节: 确定并使用正弦比率 | 7.16 确定并使用正弦比率

章节大纲

Jeremy was in the park flying the kite. It was a windy day and the kite was airborne within seconds. The kite was flying at the end of a string 48 meters long, making an angle of 75° with the ground. How can Jeremy figure out the altitude of his kite?
::Jeremy当时在公园里放风筝,风筝在几秒钟内就升空了。风筝在48米长的一条线的尽头飞行,与地面的距离为75度。Jeremy怎么知道风筝的高度?

In this concept, you will learn to determine and use the Sine ratio.
::在此概念中,您将学会确定并使用辛尼比。

Sine Ratio
::线间比比

You have used the TI-calculator to determine the measure of an angle using the inverse sine function $\begin{align*}(\sin^{-1})\end{align*}$ when value of the ratio was known. If $\begin{align*}\sin A=0.7845\end{align*}$ then using the TI-calculator displayed the measure of 51.7° for the angle.
::您已经使用 TI 计算器来确定一个角度的度量, 当知道该比例值时使用反正正弦函数( sin-1) 。如果 sin *A= 0.7845, 则使用 TI 计算器显示该角度的度量为51.7°。

The TI-calculator can also be used to find the ratio when the measure of the angle is known. If the measure of $\begin{align*}\angle B = 48^\circ\end{align*}$ then the value of the ratio can be found by following the Key Press History:
::也可以使用 TI- 计算器来在已知角度测量值时找到比例。如果测量 {B=48} , 则此比例的值可以在 Key Press History 后面找到 :

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)。

$\begin{aligned} \sin 48 = 0.7431 \end{aligned}$
$\begin{align*}\sin 48=0.7431\end{align*}$
::=0.7431 =0.7431 = 0.7431 = 0.7431

This sine ratio for an angle of 48° will remain constant regardless of the size of the triangle. The sides will always be in the same proportion to each other.
::角为 48 度的正弦比将保持不变, 不论三角形大小。边之间的比例总是相同。

The value of the Sine ratio will always be between zero and one.
::Sine 比例的值总是介于零和一之间。

The Sine ratio is related to one of the acute angles of a right triangle such that the sine of the acute angle is the ratio of the side opposite the specific acute angle (the reference angle) to the hypotenuse. The Sine ratio can be written as $\begin{align*}\sin \angle = \frac{\text{opposite}}{\text{hypotenuse}}\end{align*}$ . 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.
::Sine 比例与右三角形的急性角度之一有关,这样,急性角的正弦就是对准特定急性角(参考角)的侧边对角(参考角)对下角的比。 Sine 比例可以写成sin oppositehypotenuse。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,可以计算角的量度。当知道角和一边的长度时,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 sine ratio, the opposite 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 sine ratio. If the length of a side is to be calculated using the sine 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, 用来计算角的度量或边的长度。要使用正弦比, 必须标明右侧和右三角形的相对值。如果要使用正弦比计算角的度量, 则这两边的侧面将具有数值。如果用正弦比计算侧面的长度, 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 $\begin{align*}\angle B\end{align*}$ is 65°. The length of the hypotenuse is 28 meters. 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.
::@B 的度量为 65 度。下限长度为 28 米。上方的 AC 带有变量 'X ' , 这意味着这是未知的一面, 必须计算其长度。

The sides of the triangle can be named using the acute angle $\begin{align*}B\end{align*}$ 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 opposite. The Sine ratio is the ratio of the opposite side to the hypotenuse.
::这个三角形上显示的两边是下限和相反的。辛尼比是对面与下限的比。

First, write the sine ratio using words.
::首先,用字词写正弦比。

$\begin{aligned} \sin B = \frac{opposite}{hypotenuse} \end{aligned}$
$\begin{align*}\sin B = \frac{\text{opposite}}{\text{hypotenuse}}\end{align*}$
::Sin B = 对面日光化

Next, write the sine ratio using symbols.
::接下来,请使用符号写下正弦比例。

$\begin{aligned} \sin B = \frac{A C}{A B} \end{aligned}$
$\begin{align*}\sin B = \frac{AC}{AB}\end{align*}$
::*B=行预咨委会

