节: 确定并使用 Cosine 比例 | 7.17 确定并使用余弦比率

章节大纲

Arnold needs to open the door into the hayloft of the barn. He leans the 20-foot ladder against the outside wall and uses a measuring device to determine the size of the angle the ladder makes with the ground.
::阿诺德需要把门打开开到谷仓的干草棚里他把20英尺的梯子靠在外墙上

Arnold is happy because the angle of $\begin{align*}75^\circ\end{align*}$ means it is safe to climb the ladder. The ladder must be left in place for everyone to get in and out of the loft. If the ladder is at least 5 feet from the barn, Arnold can hook up the portable food and drink station.
::阿诺德之所以高兴,是因为75的角意味着可以安全地爬上梯子。梯子必须留在位置上, 所有人都可以进出阁楼。如果梯子至少离谷仓5英尺, 阿诺德可以连接便携式食品和饮料站。

How can Arnold figure out the distance from the base of the barn to the foot of the ladder before the portable station arrives?
::阿诺德如何在便携式车站到达之前找到从谷仓底到梯子脚的距离?

In this concept, you will learn to determine and use the cosine ratio.
::在此概念中,您将学会确定并使用余弦比。

Cosine Ratio
::余弦比

The TI-calculator can be used to find the ratio when the measure of the angle is known. If the measure of $\begin{align*} \angle B= 68^ \circ\end{align*}$ then the value of the cosine ratio can be found by following the Key Press History:
::当已知角度的测量值时, 可以用 TI 计算器来查找比例。如果测量了 {B=68}} , 则可以通过 Key Press History 找到余弦比的值 :

On the calculator screen the following is displayed:
::在计算器屏幕上显示以下显示:

This decimal should always be rounded to the nearest ten thousandth (four places after the decimal).
::此小数点应总是四舍五入到最接近的万分之一(小数点后四位)。

$\begin{align*}\cos 68^ \circ = 0.3746\end{align*}$
::COS680.3746

This cosine ratio for an angle of $\begin{align*}68 ^\circ\end{align*}$ will remain constant regardless of the size of the triangle. The sides will always be in the same proportion to each other. The value of the Cosine ratio will always be between zero and one.
::角为 68 的余弦比将保持不变, 不论三角形大小。边将始终保持相同比例。共弦比的值总是介于零和一之间。

The Cosine ratio is related to one of the acute angles of a right triangle such that the cosine of the acute angle is the ratio of the side adjacent to the specific acute angle (the reference angle) to the hypotenuse.
::余弦比与右三角形的急性角之一有关,这样,急性角的余弦就是与特定急性角(参考角)相邻的侧面与下限的对角(参考角)的对角。

The Cosine ratio can be written as $\begin{align*}\cos \angle = \frac{\text {adjacent}}{\text {hypotenuse}}\end{align*}$ . This equation has three parts to it – an angle and two sides. When the lengths of the two sides are known, the measure of the angle can be calculated. When the measure of an angle and the length of one side is known, the length of the other side can be calculated.
::共弦比可以写成“ cosáadjahcenthypotenuse ” 。这个方程式有三个部分 — — 一个角度和两边。当知道两边的长度时,可以计算角的度量。当知道一个角度的度量和一边的长度时,可以计算另一边的长度。

It is the sides of the triangle that determine the trigonometric ratio that will be used to calculate the measure of an angle or the length of a side. To use the cosine ratio, the adjacent side and the hypotenuse of the right triangle must be indicated. These two sides will have values on them if the measure of an angle is to be calculated using the cosine ratio. If the length of a side is to be calculated using the cosine ratio then one of the sides will display a value and the other will display a variable (the side that is unknown).
::三角形的侧面决定三角比, 用来计算角或边长的度量。要使用余弦比, 必须标明右三角形的相邻侧和下限。如果要使用余弦比计算角的度量, 则这两边的侧面将具有数值。如果使用余弦比计算边的长度, 则其中一方将显示一个数值, 而另一方将显示一个变量( 未知的侧面 )。

Let’s look at the following right triangle to see how this works.
::让我们看看下面的右三角形,看看它是如何运作的。

