Course: 7.14 了解孔径 | The Way To Learn

Section outline

Select section General

Collapse Expand
General

Collapse all Expand all
- Select activity 新闻通告
  
  新闻通告 Forum
Select section A. 理解科松

Collapse Expand
A. 理解科松
Roger has placed a ladder that is 12 feet in length against the side of the house such that the foot of the ladder is 5 feet from the base of the house. As Roger prepares to climb the ladder to clean the rain gutters of the house, his neighbor yells to him, “You had better adjust that ladder. The angle between the ladder and the ground has to be less than 75°.”
::罗杰在房屋侧方架设了12英尺长的梯子,梯子脚离房屋底部5英尺。当罗杰准备爬上梯子清理房屋的雨水槽时,他的邻居对他大喊 : “ 你最好调整梯子。梯子与地面之间的角必须小于75度。 ”

Roger stepped back and looked at the position of the ladder. “He could be right,” thought Roger. “I’ve got to figure out the size of that angle before I climb up that ladder.”
::罗杰退后看了梯子的位置。 “他可能是对的 , ” 罗杰认为。 “ 在我爬上梯子之前,我必须先弄清楚这个角度的大小。 ”

How can Roger calculate the measure of the angle?
::罗杰如何计算角度的度量?

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

Cosine
::余弦

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 alongside the reference angle to the length of the hypotenuse is known as the Cosine 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 Cosine ratio is the ratio of the adjacent side to the hypotenuse.
::三角测量是一个数学分支,用来以非常精确的方式确定边的长度和角的测量。右三角的边的长度比称为三角比。侧的长度与参考角的长度比与下拉度长度之比称为“科斯尼比”。右三角的急性角由三角的下拉度和腿组成。这条腿称为参考角的相邻边。科斯尼比为邻边与下拉度之比。科斯尼比为近邻边与下拉度之比。

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).
:h) 侧对角参考角(O);侧对角参考角(A)。

The Cosine ratio for the first triangle can be written in words as:
::第一个三角形的余弦比可以以下列文字写入:

cosine of $\begin{align*}\angle A = \frac{\text{side adjeacent}\ \angle A}{\text{hypotenuse}} \end{align*}$ Or in a shortened form as cosine of $\begin{align*}\angle A= \frac{\text{adjacent}}{\text{hypotenuse}}\end{align*}$

::A 的 consine = 旁的 adjeacent Ahypotenteuse 或缩缩为 A 的 cosine = 邻接hypotenuse

The cosine ratio for the first triangle can be written in symbols as:
::第一个三角形的余弦比可以用符号写成 :

$\begin{align*}\cos A = \frac{AC}{AB}\end{align*}$
::CASA=行预咨委会

The Cosine ratio for the second triangle can be written in words as:
::第二个三角形的余弦比可以以下列文字写成:

cosine of $\begin{align*}\angle F = \frac{\text{side adjeacent}\ \angle F}{\text{hypotenuse}}\end{align*}$ Or in a shortened form as cosine of $\begin{align*}\angle F = \frac{\text{adjacent}}{\text{hypotenuse}}\end{align*}$
::F = F = F = F F F F F = F F F F F F Fhypotenteuse 或缩写为 F F F F F = F= Hypotenuse

The Cosine ratio for the second triangle can be written in symbols as:
::第二个三角形的余弦比可以用符号写成:

$\begin{align*}\cos F = \frac{DF}{EF}\end{align*}$
::COSF=DFEF =FFEF =FFEF =FFF =FFEF =FFEF =FFEF =FFFF =FFEF =FFF =FFEF =FFFF =FFFEF =FFFFF =FFFFF =FFFFF =FFFFF

Let’s determine the value for the cosine ratio of each of the acute angles using the following right triangle. Express the cosine 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余弦比值。用文字和符号表达余弦比。然后,使用对应边上的值,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 cosine 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 的余弦比。

Words:
::单词 :

cosine of $\begin{align*}\angle A = \frac{\text{adjacent}}{\text{hypotenuse}}\end{align*}$
::A=近邻相合聚苯乙烯的consine