Next, fill all known values into the equation. $\begin{align*}\angle B=65^\circ;\ \overline{AC}=X; \ \overline{AB}=28\end{align*}$ .
::下一步,将所有已知值填入方程。 @B=65;AC=X;AB=28。

$\begin{align*}\sin 65^\circ = \frac{X}{28}\end{align*}$
::X28(第65章)

Next, use the TI-calculator to find the value of $\begin{align*}\sin 65^\circ\end{align*}$ . Round the decimal to the nearest ten thousandth.
::接下来,使用 TI 计算器来查找 sin65。将小数点舍入小数点到最接近的万分之一。

$\begin{aligned} \sin 65^{\circ} = 0.9063 \end{aligned}$
$\begin{align*}\sin 65^\circ = 0.9063\end{align*}$
:九九九九年九月九日)

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

$\begin{aligned} \begin{array}{rcl} \sin 65^{\circ} & = & \frac{X}{28} \\ 0.9063 & = & \frac{X}{28} \end{array} \end{aligned}$
$\begin{align*}\begin{array}{rcl} \sin 65^\circ &=& \frac{X}{28} \\ 0.9063 &=& \frac{X}{28} \end{array}\end{align*}$
::X280.9063=X28

Next, multiply both sides of the equation by 28 to solve for the variable.
::接下来,将方程的两侧乘以 28 来解析变量。

$\begin{aligned} \begin{array}{rcl} 0.9063 & = & \frac{X}{28} \\ 28 (0.9063) & = & 28 (\frac{X}{28}) \\ 25.3764 & = & \overset{1}{28} (\frac{X}{28}) \\ 25.38 & = & X \end{array} \end{aligned}$
$\begin{align*}\begin{array}{rcl} 0.9063 &=& \frac{X}{28} \\ 28(0.9063) &=& 28 \left( \frac{X}{28} \right) \\ 25.3764 &=& { \overset{1} {\cancel{28}} } \left ( \frac{X}{ \cancel{28}} \right) \\ 25.38 &=& X \end{array}\end{align*}$
::0.9063=X282828(0.9063)=28(X2825.3764=281(X2825.38=X)

The answer is 25.38 meters.

The length of the opposite side of the right triangle is 25.38 meters.
::右三角形对面的长度为25.38米。

When calculating the length of a side of a right triangle, the answer is usually rounded to the nearest hundredth unless otherwise stated.
::在计算右三角形侧边的长度时,答案通常四舍五入到最接近的一百分之一,除非另有说明。

When calculating the measure of an angle of a right triangle, the answer is usually rounded to the nearest tenth unless otherwise stated.
::在计算右三角角的度量时,答案通常四舍五入到最接近的十分之一,除非另有说明。

Examples
::实例

Example 1
::例1

Earlier, you were given a problem about Jeremy and his kite. He needs to figure out how high his kite is above the ground.
::他需要弄清楚他的风筝在地面上有多高

Jeremy can use the sine ratio to calculate the answer.
::Jeremy可以使用正弦比来计算答案。

First, draw and label a right triangle to model the kite flying.
::首先,绘制并标出一个右三角形,以模拟风筝飞行。

First, using the reference angle $\begin{align*}A\end{align*}$ , name the sides of the right triangle.
::首先,使用参考角度 A, 命名右三角形的两边。

First, write the sine ratio using words.
::首先,用字词写正弦比。

$\begin{aligned} \sin A = \frac{opposite}{hypotenuse} \end{aligned}$
$\begin{align*}\sin A = \frac{\text{opposite}}{\text{hypotenuse}}\end{align*}$
::A=对面的日光化

Next, write the sine ratio using symbols.
::接下来,请使用符号写下正弦比例。

$\begin{aligned} \sin A = \frac{B C}{A B} \end{aligned}$
$\begin{align*}\sin A = \frac{BC}{AB}\end{align*}$
::=BCAB =BCAB =BCAB =BBB =BB =BB =BB =BB =BB =BB =BB =BB =BB =BB =BB =BB =BB

