} α = {21.567}^{\circ} \end{aligned}$ . They move 507 meters closer to the cliff and find that the angle to the top of the cliff is now $\begin{aligned} β = {25.683}^{\circ} \end{aligned}$ . How tall is the cliff?
::测量组被授予核查悬崖高度的任务。从A点, 他们测量悬崖顶部的高度角为 21.567。他们移动507米接近悬崖, 发现悬崖顶部的角现在是25.683。悬崖有多高?

Note that $\begin{aligned} α \end{aligned}$ is just the Greek letter alpha and in this case it stands for the number $\begin{aligned} {21.567}^{\circ} \end{aligned}$ . $\begin{aligned} β \end{aligned}$ is the Greek letter beta and it stands for the number $\begin{aligned} {25.683}^{\circ} \end{aligned}$ .
::请注意,α只是希腊字母阿尔法,在此情况下,它代表数字21.567。β是希腊字母贝特,它代表数字25.683。

First, sketch the image and label what you know.
::首先,绘制图像,标上你所知道的标签。

Next, because the height is measured at a right angle with the ground, set up two equations. Remember that $\begin{aligned} α \end{aligned}$ and $\begin{aligned} β \end{aligned}$ are just numbers, not variables.
::下一步, 因为高度与地面以右角测量, 设置两个方程。记住 α 和 β 仅仅是数字, 而不是变量。

\begin{aligned} \tan α & = \frac{h}{507 + x} \\ \tan β & = \frac{h}{x} \end{aligned}

::tanh507+xtanhx

Both of these equations can be solved for $\begin{aligned} h \end{aligned}$ and then set equal to each other to find $\begin{aligned} x \end{aligned}$ .
::这两种方程式都可以为h解答,然后对等方程式来找到x。

\begin{aligned} h = \tan α (507 + x) & = x \tan β \\ 507 \tan α + x \tan α & = x \tan β \\ 507 \tan α & = x \tan β - x \tan α \\ 507 \tan α & = x (\tan β - \tan α) \\ x = \frac{507 \tan α}{\tan β - \tan α} & = \frac{507 \tan {21.567}^{\circ}}{\tan {25.683}^{\circ} - \tan {21.567}^{\circ}} \approx 2340 m e t e r s \end{aligned}

507+xx) =xtan 507tan 507tan 507tan 507tan (tantan )x= 507tan 507tan 51.567 25.683 tan 21.567 2340米

Since the problem asked for the height, you need to substitute $\begin{aligned} x \end{aligned}$ back and solve for $\begin{aligned} h \end{aligned}$ .
::由于问题要求的高度, 您需要替换 x 并解决 h 。

$\begin{aligned} h = x \tan β = 2340 \tan {25.683}^{\circ} \approx 1125.31 m e t e r s \end{aligned}$
::=xtan2340tan25.6831125.31米

Example 3
::例3

Given a triangle with SSS or SAS you know to use the Law of Cosines. In triangles where there are corresponding angles and sides like AAS or SSA it makes sense to use the Law of Sines. What about ASA?
::根据SSS或SAS的三角关系,您可以使用Cosines定律。在有AAS或SSA等相应角度和侧面的三角关系中,使用Sines定律是有道理的。 ASA呢?

Given $\begin{aligned} Δ A B C \end{aligned}$ with $\begin{aligned} A = \frac{π}{4} r a d i a n s, C = \frac{π}{6} r a d i a n s \end{aligned}$ and $\begin{aligned} b = 10 i n \end{aligned}$ what is $\begin{aligned} a \end{aligned}$ ?
::鉴于 ABC 具有 A4 弧度, C6 弧度和 b= 10 值是什么?