Symbols:
::符号 :

$\begin{align*}\cos A = \frac{AC}{AB}\end{align*}$
::CASA=行预咨委会

Fraction:
::分数 :

$\begin{align*}\cos A = \frac{5}{13}\end{align*}$
::COA=513 COA=513

Decimal:
::十进制 :

$\begin{align*}\cos A = 0.3846\end{align*}$
::COA=0.3846 COA=0.3846

$\begin{align*}\triangle ABC\end{align*}$ 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 cosine 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 的余弦比。

Words:
::单词 :

cosine of $\begin{align*}\angle B = \frac{\text{adjacent}}{\text{hypotenuse}}\end{align*}$
::B=近邻相聚物的cosine

Symbols:
::符号 :

$\begin{align*}\cos B = \frac{BC}{AB}\end{align*}$
::COSB=BBAB =BBAB COSB =BBAB =BBAB =BBBB =BBBB =BBBBB =BBBBB =BBBBB =BBBBBB =BBBBB =BBBBB =BBBBB =BBBB =BBBB =BBBB =BBBB =BBBB =BBBBBB =BBBBB =BBBB =BBB =BBBB =BBB =BBB =BBB =BBB =BBBBB =BBBBB =BBBBB =BBBBB =BBB BBBB =BBBBB =BBBBB =BBBBBBB =B BBBBB =BBBBBB =BBBBB B B B B B =BBBB BBBBBBBBBBBBBBBB =B =BBBBBBBBBBBBBBB =B =B =BBBBBBBBBB =B =BBB BBBBBBBBBB B B BBBBBBB =B =B =B =B =B =BBBBBBB =B B B B B BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB

Fraction:
::分数 :

$\begin{align*}\cos B = \frac{12}{13}\end{align*}$
::COSB=1213 COSB=1213

Decimal:
::十进制 :

$\begin{align*}\cos B = 0.9231\end{align*}$
::COSB=0.9231 COSB=0.9231

If $\begin{align*}\cos A = 0.3846\end{align*}$ , then the measure of $\begin{align*}\angle A\end{align*}$ can be found using the inverse cosine function on the TI- calculator.
::如果 cosA=0. 3846,那么使用TI-计算器上的反余弦函数可以找到 A 的量度。

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

Next, look at the calculator screen to see the measure of $\begin{align*}\angle A\end{align*}$ .
::接下来,看看计算器屏幕,看看 QA 的度量。

Then, write the measure of $\begin{align*}\angle A\end{align*}$ to the nearest tenth.
::然后,将A的量量写在最接近的十分之一。

$\begin{align*}\angle A = 67.4^\circ\end{align*}$
::A=67.4

The answer is 67.4°
::答案是67.4摄氏度

Examples
::实例

Example 1
::例1

Earlier, you were given a problem about Roger and the ladder. He needs to figure out the angle the ladder makes with the ground.
::早些时候,有人给了你罗杰和梯子的问题。他需要找出梯子与地面的角。

How can Roger figure out the measure of the angle?
::罗杰怎么知道角度的大小?

He can use the cosine ratio.
::他可以用余弦比法

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.
::下一位是三角形两边的名字

Next, write the cosine ratio in words.
::接下来,用单词写下余弦比。

$\begin{align*}\cos A = \frac{\text{adjacent}}{\text{hypotenuse}} \end{align*}$
::COSA = 相邻合合质

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

$\begin{align*}\cos A = \frac{AB}{AC}\end{align*}$
::COA=ABAC

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。

$\begin{align*}\cos A = \frac{5}{12}\end{align*}$
::COA=512 COA=512

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。

$\begin{align*}\cos A = 0.4167\end{align*}$
::COA=0.4167 COA=0.4167

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

Then, write the measure of $\begin{align*} \angle A\end{align*}$ to the nearest tenth.
::然后,将A的量量写在最接近的十分之一。