The measure of $\begin{align*} \angle A \text{ is } 52^ \circ\end{align*}$ . The length of the hypotenuse is $\begin{align*}18 \text{ centimeters}\end{align*}$ . The side $\begin{align*}\overline{AC}\end{align*}$ has the variable ‘ $\begin{align*}X\end{align*}$ ’ on it which means this is the side that is unknown and its length must be calculated.
::QA 的度量为 52 。下限长度为 18 厘米。侧面的 AC 有变量 'X ' , 这意味着这是未知的侧面, 必须计算其长度。

The sides of the triangle can be named using the acute angle B which is the reference angle for this triangle.
::三角形的两边可以使用急性角B命名,B是此三角形的参考角。

The two sides that are indicated on this triangle are the hypotenuse and the adjacent. The Cosine ratio is the ratio of the adjacent side to the hypotenuse.
::三角形上标明的两边是下限和相邻。共弦比是相邻侧与下限之比。

First, write the cosine ratio using words.
::首先,用单词来写余弦比。

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

Next, write the cosine ratio using symbols.
::接下来,用符号写下余弦比。

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

Next, fill all known values into the equation. $\begin{align*}\angle A = 52^\circ; \ \overline{AC} = X; \ \overline{AB} = 18\end{align*}$ .
::接下来,将所有已知的值填入方程中。 @A=52;'AC=X;'AB=18。

$\begin{align*}\cos 52^\circ = \frac{X}{18}\end{align*}$

::COS52X18

Next, use the TI-calculator to find the value of $\begin{align*} \cos 52^\circ\end{align*}$ . Round the decimal to the nearest ten thousandth.
::下一步,使用 TI 计算器查找 COS52 的值。将小数点舍入小数点到最接近的万分之一。

$\begin{align*}\cos 52^\circ = 0.6157\end{align*}$
:cos520.6157)

Next, substitute this value into the equation.
::接下来,将这个值替换为等式。

$\begin{align*}\begin{array}{rcl} \cos 52^\circ &=& \frac{X}{18} \\ 0.6157 &=& \frac{X}{18} \\ \end{array}\end{align*}$
::COS52X180.6157=X18

Next, multiply both sides of the equation by 18 to solve for the variable.
::接下来,将方程的两边乘以 18 来解析变量。

$\begin{align*}\begin{array}{rcl} 0.6157 &=& \frac{X}{18} \\ 18(0.6157) &=& 18 \left (\frac{X}{18} \right) \\ 11.0826 &=& { \overset{1} {\cancel{18}} } \left ( \frac{X}{ \cancel{18}} \right) \\ 11.08 &=& X \\ \end{array}\end{align*}$

::0.6157=X1818(0.6157)=18(X18)11.0826=118(X18)11.08=X

The answer is 11.08.
::答案是11.08

The length of the adjacent side of the right triangle is 11.08 centimeters.
::右三角形的相邻侧长度为11.08厘米。

The calculated lengths of sides are usually expressed to the nearest hundredth while the calculated measure of an angle is usually expressed to the nearest tenth, unless otherwise stated.
::边的计算长度通常表示为最接近的一百,而角的计算尺度通常表示为最接近的十分之一,除非另有说明。

Examples
::实例

Example 1
::例1

Earlier, you were given a problem about Arnold and the hayloft door. He needs to know if there is enough room between the barn and the ladder for the portable station. How can he figure this out?
::他需要知道谷仓和梯子之间是否有足够的空间

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 using the reference $\begin{align*}\angle A\end{align*}$ .
::下一步,使用引用 {A} 命名三角形的两边。

Next, write the cosine ratio using words.
::接下来,用文字写下余弦比。

$\begin{align*}\cos A = \frac{\text{ adjacent }}{\text { hypotenuse }}\end{align*}$
::COAA = 相邻的下限

Next, write the cosine ratio using symbols.
::接下来,用符号写下余弦比。

$\begin{align*}\cos 75 = \frac{AC}{AB}\end{align*}$
::cos75=行预咨委会

Next, fill all known values into the equation. $\begin{align*}\angle A = 75^\circ; \ \overline{AC} = X; \ \overline{AB} =20 \ ft\end{align*}$ .
::接下来,将所有已知的值填入方程。 @A=75;'AC=X;'AB=20英尺。