First,draw a picture.
::首先,画一张照片。

The sum of the angles in a triangle is $\begin{aligned} 180^{\circ} \end{aligned}$ . Since this problem is in radians you either need to convert this rule to radians, or convert the picture to degrees.
::三角形中角度的总和是 180 。由于这个问题是在弧度中, 您需要将此规则转换为弧度, 或者将图片转换为度。

\begin{aligned} A & = \frac{π}{4} \cdot \frac{180^{\circ}}{π} = 45^{\circ} \\ C & = \frac{π}{6} \cdot \frac{180^{\circ}}{π} = 30^{\circ} \end{aligned}

::418045C683030

The missing angle must be $\begin{aligned} ∠ B = 105^{\circ} \end{aligned}$ . Now you can use the Law of Sines to solve for $\begin{aligned} a \end{aligned}$ .
::缺失的角度必须是 {B=105

\begin{aligned} \frac{\sin 105^{\circ}}{10} & = \frac{\sin 45^{\circ}}{a} \\ a & = \frac{10 \sin 45^{\circ}}{\sin 105^{\circ}} \approx 7.32 i n \end{aligned}

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

The angle of depression of a boat in the distance from the top of a lighthouse is $\begin{aligned} \frac{π}{10} \end{aligned}$ . The lighthouse is 200 feet tall. Find the distance from the base of the lighthouse to the boat.
::在离灯塔顶部的距离内,船的压抑角度是10°C,灯塔高200英尺,从灯塔底部到船的距离找到。

When you draw a picture, you see that the given angle $\begin{aligned} \frac{π}{10} \end{aligned}$ is not directly inside the triangle between the lighthouse, the boat and the base of the lighthouse. It is complementary to the angle you need.
::当您绘制图片时, 您可以看到给定角度 10 不是直接在灯塔、船和灯塔底部之间的三角形内。它与您需要的角度是互补的。

\begin{aligned} \frac{π}{10} + θ & = \frac{π}{2} \\ θ & = \frac{2 π}{5} \end{aligned}

Now that you have the angle, use tangent to solve for $\begin{aligned} x \end{aligned}$ .
::现在您有了角度, 使用切线解析 x 。

\begin{aligned} \tan \frac{2 π}{5} & = \frac{x}{200} \\ x & = 200 \tan \frac{2 π}{5} \approx 615.5 f t \end{aligned}

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

Alternatively, you could have noticed that $\begin{aligned} \frac{π}{10} \end{aligned}$ is alternate interior angles with the angle of elevation of the lighthouse from the boat’s perspective. This would yield the same distance for $\begin{aligned} x \end{aligned}$ .
::换句话说,你本可以注意到 10 是另外的内部角度, 从船的角度看是灯塔的高度角度。这将给 x 带来相同的距离。

Example 5
::例5

From the third story of a building (50 feet) David observes a car moving towards the building driving on the streets below. If the angle of depression of the car changes from $\begin{aligned} 21^{\circ} \end{aligned}$ to $\begin{aligned} 45^{\circ} \end{aligned}$ while he watches, how far did the car travel?
::大卫看到一辆汽车在下面的街道上驶向大楼。如果汽车的压抑角度在他看着的时候从21°Q变为45°C,车程有多远?

Draw a very careful picture:
::绘制一个非常仔细的图片:

In the upper right corner of the picture there are four important angles that are marked with angles. The measures of these angles from the outside in are $\begin{aligned} 90^{\circ}, 45^{\circ}, 21^{\circ}, 69^{\circ} \end{aligned}$ . There is a 45-45-90 right triangle on the right, so the base must also be 50. Therefore you can set up and solve an equation for $\begin{aligned} x \end{aligned}$ .
::在图片的右上角有四个以角度标记的重要角度。这些角度从外部的度量为 90 , 45 , 21 , 69 。右边有一个45- 45- 90 的右三角形, 所以基数也必须是 50 。因此您可以设置并解析 x 的方程式。

\begin{aligned} \tan 69^{\circ} & = \frac{x + 50}{50} \\ x & = 50 \tan 69^{\circ} - 50 \approx 80.25 f t \end{aligned}

