Course: 7.15 了解相切物 | The Way To Learn

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理解对象
The members of the Volunteer Fire Department have been busy building a ramp at the entrance to the local arena to accommodate a young man who is confined to a wheelchair. Joey, who has recently moved to the small community, is about to try out the new ramp.
::志愿消防局成员一直忙于在当地竞技场入口处修建坡道,以容纳一名被困在轮椅上的年轻人。最近搬到小社区的乔伊即将尝试新的坡道。

If the ramp covers a horizontal distance of 8.4 meters along the ground and rises to a height of 1.2 meters, how can Joey figure out what angle the ramp makes with the ground?
::如果坡道覆盖地上8.4米的横向距离, 并升至1.2米高, Joey怎么会知道坡道与地面之间的角?

In this concept, you will learn to understand the trigonometric ratio Tangent.
::在这个概念中,你会学会理解三角比Tangent。

Tangent Ratio
::相近比率

Trigonometry is a branch of mathematics used to determine the lengths of sides and the measure of angles with great accuracy. A ratio of the lengths of the sides of a right triangle is called a trigonometric ratio . The ratio of the length of the side opposite the reference angle to the length of the side alongside the reference angle is the Tangent ratio . An acute angle of a right triangle is formed by the hypotenuse and one of the legs of the triangle. This leg is called the adjacent side of the reference angle. The Tangent ratio is the ratio of the opposite side to the adjacent side.
::三角形是用于确定侧面长度和角度测量的数学分支。右三角形侧面长度的比例称为三角比。参考角对面的侧面长度与侧面长度之比为Tangent比。右三角形的急性角由三角形的下压和腿部组成。此腿称为参考角的相邻侧。 Tangent 比例为对面与相邻侧的比例。

In each given diagram, label each side of the right triangle as the:
::在每个给定图表中,右三角形的每侧标记为:

Hypotenuse (H); Side Opposite reference angle (O); Side Adjacent to reference angle (A)
::日光(H);侧对面参考角(O);侧对近参考角(A)

The Tangent ratio for the first triangle can be written in words as:
::第一个三角形的切线比可以以下列文字写成:

tangent of $\begin{align*}\angle A = \frac{\text{side opposite } \angle A}{\text{side adjacent } \angle A}\end{align*}$ Or in a shortened form as $\begin{align*}\tan A = \frac{\text{opposite}}{\text{adjacent}}\end{align*}$
::A=对面 A 旁边 A 或缩写为 tanA = 对面近地点

The tangent ratio for the first triangle can be written in symbols as:
::第一个三角形的正切比可以用以下符号表示:

$\begin{aligned} \tan A = \frac{B C}{A C} \end{aligned}$
$\begin{align*}\tan A = \frac{BC}{AC}\end{align*}$
::tanA=BCAC(BCAC)

The tangent ratio for the second triangle can be written in words as:
::第二个三角形的正切比可以以下列文字写入:

tangent of $\begin{align*}\angle F = \frac{\text{side opposite } \angle F}{\text{side adjacent } \angle F}\end{align*}$ Or in a shortened form as $\begin{align*}\tan F = \frac{\text{opposite}}{\text{adjacent}}\end{align*}$
::F=对面 F=对面 F 或缩写为 tanF = 对面近地点

The Tangent ratio for the second triangle can be written in symbols as:
::第二个三角形的切线比可以以下列符号写成:

$\begin{aligned} \tan F = \frac{D E}{E F} \end{aligned}$
$\begin{align*}\tan F = \frac{DE}{EF}\end{align*}$
::F=DEEF

Let’s determine the value for the tangent ratio of each of the acute angles using the following right triangle. Express the tangent ratio in words and symbols. Then, using the values on the corresponding sides, replace the symbols with the numbers and express the ratio first as a fraction and then as a decimal rounded to the nearest ten thousandth (four places after the decimal).
::让我们使用以下右三角确定每个急角的正切比值。用文字和符号表示正切比。然后,使用对应边上的数值,用数字替换符号,先以小数表示比率,然后以小数四舍五入到最接近的万分之一(小数后四位数)。

$\begin{align*}\triangle ABC\end{align*}$ with $\begin{align*}\angle A\end{align*}$ as the reference angle.
::以 A 为参考角度的 ABC 。

First, name the sides of the triangle.
::首先,给三角形两边取个名字

The hypotenuse is across from the right angle. The opposite side is across from $\begin{align*}\angle A\end{align*}$ . The side alongside $\begin{align*}\angle A\end{align*}$ is the adjacent. The sides are labeled with the letters H, O, A respectively.
::下方是右角的对面。反面是 A 的对面。 A 旁边是 A 的侧面是相邻的。侧面分别标有字母H、 O、 A 的标签。