$\begin{align*}\cos75 = \frac{X}{20}\end{align*}$
::COS75=X20

Next, use the TI-calculator to find the value of $\begin{align*}\cos75^\circ\end{align*}$ .
::接下来,使用 TI 计算器来查找 COS75 的值。

$\begin{align*}\cos75^\circ = 0.2588\end{align*}$
::COS750.2588

Next, substitute this value into the equation.
::接下来,将这个值替换为等式。

$\begin{align*}\begin{array}{rcl} \cos75^\circ &=& \frac{X}{20} \\ 0.2588 &=& \frac{X}{20} \\ \end{array}\end{align*}$
::COS75X200.2588=X20

Next, multiply both sides of the equation by 20 to solve for the variable.

$\begin{align*}\begin{array}{rcl} 0.2588 &=& \frac{X}{20} \\ 20(0.2588) &=& 20 \left (\frac{X}{20} \right) \\ 5.176 &=& { \overset{1} {\cancel{20}} } \left ( \frac{X}{ \cancel{20}} \right) \\ 5.18 &=& X \end{array}\end{align*}$
The answer is 5.18.
::接下来,将方程的两边乘以 20 以解答变数。 0.2588=X2020(0.2588)=20(X20)5.176=120(X205.18=X)5.18=X 答案是5.18。

The distance is 5.18 feet, so Arnold can put the portable station in place.
::距离是5.18英尺阿诺德可以把便携式车站装好

Example 2
::例2

For the following right triangle calculate the length of side ‘ $\begin{align*}X\end{align*}$ ’ to the nearest hundredth.
::以下的右三角形计算“X”到最接近的百分百的侧边长度。

First, use the reference angle $\begin{align*}D\end{align*}$ to name the sides of the triangle.
::首先,使用引用角度 D 来命名三角形的侧面。

The two sides that are indicated on this triangle are the adjacent and the hypotenuse. The Cosine ratio is the ratio of the adjacent side to the hypotenuse.
::这个三角形上标出的两边是相邻和下限。共弦比是相邻侧与下限之比。

First, write the cosine ratio using words.
::首先,用单词来写余弦比。

$\begin{align*}\cos D = \frac{\text{adjacent}}{\text{hypotenuse}}\end{align*}$
::COsD = 近邻合用

Next, write the cosine ratio using symbols.
::接下来,用符号写下余弦比。

$\begin{align*}\cos D = \frac{DE}{DF}\end{align*}$

::COsD=DEDF

Next, fill all known values into the equation. $\begin{align*}\angle D = 35^\circ; \ \overline{DF} = X; \ \overline{DE} = 8.2\end{align*}$ .
::接下来,将所有已知的值填入方程。 @ D=35 {; 'DF=X; =8. 2 。

$\begin{align*}\begin{array}{rcl} \cos D &=& \frac{DE}{DF} \\ \cos 35^\circ &=& \frac{8.2}{X} \end{array}\end{align*}$

::COsD=DEDFcos358.2X

Next, use the TI-calculator to find the value of $\begin{align*}\cos35^\circ\end{align*}$ .
::接下来,使用 TI 计算器来查找 COs35 的值。

$\begin{align*}\cos35^\circ = 0.8192\end{align*}$

::COS350.8192

Next, substitute this value into the equation.
::接下来,将这个值替换为等式。

$\begin{align*}\begin{array}{rcl} \cos35^\circ &=& \frac{8.2}{X} \\ 0.8192 &=& \frac{8.2}{X} \end{array}\end{align*}$
::COs358.2X0.8.192=8.2X

Next, multiply both sides of the equation by $\begin{align*}X\end{align*}$ to clear the denominator.
::接下来,将方程的两侧乘以 X 来清除分母。

$\begin{align*}\begin{array}{rcl} 0.8192 &=& \frac{8.2}{X} \\ X(0.8192) &=& X \left (\frac{8.2}{X} \right) \\ 0.8192 \ X &=& { \overset{1} {\cancel{X}} } \left ( \frac{8.2}{ \cancel{X}} \right) \\ 0.8192 \ X &=& 8.2 \\ \end{array}\end{align*}$