Next, fill all known values into the equation. $\begin{align*}\angle A=75^\circ; \ \overline{BC}=X; \ \overline{AB}=48 \ m\end{align*}$ .
::下一步,将所有已知值填入方程。 @A=75;BC=X;AB=48米。

$\begin{aligned} \sin 75^{\circ} = \frac{X}{48} \end{aligned}$
$\begin{align*}\sin 75^\circ = \frac{X}{48}\end{align*}$
::X48(西班牙语)

Next, use the TI-calculator to find the value of $\begin{align*}\sin 75^\circ\end{align*}$ .
::接下来,使用 TI 计算器来寻找 sinQQQ75 的值。

$\begin{aligned} \sin 75^{\circ} = 0.9659 \end{aligned}$
$\begin{align*}\sin 75^\circ =0.9659\end{align*}$
:九九九九九年九月九日)

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

$\begin{aligned} \begin{array}{rcl} \sin 75^{\circ} & = & \frac{X}{48} \\ 0.9659 & = & \frac{X}{48} \end{array} \end{aligned}$
$\begin{align*}\begin{array}{rcl} \sin 75^\circ &=& \frac{X}{48} \\ 0.9659 &=& \frac{X}{48} \end{array}\end{align*}$
::X480.9659=X48

Next, multiply both sides of the equation by 48 to solve for the variable.
::接下来,将方程的两边乘以48,以解答变量。

$\begin{aligned} \begin{array}{rcl} 0.9659 & = & \frac{X}{48} \\ 48 (0.9659) & = & 48 (\frac{X}{48}) \\ 46.36 & = & \overset{1}{48} (\frac{X}{48}) \\ 46.36 & = & X \end{array} \end{aligned}$
$\begin{align*}\begin{array}{rcl} 0.9659 &=& \frac{X}{48} \\ 48(0.9659) &=& 48 \left( \frac{X}{48} \right) \\ 46.36 &=& { \overset{1} {\cancel{48}} } \left ( \frac{X}{ \cancel{48}} \right) \\ 46.36 &=& X \end{array}\end{align*}$
::0.9659=X4848(0.9659)=48(X4846.36=481(X4846.36=X)

The answer is 46.36.
::答案是46.36

Jeremy’s kite is 46.36 meters above the ground.
::Jeremy的风筝距离地面46.36米。

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, write down what you know from the right triangle.
::首先,从正确的三角形写下你所知道的

$\begin{aligned} \begin{array}{rcl} ∠ F & = & 30^{\circ} \\ \bar{D E} & = & 17.05 units \\ \bar{E F} & = & X \end{array} \end{aligned}$
$\begin{align*}\begin{array}{rcl} \angle F &=& 30^\circ \\ \overline{DE} &=& 17.05 \ \text{units} \\ \overline{EF} &=& X \end{array}\end{align*}$
::F=30 DE =17.05单位EF =X

Next, use the reference angle $\begin{align*}F\end{align*}$ to name the sides of the triangle.
::下一步,使用引用角度 F 来命名三角形的侧面。

The two sides that are indicated on this triangle are the hypotenuse and the opposite. The Sine ratio is the ratio of the opposite side to the hypotenuse.
::这个三角形上显示的两边是下限和相反的。辛尼比是对面与下限的比。

First, write the sine ratio using words.
::首先,用字词写正弦比。

$\begin{aligned} \sin F = \frac{opposite}{hypotenuse} \end{aligned}$
$\begin{align*}\sin F = \frac{\text{opposite}}{\text{hypotenuse}} \end{align*}$
::SinF=对面的日光化

Next, write the sine ratio using symbols.
::接下来,请使用符号写下正弦比例。

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

Next, fill all known values into the equation. $\begin{align*}\angle F=30^\circ; \ \overline{EF}=X; \ \overline{DE}=17.05\end{align*}$ .
::接下来,将所有已知值填入方程。 @F=30;EF=X;DE=17.05。