Next, write the tangent ratio for $\begin{align*}\angle A\end{align*}$ in all of the required forms.
::接下来,请在所有要求的表格中填入 A 的正切比例。

Words:
::单词 :

tangent of $\begin{align*}\angle A = \frac{\text{opposite}}{\text{adjacent}}\end{align*}$
::A=对面近地点

Symbols:
::符号 :

$\begin{align*}\tan A = \frac{BC}{AC}\end{align*}$
::tanA=BCAC(BCAC)

Fraction:
::分数 :

$\begin{align*}\tan A = \frac{12}{5}\end{align*}$
::泰纳=125

Decimal:
::十进制 :

$\begin{align*}\tan A = 2.4\end{align*}$
::泰纳=2.4

$\begin{align*}\triangle ABC\end{align*}$ with $\begin{align*}\angle B\end{align*}$ as the reference angle.
::ABC 以 B 作为参考角度。

Notice that the locations of the opposite and adjacent sides have changed from where they were when $\begin{align*}\angle A\end{align*}$ was the reference angle.
::请注意,对面和相邻两侧的位置已经从ZA为参考角时的位置发生变化。

First, name the sides of the triangle.
::首先,给三角形两边取个名字

The hypotenuse is across from the right angle. The opposite side is across from $\begin{align*}\angle B\end{align*}$ . The side alongside $\begin{align*}\angle B\end{align*}$ is the adjacent. The sides are labeled with the letters H, O, A respectively.
::下方是右角的对面。反面是 {B} 的对面。 {B} 旁边是 {B} 的对面是相邻的。这两边分别标有字母H, O, A 的标签。

Next, write the tangent ratio for $\begin{align*}\angle B\end{align*}$ in all of the required forms.
::下一步,请在所有要求的表格中填入 B 的正切比。

Words:
::单词 :

tangent of $\begin{align*}\angle B = \frac{\text{opposite}}{\text{adjacent}}\end{align*}$
::B=对面相近点的正切值

Symbols:
::符号 :

$\begin{align*}\tan B = \frac{AC}{BC}\end{align*}$
::tanB = ACBC

Fraction:
::分数 :

$\begin{align*}\tan B = \frac{5}{12}\end{align*}$
::tan_B=512

Decimal:
::十进制 :

$\begin{align*}\tan B = 0.4167\end{align*}$
::tan_B=0.4167

If $\begin{align*}\tan B = 0.4167\end{align*}$ , then the measure of $\begin{align*}\angle B\end{align*}$ can be found using the inverse tangent function $\begin{align*}(\tan^{-1})\end{align*}$ on the TI- calculator.
::如果 TanB = 0. 4167, 则使用TI- 计算器上的反正切函数( tan-1) 来找到 B 的测量值。

First, follow the Key Press History below to calculate the measure of $\begin{align*}\angle B\end{align*}$ .
::首先,遵循下面的 Key Press Histics 来计算 {B} 的度量。

Next, look at the calculator screen to see the measure of $\begin{align*}\angle B\end{align*}$ .
::下一步,看看计算器屏幕以查看 B 的度量。

Then, write the measure of $\begin{align*}\angle B\end{align*}$ to the nearest tenth.
::然后,写下B的量子,到最近的十分之一。

$\begin{aligned} ∠ B = {22.6}^{\circ} \end{aligned}$
$\begin{align*}\angle B = 22.6^\circ\end{align*}$
::B=22.6

The answer is 22.6°.
::答案是22.6摄氏度

Examples
::实例

Example 1
::例1

Earlier, you were given a problem about Joey and the ramp. He is wondering what angle the ramp makes with the ground. How can Joey figure out the measure of this angle?
::早些时候,你遇到了关于乔伊和斜坡的问题。他想知道斜坡与地面之间的角。Joey如何了解这个角度的大小?

Joey can use the Tangent ratio and round his answer to the nearest hundredth.
::Joey可以使用Tangent比率将他的回答绕到最近的第一百次

First, draw and label a right triangle to model the problem.
::首先,绘制并标出一个右三角形来模拟问题。

Next, name the sides of the triangle using the reference angle A.
::下一步,使用参考角度 A 命名三角形的边。

Next, write the tangent ratio in words.
::接下来,用文字写下正切比。

$\begin{aligned} \tan A = \frac{opposite}{adjacent} \end{aligned}$
$\begin{align*}\tan A = \frac{\text{opposite}}{\text{adjacent}}\end{align*}$
::tana=对面近地点

Next, write the ratio using symbols.
::接下来,用符号来写这个比例。

$\begin{aligned} \tan A = \frac{B C}{A C} \end{aligned}$
$\begin{align*}\tan A = \frac{BC}{AC}\end{align*}$
::tanA=BCAC(BCAC)