Then, divide both sides of the equation by 0.8192 to solve for the variable.
::然后将方程的两边除以 0.8192 来解析变量。

$\begin{align*}\begin{array}{rcl} 0.8192X &=& 8.2 \\ \frac{\overset{1}{\cancel{0.8192X}}}{\cancel{0.8192}} &=& \frac{8.2}{0.8192} \\ X &=& 10.01 \\ \end{array}\end{align*}$
::0.8192X=8.210.8192X0.8192=8.20.8192X=10.01

The answer is 10.01.

The length of the hypotenuse of the right triangle is 10.01 inches.
::右三角形的下限长度为10.01英寸。

Example 3
::例3

For the following right triangle calculate the length of side ‘ $\begin{align*}X\end{align*}$ ’.
::以下右三角形计算侧边“X”的长度。

First, use the reference angle $\begin{align*}C\end{align*}$ to name the sides of the triangle.
::首先,使用引用角度C来命名三角形的侧面。

The two sides that are indicated on this triangle are the adjacent and the hypotenuse. The Cosine ratio is the ratio of the adjacent side to the hypotenuse.
::这个三角形上标出的两边是相邻和下限。共弦比是相邻侧与下限之比。

First, write the cosine ratio using words.
::首先,用单词来写余弦比。

$\begin{align*}\cos C = \frac{\text{adjacent}}{\text{hypotenuse}}\end{align*}$
::COSC = 相邻合合质化

Next, write the cosine ratio using symbols.
::接下来,用符号写下余弦比。

$\begin{align*}\cos C = \frac{BC}{AC}\end{align*}$
::COSC = BCC

Next, fill all known values into the equation. $\begin{align*}\angle C = 59^\circ; \ \overline{BC} = X; \ \overline{AC} =38\end{align*}$ .
::接下来,将所有已知的值填入方程。 @C=59;'BC=X;'AC=38。

$\begin{align*}\begin{array}{rcl0} \cos C &=& \frac{BC}{AC} \\ \cos 59^\circ &=& \frac{X}{38} \\ \end{array}\end{align*}$

::COSC = BCCACcos59X38

Next, use the TI-calculator to find the value of $\begin{align*}\cos59^\circ\end{align*}$ .
::下一步,使用 TI 计算器来查找 cos59 的值。

$\begin{align*}\cos59^\circ = 0.5150\end{align*}$
::COS590.5150

Next, substitute this value into the equation.
::接下来,将这个值替换为等式。

$\begin{align*}\begin{array}{rcl} \cos 59^\circ &=& \frac{X}{38} \\ 0.5150 &=& \frac{X}{38} \\ \end{array}\end{align*}$

::COS59X380.5150=X38

Next, multiply both sides of the equation by 38 to solve for the variable.
::接下来,将方程的两边乘以38,以解答变量。

$\begin{align*}\begin{array}{rcl} 0.5150 &=& \frac{X}{38} \\ 38(0.5150) &=& 38 \left (\frac{X}{38} \right) \\ 38(0.5150) &=& { \overset{1} {\cancel{38}} } \left ( \frac{X}{ \cancel{38}} \right) \\ 19.57 &=& X \\ \end{array}\end{align*}$
::0.5150=X3838(0.5150)=38(X38)38(0.5150)=138(X38)19.57=X

The answer is 19.57.
::答案是19.57。

The length of the adjacent side of the right triangle is 19.57 feet.
::右三角形的相邻侧长度为19.57英尺。

Example 4
::例4

Answer the questions below for the following right triangle.
::回答以下右三角的问题。

What angle is the reference angle of $\begin{align*}\triangle JKL\end{align*}$ ?
::@JKL 的引用角度是什么 ?

What is the measure of the reference angle?
::参考角度的度量是多少?

What two sides of the triangle are indicated in $\begin{align*}\triangle JKL\end{align*}$ ?
::三角形的两边在 @JKL中标明的是什么?

Write the trigonometric ratio in words that would be used to calculate the length of $\begin{align*}\overline{JL}\end{align*}$ ?
::将三角比写成文字用来计算JL的长度?