$\begin{align*}\sin 30^\circ = \frac{17.05}{X}\end{align*}$
::七零五年五月三十日

Next, use the TI-calculator to find the value of $\begin{align*}\sin 30^\circ\end{align*}$ .
::接下来,请使用 TI 计算器来找到 siná30 的值。

$\begin{aligned} \sin 30^{\circ} = 0.5 \end{aligned}$
$\begin{align*}\sin 30^\circ = 0.5\end{align*}$
::-=YTET -伊甸园字幕组=-伊甸园字幕组=-伊甸园字幕组=-伊甸园字幕组=-伊甸园字幕组=-伊甸园字幕组=-伊甸园字幕组=-伊甸园字幕组=-伊甸园字幕组=-伊甸园字幕组=-伊甸园字幕组=-伊甸园字幕组=- 翻译:

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

$\begin{aligned} \begin{array}{rcl} \sin 30^{\circ} & = & \frac{17.05}{X} \\ 0.5 & = & \frac{17.05}{X} \end{array} \end{aligned}$
$\begin{align*}\begin{array}{rcl} \sin 30^\circ &=& \frac{17.05}{X} \\ 0.5 &=& \frac{17.05}{X} \end{array}\end{align*}$
::=17.05x0.5=17.05X =17.05X =17.05X =17.05X =17.05X =17.05X =17.05X =17.05X

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。

$\begin{aligned} \begin{array}{rcl} 0.5 & = & \frac{17.05}{X} \\ X (0.5) & = & X (\frac{17.05}{X}) \\ X (0.5) & = & \overset{1}{X} (\frac{17.05}{X}) \\ 0.5 X & = & 17.05 \end{array} \end{aligned}$
$\begin{align*} \begin{array}{rcl} 0.5 &=& \frac{17.05}{X} \\ X(0.5) &=& X \left( \frac{17.05}{X} \right) \\ X(0.5) &=& { \overset{1} {\cancel{X}} } \left ( \frac{17.05}{ \cancel{X}} \right) \\ 0.5 \ X &=& 17.05 \end{array}\end{align*}$

Then, divide both sides of the equation by 0.5 to solve for the variable.
::然后,将方程两边除以0.5,解决变量。

$\begin{aligned} \begin{array}{rcl} 0.5 X & = & 17.05 \\ \frac{\overset{1}{0.5} X}{0.5} & = & \frac{17.05}{0.5} \\ X & = & 3.41 \end{array} \end{aligned}$
$\begin{align*}\begin{array}{rcl} 0.5 \ X &=& 17.05 \\ \frac{ { \overset{1} { \cancel{0.5}}} X }{ \cancel{0.5}} &=& \frac{17.05}{0.5} \\ X &=& 3.41 \end{array}\end{align*}$
::0.5X=17.50.51X0.5=17.050.50X=3.41

The answer is 34.10.
::答案是34.10。

The length of the hypotenuse of the right triangle is 34.10 units.
::右三角形的下限长度为34.10个单位。

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, write down what you know from the right triangle.
::首先,从正确的三角形写下你所知道的

$\begin{aligned} \begin{array}{rcl} ∠ C & = & 32^{\circ} \\ \bar{B C} & = & 15 feet \\ \bar{A B} & = & X \end{array} \end{aligned}$
$\begin{align*}\begin{array}{rcl} \angle C &=& 32^\circ \\ \overline{BC} &=& 15 \ \text{feet} \\ \overline{AB} &=& X \end{array} \end{align*}$
::C=32BC=15英尺AB=X

Next, use the reference angle $\begin{align*}C\end{align*}$ to name the sides of the triangle.
::下一步,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 hypotenuse and the opposite. The Sine ratio is the ratio of the opposite side to the hypotenuse.
::这个三角形上显示的两边是下限和相反的。辛尼比是对面与下限的比。

First, write the sine ratio using words.
::首先,用字词写正弦比。

$\begin{aligned} \sin C = \frac{opposite}{hypotenuse} \end{aligned}$
$\begin{align*}\sin C = \frac{\text{opposite}}{\text{hypotenuse}}\end{align*}$
::C=对面的日光化