$\begin{align*}\angle A = 65.4^\circ \end{align*}$
::A=65.4

The answer is 65.4°
::答案是65.4摄氏度

The measure of $\begin{align*}\angle A\end{align*}$ is 65.4°. It is less than 75° so Roger can repair the rain gutters.
::A的度量为65.4°,小于75°,罗杰可以修补雨水槽。

Example 2
::例2

Use the cosine ratio to calculate the measure of $\begin{align*}\angle A\end{align*}$ to the nearest tenth.
::使用余弦比来计算 A 到最近的十分之一的度量。

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

Next, write the cosine ratio in words.
::接下来,用单词写下余弦比。

$\begin{align*}\cos A = \frac{\text{adjacent}}{\text{hypotenuse}}\end{align*}$
::COSA = 相邻合合质

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

$\begin{align*}\cos A = \frac{AB}{AC}\end{align*}$
::COA=ABAC

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。

$\begin{align*}\cos A = \frac{38}{50}\end{align*}$
::COA=3850 COSA=3850

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。

$\begin{align*}\cos A = 0.76\end{align*}$
::COA=0.76 COSA=0.76

The decimal terminated. You could write the decimal as 0.7600 but this is not necessary.
::小数点终止。您可以将小数点写为 0. 7600, 但这没有必要。

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

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

$\begin{align*}\angle A = 40.5^\circ\end{align*}$
::A=40.5

The answer is 40.5°
::答案是40.5摄氏度

Example 3
::例3

If $\begin{align*}\cos B=0.7984\end{align*}$ , what is the measure of $\begin{align*}\angle B\end{align*}$ to the nearest tenth.
::如果 COSB = 0.7984, 则 B 到最近的十分之一的量度是多少。

First, use the TI-calculator to determine the measure of $\begin{align*}\angle B\end{align*}$ by using the inverse cosine function $\begin{align*}(\cos^{-1})\end{align*}$ on the calculator. This function is above the $\begin{align*}\boxed{\cos}\end{align*}$ button on the calculator. To access this function press
::首先,使用 TI 计算器来确定 B 的度量, 方法是在计算器上使用反余弦函数( cos-1) 。此函数高于计算器上的 cos- 1 按钮。要访问此函数, 请按此函数

Next, enter the decimal 0.7984 in the parenthesis where the cursor is flashing.
::下一步,在光标闪烁所在的括号中输入小数 0.7984。

Next, press enter and the calculator screen will display the measure of the angle.
::下一步,按下输入,计算器屏幕将显示角度的度量。

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。

$\begin{align*}\angle B = 32.0^\circ\end{align*}$ .
::B=32. 0

The answer is 32.0°
::答案是32.0摄氏度

Example 4
::例4

For the following solution that shows using the cosine ratio to calculate the measure of $\begin{align*}\angle B\end{align*}$ , draw and label the right triangle ABC that was used to do the solution.
::以下的解决方案显示使用余弦比计算 B 的度量, 绘制并标出用于解析的右三角ABC 。

$\begin{align*} \begin{array}{rcl} \cos B &=& \frac{\text{adjacent}}{\text{hypotenuse}} \\ \cos B &=& \frac{BC}{AB}\\ \cos B &=& \frac{36}{76.7} \\ \cos B &=& 0.4694 \\ \cos^{-1}(\cos B) &=& \cos^{-1}(0.4694) \\ \angle B &=& 62.00454372 \\ \angle B &=& 62.0^\circ \end{array}\end{align*}$
::COSB = 相邻近地环球生物B = BBABcosB= 3676.7cosB= 0. 4694cos-1(cosB) = cos-1(0. 4694) B = 62. 00454372B=62.0

First, write down what you know from the solution.
::首先,写下你从解决方案中知道的东西。

$\begin{align*}\triangle ABC\end{align*}$ is a right triangle.
::ABC是一个右三角形。

$\begin{align*}\overline{BC}\end{align*}$ is the adjacent side of the triangle.
::BC是三角形的相邻一面

$\begin{align*}\overline{BC}\end{align*}$ has a length of 36.
::BC的长度是36

$\begin{align*}\overline{AB}\end{align*}$ is the hypotenuse of the triangle.
::AB是三角形的抵押物