What is the value of $\begin{align*} \cos40^\circ\end{align*}$ ?
::COS40值是多少?

The answers to the above questions for $\begin{align*}\triangle JKL\end{align*}$ are:
::对上述问题的答复是:

The acute angle referred to in $\begin{align*}\triangle JKL\end{align*}$ is $\begin{align*}\angle J\end{align*}$ .
::“JKL”中提到的急性角是“J”。

The measure of the reference angle is $\begin{align*}40^\circ\end{align*}$ .
::参考角的度量为 40 。

The two sides of $\begin{align*}\triangle JKL\end{align*}$ that are indicated are $\begin{align*}\overline{JK}\end{align*}$ the adjacent side and $\begin{align*}\overline{JL}\end{align*}$ the hypotenuse.
::标注的JKL两边是邻边的JK,

$\begin{align*}\cos J = \frac{\text{ adjacent }}{\text{ hypotenuse }}\end{align*}$
::COsJ = 相邻的下拉度

The value of $\begin{align*} \cos40^\circ \text { is } 0.7660.\end{align*}$
::COS40 值为 0.7660 。

Review
::回顾

Use a calculator to find each Cosine.You may round to the nearest hundredth.
::使用计算器找到每个科内。你可以绕到最近的第一百。

1. $\begin{align*} \text{Cosine} \ 33^{\circ}\end{align*}$
::1. 脊弦33_______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

2. $\begin{align*}\text{Cosine} \ 29^{\circ}\end{align*}$
::2. 脊弦29____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

3. $\begin{align*}\text{Cosine} \ 73^{\circ}\end{align*}$
::3. 73+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

4. $\begin{align*}\text{Cosine} \ 88^{\circ}\end{align*}$
::4. 脊髓88

5. $\begin{align*}\text{Cosine} \ 50^{\circ}\end{align*}$
::5. 余弦50______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

6. $\begin{align*}\text{Cosine} \ 67^{\circ}\end{align*}$
::6. 共弦67________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

7. $\begin{align*}\text{Cosine} \ 42^{\circ}\end{align*}$
::7. 脊弦 42______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

8. $\begin{align*}\text{Cosine} \ 18^{\circ}\end{align*}$
::8. 脊弦18___________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

9. $\begin{align*}\text{Cosine} \ 9^{\circ}\end{align*}$
::9. 脊髓 9______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

Find the length of the adjacent side. Round your answer to the nearest hundredths place.
::查找相邻侧的长度。将您的答案四舍五入到最近的一百个位置。

10. A triangle has a hypotenuse of 7 inches. Angle A is equal to 60 degrees. Find the length of the adjacent side.
::10. 三角形的下限为7英寸,角A等于60度,查找相邻侧的长度。

11. A triangle has a hypotenuse of 12 inches. Angle B is equal to 45 degrees. Find the length of the adjacent side.
::11. 三角形的下限为12英寸,角B等于45度,查找相邻侧的长度。

12. A triangle has a hypotenuse of 8 inches. Angle A is equal to 35 degrees. Find the length of the adjacent side.
::12. 三角形的下限为8英寸。角A等于35度。查找相邻侧的长度。

13. A triangle has a hypotenuse of 12 inches. Angle A is equal to 28 degrees. Find the length of the adjacent side.
::13. 三角形的下限为12英寸,角A等于28度,查找相邻侧的长度。

14. A triangle has a hypotenuse of 6 inches. Angle A is equal to 33 degrees. Find the length of the adjacent side.
::14. 三角形的下限为6英寸,角A等于33度,查找相邻侧的长度。

15. A triangle has a hypotenuse of 14 inches. Angle A is equal to 72 degrees. Find the length of the adjacent side.
::15. 三角形的下限为14英寸,角A等于72度,查找相邻侧的长度。

16. A triangle has a hypotenuse of 11 inches. Angle A is equal to 80 degrees. Find the length of the adjacent side.
::16. 三角形的下限为11英寸,角A等于80度,查找相邻侧的长度。

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。

章节大纲

Cosine Ratio ::余弦比

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)

Cosine Ratio
::余弦比

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)