Next, express the ratio as a fraction using the values of the corresponding sides.
::接下来,用对应方的数值将比率表示成一个分数。

$\begin{align*}\tan A = \frac{1.2}{8.4}\end{align*}$
::A=1.28.4

Next, express the ratio as a decimal rounded to four places after the decimal.
::接下来,以小数点后四位位四位四位四位四位四位四位四位表示该比率。

$\begin{align*}\tan A = 0.1429\end{align*}$
::泰纳=0.1429

Next, use the inverse tangent function $\begin{align*}(\tan^{-1})\end{align*}$ on the TI calculator to find the measure of $\begin{align*}\angle A\end{align*}$ .
::其次,在 TI 计算器上使用反切函数( tan-1) 来查找 A 的量度。

$\begin{aligned} ∠ A = {8.13}^{\circ} \end{aligned}$
$\begin{align*}\angle A = 8.13^\circ\end{align*}$
::A=8.13

The answer is 8.13°.
::答案是8.13摄氏度

The ramp makes an angle of 8.13° with the ground.
::坡道与地面角为8.13度。

Example 2
::例2

For the following right triangle, use the tangent ratio to calculate the measure of $\begin{align*}\angle B\end{align*}$ , the reference angle, to the nearest tenth.
::对于以下的右三角形,使用正切比来计算 B 的度量,即参考角度,至最近的十分。

First, using the reference angle to name the sides of the triangle.
::首先,使用引用角度来命名三角形的侧面。

Next, write the tangent ratio in words.
::接下来,用文字写下正切比。

$\begin{aligned} \tan B = \frac{opposite}{adjacent} \end{aligned}$
$\begin{align*}\tan B = \frac{\text{opposite}}{\text{adjacent}}\end{align*}$
::tanB=对面近地点

Next, write the ratio using symbols.
::接下来,用符号来写这个比例。

$\begin{aligned} \tan B = \frac{A C}{B C} \end{aligned}$
$\begin{align*}\tan B = \frac{AC}{BC}\end{align*}$
::tanB = ACBC

Next, express the ratio as a fraction using the values of the corresponding sides.
::接下来,用对应方的数值将比率表示成一个分数。

$\begin{align*}\tan B = \frac{32}{21}\end{align*}$
::丹尼B=3221

Next, express the ratio as a decimal rounded to four places after the decimal.
::接下来,以小数点后四位位四位四位四位四位四位四位四位表示该比率。

$\begin{align*}\tan B = 1.5238\end{align*}$
::B=1.5238

Next, use the inverse tangent function $\begin{align*}(\tan^{-1})\end{align*}$ on the TI calculator to find the measure of $\begin{align*}\angle B\end{align*}$ .
::下一步,在 TI 计算器上使用反切函数( tan-1) 来查找 {B} 的量度。

Then, write the value of $\begin{align*}\angle B\end{align*}$ to the nearest tenth.
::然后,将 B 的值写到最近的十分之一。

$\begin{aligned} ∠ B = {56.7}^{\circ} \end{aligned}$
$\begin{align*}\angle B = 56.7^\circ\end{align*}$
::B=56.7

The answer is 56.7°.
::答案是56.7摄氏度

Example 3
::例3

If $\begin{align*}\tan C = 3.6285\end{align*}$ , then what is the measure of $\begin{align*}\angle C\end{align*}$ to the nearest tenth.
::如果 tanC = 3.6285,那么 C 到最近的十分之一的量度是多少。

First, use the TI-calculator to find the measure of the angle by using the inverse tangent function $\begin{align*}(\tan^{-1})\end{align*}$ located above the $\begin{align*}\boxed{\tan}\end{align*}$ button on the calculator.
::首先,使用 TI 计算器,通过使用位于计算器上棕色按钮上方的反正切函数(tan-1), 找到角的量值。

Next, to access this function press:
::下一步,访问此函数新闻 :

Next, enter the given decimal value inside the parenthesis where the cursor is flashing.
::下一步,在光标闪烁的括号中输入给定的小数值。

Next, press enter to see the measure of the angle on the screen.
::下一步,按下 Enter 键可以看到屏幕上角的度量。

Then, write the value of $\begin{align*}\angle C\end{align*}$ to the nearest tenth.
::然后,将 C 的值写到最近的十分之一。

$\begin{aligned} ∠ C = {74.6}^{\circ} \end{aligned}$
$\begin{align*}\angle C = 74.6^\circ\end{align*}$
::C=74.6