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

Summary

Use the collection of all the mathematical techniques we’ve learned to help solve applications with trigonometry.
::使用我们学到的所有数学技术, 帮助解决三角测量的应用。

Review
::回顾

The angle of depression of a boat in the distance from the top of a lighthouse is $\begin{aligned} \frac{π}{6} \end{aligned}$ . The lighthouse is 150 feet tall. You want to find the distance from the base of the lighthouse to the boat.
::与灯塔顶部相距的一艘船的压抑角度是 6 。灯塔有150英尺高。您想要找到灯塔底部至船的距离。

1. Draw a picture of this situation.
::1. 描绘一下这种情况。

2. What methods or techniques will you use?
::2. 你会使用何种方法或技术?

3. Solve the problem.
::3. 解决问题。

From the third story of a building (60 feet) Jeff observes a car moving towards the building driving on the streets below. The angle of depression of the car changes from $\begin{aligned} 34^{\circ} \end{aligned}$ to $\begin{aligned} 62^{\circ} \end{aligned}$ while he watches. You want to know how far the car traveled.
::Jeff看到一辆汽车驶向楼层, 在下面的街道上行驶。汽车的压抑角度在他看着的时候从34英寸变为62英尺。您想知道车行驶有多远。

4. Draw a picture of this situation.
::4. 描绘一下这种情况。

5. What methods or techniques will you use?
::5. 你会使用何种方法或技术?

6. Solve the problem.
::6. 解决问题。

A boat travels 6 miles NW and then 2 miles SW. You want to know how far away the boat is from its starting point.
::一艘船行驶6英里长,然后2英里长。你想知道船离起点有多远。

7. Draw a picture of this situation.
::7. 描述一下这种情况。

8. What methods or techniques will you use?
::8. 你会使用何种方法或技术?

9. Solve the problem.
::9. 解决问题。

You want to figure out the height of a building. From point $\begin{aligned} A \end{aligned}$ , you measure an angle of elevation to the top of the building to be $\begin{aligned} α = 10^{\circ} \end{aligned}$ . You move 50 feet closer to the building to point $\begin{aligned} B \end{aligned}$ and find that the angle to the top of the building is now $\begin{aligned} β = 60^{\circ} \end{aligned}$ .
::您想要弄清楚建筑物的高度。从 A点, 您测量建筑物顶部的高度角为 10 。您将距离建筑物更近50英尺的角移动到 B点, 并发现大楼顶部的角现在是 60 。

10. Draw a picture of this situation.
::10. 描绘一下这种情况。

11. What methods or techniques will you use?
::11. 你会使用何种方法或技术?

12. Solve the problem.
::12. 解决问题。

13. Given $\begin{aligned} Δ A B C \end{aligned}$ with $\begin{aligned} A = 40^{\circ}, C = 65^{\circ} \end{aligned}$ and $\begin{aligned} b = 8 i n \end{aligned}$ , what is $\begin{aligned} a \end{aligned}$ ?
::13. 鉴于A=40,C=65,B=8,A=40,C=65,B=8,什么是A?

14. Given $\begin{aligned} Δ A B C \end{aligned}$ with $\begin{aligned} A = \frac{π}{3} r a d i a n s, C = \frac{π}{8} r a d i a n s \end{aligned}$ and $\begin{aligned} b = 12 i n \end{aligned}$ what is $\begin{aligned} a \end{aligned}$ ?
::14. 以AQQ3弧度、CQQQ8弧度和B=12的A=3弧度计算,

15. Given $\begin{aligned} Δ A B C \end{aligned}$ with $\begin{aligned} A = \frac{π}{6} r a d i a n s, C = \frac{π}{4} r a d i a n s \end{aligned}$ and $\begin{aligned} b = 20 i n \end{aligned}$ what is $\begin{aligned} a \end{aligned}$ ?
::15. 以A++6弧度、C++4弧度和B=20弧度的 ABC 表示什么?

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

基本三角测量应用

Trigonometry Applications ::三角测量应用

Toolbox ::工具箱

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)