Next, write the sine ratio using symbols.
::接下来,请使用符号写下正弦比例。

$\begin{aligned} \sin C = \frac{A B}{B C} \end{aligned}$
$\begin{align*}\sin C = \frac{AB}{BC}\end{align*}$
::C=ABBC 缩略语

Next, fill all known values into the equation. $\begin{align*}\angle C=32^\circ; \ \overline{AB}=X;\ \overline{BC}=15\end{align*}$ .
::下一步,将所有已知值填入方程。 @C=32;AB=X;BC=15。

$\begin{align*}\sin 32^\circ = \frac{X}{15}\end{align*}$
:三) 三、三、十五、四、五、六、六、六、六、六、六、六、六、六、六、六、六、六、七、六、六、六、七、六、七、六、七、七、七、七、七、七、七、七、七、七、七、十、十、十、十、十、十、十、十、十、十、十、十、十、十、十、十、十、十、十、十、十、十、十、十、十、十、十、十、十、十、十、十、十、十、十、十、十、十、十、十、十、十、十、十、十、十、十、十、十、十、十、十、十、十、十、十、十、十、十、十、十、十、十、十、十、十、十、十、十、十、十、十、十、十、十、十、十、十、十、十、十、十、十、十、十、十、十、十、十、十、十、十、十、十、十、十、十、十、十、

Next, use the TI-calculator to find the value of $\begin{align*}\sin 32^\circ\end{align*}$ .
::接下来,请使用 TI 计算器来寻找 sinQQQQQQ的值。

$\begin{aligned} \sin 32 = 0.5299 \end{aligned}$
$\begin{align*}\sin 32=0.5299\end{align*}$
::第32条=0.5299

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

$\begin{aligned} \begin{array}{rcl} \sin 32^{\circ} & = & \frac{X}{15} \\ 0.5299 & = & \frac{X}{15} \end{array} \end{aligned}$
$\begin{align*}\begin{array}{rcl} \sin 32^\circ &=& \frac{X}{15} \\ 0.5299 &=& \frac{X}{15} \end{array}\end{align*}$
::X150.5299=X15

Next, multiply both sides of the equation by 15 to solve for the variable.
::接下来,将方程的两边乘以 15 来解答变量。

$\begin{aligned} \begin{array}{rcl} 0.5299 & = & \frac{X}{15} \\ 15 (0.5299) & = & 15 (\frac{X}{15}) \\ 15 (0.5299) & = & \overset{1}{15} (\frac{X}{15}) \\ 7.9485 & = & X \\ 7.95 & = & X \end{array} \end{aligned}$
$\begin{align*} \begin{array}{rcl} 0.5299 &=& \frac{X}{15} \\ 15(0.5299) &=& 15 \left( \frac{X}{15} \right) \\ 15(0.5299) &=& { \overset{1} {\cancel{15}} } \left ( \frac{X}{ \cancel{15}} \right) \\ 7.9485 &=& X \\ 7.95 &=& X \end{array}\end{align*}$
::0.5299=X151515(0.5299)=15(X151515(0.5299)=151(X157.9485=X7.95=X)

The answer is 7.95.
::答案是7.95。

The length of the opposite side of the right triangle is 7.95 feet.
::右三角形对面的长度为7.95英尺。

Example 4
::例4

For the following mathematical solution, using the sine ratio to determine the length of the hypotenuse of the right triangle, briefly tell what is happening in each part of the solution.
::对于以下的数学解决方案,使用正弦比来确定右三角的下限长度,简要描述解决方案的每个部分正在发生的情况。

$\begin{aligned} \sin B = \frac{opposite}{hypotenuse} \end{aligned}$
$\begin{align*}\sin B = \frac{\text{opposite}}{\text{hypotenuse}}\end{align*}$
::Sin B = 对面日光化

$\begin{aligned} \sin B = \frac{A C}{B C} \end{aligned}$
$\begin{align*}\sin B = \frac{AC}{BC}\end{align*}$
::b=ACBC 缩略语