The answer is 74.6°.
::答案是74.6摄氏度

Example 4
::例4

For the following right triangle, use the tangent ratio to calculate the measure of $\begin{align*}\angle C\end{align*}$ to the nearest hundredth.
::对于以下的右三角形,使用正切比来计算 C 到最近的一百度的度量。

First, using the reference angle to name the sides of the triangle.
::首先,使用引用角度来命名三角形的侧面。

Next, write the tangent ratio in words.
::接下来,用文字写下正切比。

$\begin{aligned} \tan C = \frac{opposite}{adjacent} \end{aligned}$
$\begin{align*}\tan C = \frac{\text{opposite}}{\text{adjacent}}\end{align*}$
::tanc=对面近地点

Next, write the ratio using symbols.
::接下来,用符号来写这个比例。

$\begin{aligned} \tan C = \frac{A B}{B C} \end{aligned}$
$\begin{align*}\tan C = \frac{AB}{BC}\end{align*}$
::tanC =ABBC C=ABBC C=ABBC C=ABBC C=ABBC

Next, express the ratio as a fraction using the values of the corresponding sides.
::接下来,用对应方的数值将比率表示成一个分数。

$\begin{align*}\tan C = \frac{7}{12}\end{align*}$
::tanc=712

Next, express the ratio as a decimal rounded to four places after the decimal.
::接下来,以小数点后四位位四位四位四位四位四位四位四位表示该比率。

$\begin{align*}\tan C = 0.5833\end{align*}$
::tanc=0.5833

Next, use the inverse tangent function $\begin{align*}(\tan^{-1})\end{align*}$ on the TI calculator to find the measure of $\begin{align*}\angle C\end{align*}$ .
::下一步,在 TI 计算器上使用反切函数( tan-1) 来查找 C 的量度。

Then, write the value of $\begin{align*}\angle C\end{align*}$ to the nearest hundredth.
::然后,将 C 的值写到最近的第一百。

$\begin{aligned} ∠ C = {30.26}^{\circ} \end{aligned}$
$\begin{align*}\angle C = 30.26^\circ\end{align*}$
::C=30.26

The answer is 30.26°.
::答案是30.26摄氏度

Review
::回顾

Solve each problem. (Round your answer to the nearest hundredths place where needed)
::解决每一个问题。 (在需要的地方,把答案转到最近的一百个地方)

1. What is the tangent of $\begin{align*}\angle G?\end{align*}$
::1. QG的相切性是什么?

2. What is the tangent of $\begin{align*}\angle H\end{align*}$ ?
::2. 什么是ZH的相切点?

3. True or False. The sine and the tangent use the same ratios.
::3. 真实或假,正弦和正弦使用相同比率。

? $\begin{align*}\angle R\end{align*}$ 4. What is the tangent of
::"R4,什么是相切的"

5. What is the tangent of $\begin{align*}\angle S\end{align*}$ ?
::5. 什么是相切的?

6. What is the tangent of $\begin{align*}\angle A\end{align*}$ ?
::6. 什么是ZA的相切点?

7. What is the tangent of $\begin{align*}\angle B\end{align*}$ ?
::7. 什么是ZB的相切点?

8. What is the length of the missing side rounded to the nearest hundredth?
::8. 失踪方的长度是多少,四舍五入到最近的第一百?

Answer each of the following questions. You may round to the nearest hundredth.
::回答以下每一个问题。您可以圆到最近的第一百个问题。

9. If a tangent ratio is $\begin{align*}\frac{4}{5}\end{align*}$ what is the measure of the tangent?
::9. 如果偏差比率为45,那么误差的尺度是什么?

10. If a tangent ratio is $\begin{align*}\frac{14}{20}\end{align*}$ what is the measure of the tangent?
::10. 如果偏差比率为1420,那么误差的尺度是什么?

11. If a tangent ratio is $\begin{align*}\frac{6}{7}\end{align*}$ what is the measure of the tangent?
::11. 如果偏差比率为67,那么误差的尺度是什么?

12. If a tangent ratio is $\begin{align*}\frac{9}{2}\end{align*}$ what is the measure of the tangent?
::12. 如果偏差比率为92,那么相差的量度是多少?

13. If a tangent ratio is $\begin{align*}\frac{4}{20}\end{align*}$ what is the measure of the tangent?
::13. 如果相差比率为420,那么相差的尺度是什么?

14. If a tangent ratio is $\begin{align*}\frac{7}{21}\end{align*}$ what is the measure of the tangent?
::14. 如果偏差比率为721,那么误差的尺度是什么?

15. If a tangent ratio is $\begin{align*}\frac{9}{19}\end{align*}$ what is the measure of the tangent?
::15. 如果偏差比率为919,那么误差的尺度是什么?

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

理解对象

Tangent Ratio ::相近比率

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)

Tangent Ratio
::相近比率

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)