$\begin{align*}\overline{AB}\end{align*}$ has a length of 76.7.
::他长76.7岁

$\begin{align*}\angle B\end{align*}$ is the reference angle.
::#B是参考角度。 #B是参考角度。 #B是参考角度。 #B是参考角度。 #B是参考角度。

Next, use the information you have written down to draw and label the triangle.
::下一步,使用您写下来的信息绘制三角形并标注三角形。

The above triangle represents the problem.
::上述三角形代表着问题。

Example 5
::例5

For the given right triangle, use the cosine ratio to calculate the measure of $\begin{align*} \angle A\end{align*}$ to the nearest tenth.
::对于给定的右三角形,请使用余弦比来计算 A 到最近的十分之一的度量。

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

Next, write the cosine ratio in words.
::接下来,用单词写下余弦比。

$\begin{align*}\cos A = \frac{\text{adjacent}}{\text{hypotenuse}}\end{align*}$
::COSA = 相邻合合质

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

$\begin{align*}\cos A = \frac{AC}{AB}\end{align*}$
::CASA=行预咨委会

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。

$\begin{align*}\cos A = \frac{32.4}{45.2}\end{align*}$
::COA=32.4455.2

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。

$\begin{align*}\cos A = 0.7168\end{align*}$
::COA=0.7168 COSA=0.7168

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

Then, write the measure of $\begin{align*}\angle A\end{align*}$ to the nearest tenth.
::然后,将A的量量写在最接近的十分之一。

$\begin{align*}\angle A = 44.2^\circ\end{align*}$ .
::A=44.2。

The answer is 44.2°
::答案是44.2°

Review
::回顾

Solve each problem.
::解决每一个问题。

1. What is the cosine of $\begin{align*}\angle G\end{align*}$ ?
::1. QG的余弦是什么?

2. What is the cosine of $\begin{align*}\angle H \end{align*}$ ?
::2. 什么是"H"的余弦?

3. How do you identify a cosine?
::3. 如何识别共弦?

4. What is the cosine of $\begin{align*} \angle R\end{align*}$ ?
::4. 什么是“R”的余弦?

5. What is the cosine of $\begin{align*}\angle S \end{align*}$ ?
::5. 什么是ZZS的余弦?

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

7.What is the cosine of $\begin{align*}\angle A\end{align*}$ ?
::7. 什么是A的余弦?

8. What is the cosine of $\begin{align*} \angle B\end{align*}$ ?
::8. 什么是B的余弦?

Answer each of the following questions.
::回答下列问题中的每一个问题。

9. Is the cosine related to an angle?
::9. 余弦与角度有关吗?

10. Do you need to know side lengths of a triangle to write a cosine?
::10. 您是否需要知道三角形的侧长来写余弦曲弦?

11. Which side lengths do you need?
::11. 你需要哪一侧长度?

12. If the cosine was $\begin{align*} \frac{5}{20}\end{align*}$ what would the numerical value of the cosine be? (Round your answer to the nearest hundredths place)
::12. 如果余弦为520,则余弦的数值是多少?

13. If the cosine was $\begin{align*} \frac{5}{25}\end{align*}$ what would the numerical value of the cosine be? (Round your answer to the nearest hundredths place)
::13. 如果余弦为525,则余弦的数值是多少? (将回答写到最近的一百个地方)

14. If the cosine was $\begin{align*}\frac{3}{33}\end{align*}$ what would the numerical value of the cosine be? (Round your answer to the nearest hundredths place)
::14. 如果余弦为333, 余弦的数值是多少? (您回答最接近的百分百位数)

15. If the cosine was $\begin{align*}\frac{12}{14}\end{align*}$ what would the numerical value of the cosine be? (Round your answer to the nearest hundredths place)
::15. 如果余弦为1214,则余弦的数值是多少? (将回答写到最近的100个地方)

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

A. 理解科松

Cosine ::余弦

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)

Cosine
::余弦

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)