$\begin{aligned} ∠ B = 36^{\circ}; \bar{A B} = 11 c m; \bar{B C} = X \end{aligned}$
$\begin{align*}\angle B=36^\circ;\ \overline{AB}=11 \ cm; \ \overline{BC}=X\end{align*}$
::*B=36;AB=11厘米;BC=X

$\begin{aligned} \begin{array}{rcl} \sin 36^{\circ} & = & \frac{11}{X} \\ 0.5878 & = & \frac{11}{X} \\ X (0.5878) & = & X (\frac{11}{X}) \\ 0.5878 X & = & \overset{1}{X} (\frac{11}{X}) \\ 0.5878 X & = & 11 \\ \frac{0.5878 X}{0.5878} & = & \frac{11}{0.5878} \\ X & = & 18.71 \end{array} \end{aligned}$
$\begin{align*}\begin{array}{rcl} \sin 36^\circ &=& \frac{11}{X} \\ 0.5878 &=& \frac{11}{X} \\ X(0.5878) &=& X \left( \frac{11}{X} \right) \\ 0.5878 X &=& { \overset{1} {\cancel{X}} } \left ( \frac{11}{ \cancel{X}} \right) \\ 0.5878 X &=& 11 \\ \frac{ \cancel{0.5878} X}{\cancel{0.5878}} &=& \frac{11} {0.5878} \\ X &=& 18.71 \end{array}\end{align*}$
::11X0.5878=11XX(0.5878)=X(11X)0.5878X=X1(11X)0.5878X=110.5878X0.5878X=110.5878X0.58787878=110.5878X=18.71

The answer is 18.71 cm.
::答案是18.71厘米

The length of the hypotenuse of the right triangle is 18.71 cm.
::右三角形的下限长度为18.71厘米。

Review
::回顾

Use a calculator to find each Sine. You may round to the nearest hundredth.
::使用计算器查找每条Sine。您可以绕到最近的第一百个。

1. Sine 55°
::1. 55度西内

2. Sine 25°
::2. 平线25度

3. Sine 11°
::3. 11度西内

4. Sine 60°
::4. 60度西内

5. Sine 75°
::5. 西内75度

6. Sine 12°
::6. 12度西内

7. Sine 29°
::7. 松9度29度

8. Sine 15°
::8. 15度西内

Use the information given and what you have learned about trigonometric ratios to figure out the measure of each missing side. Round your answer to the nearest hundredths place.
::使用给定的信息和您了解的关于三角比的信息来计算每个缺失方的度量。将答案转至最近的一百个位置。

9. Sine angle $\begin{align*}D \ 2^\circ\end{align*}$ , hypotenuse – 12, what is the length of the opposite side?
::9. Sine 角度 D 2 , 下限 - 12, 反面的长度是多少?

10. Sine angle $\begin{align*}E \ 65^\circ\end{align*}$ , hypotenuse – 8, what is the length of the opposite side?
::10. Sine 角度E 65, 下限8, 反面的长度是多少?

11. Sine angle $\begin{align*}F \ 45^\circ\end{align*}$ , hypotenuse – 2, what is the length of the opposite side?
::11. Sine 角度F 45, 下限-2,对面的长度是多少?

12. Sine angle $\begin{align*}D \ 25^\circ\end{align*}$ , hypotenuse – 10, what is the length of the opposite side?
::12. Sine 角度D 25, 下限- 10, 反面的长度是多少?

13. Sine angle $\begin{align*}D \ 80^\circ\end{align*}$ , hypotenuse – 8, what is the length of the opposite side?
::13. Sine 角D 80 80 , 下限8, 反面的长度是多少?

14. Sine angle $\begin{align*}D \ 45^\circ\end{align*}$ , hypotenuse – 5, what is the length of the opposite side?
::14. Sine 角度 D 45, 下限-5, 反面的长度是多少?

15. Sine angle $\begin{align*}D \ 40^\circ\end{align*}$ , hypotenuse – 18, what is the length of the opposite side?
::15. Sine 角度D 40°Z, 下限 -18, 反面的长度是多少?

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

Sine Ratio ::线间比比

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)

Sine Ratio
::线间比比

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)