7.13 了解西奈

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  理解西线

  

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  Jacob has to string lights around the eaves of his flat-roofed shed. Since he is afraid of heights he wants to plan this endeavor very carefully. He knows that for safety reasons, a ladder should make an angle of 75° or less with the ground. If Jacob has a ladder 6.5 meters long to place against the side of the shed that is 2.5 meters high, then what angle will the ladder make with the ground? Jacob needs to be certain that the angle meets safety requirements before he puts one foot on the ladder. How can he figure out the measure of the angle between the ladder and the ground?
  ::雅各必须用他的平顶屋顶的棚子的树叶上挂灯。 既然他害怕高高, 他想要非常小心地规划这一努力。 他知道, 出于安全的原因, 梯子应该用地面做一个75°或不到75°的角。 如果雅各有一个6.5米长的梯子, 可以放在2.5米高的棚子边, 那么梯子会用地面做什么角度? 雅各需要确定, 在他把一英尺的梯子放在梯子上之前, 角度符合安全要求 。 他如何找到梯子和地面之间的角 ?

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  In this concept, you will learn to understand the trigonometric ratio of sine.
  ::在这个概念中,你将学会理解正弦的三角比。

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  Sines
  ::Sine 线

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  Trigonometry is a branch of mathematics used to determine the lengths of sides and the measure of angles with great accuracy. You will only consider the results obtained by using right angled triangles.
  ::三角测量是数学的一个分支,用来非常精确地确定边的长度和角度的度量。您将只考虑通过使用右角三角形获得的结果。

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  The side of a right triangle located opposite the 90° angle is the longest side of the triangle and is called the hypotenuse. The two shorter sides of the triangle, often called legs, have specific names in reference to the location of one of the acute angles of the right triangle.
  ::直角为90度角的右三角的侧面是三角形最长的侧面,称为下限。三角形的两个较短的侧面,通常称为双腿,在右三角形的急性角之一的位置方面有具体的名称。

  

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  In both triangles the hypotenuse is opposite the right angle. In the first triangle the angle marked in red is called the reference angle and it is used to name the legs of the triangle. The measure of the reference angle can be given or its measure may have to be calculated. Whatever the case, one of the acute angle will be referred to in the question or problem. The side across from the reference angle is called the opposite side . The third side is called the adjacent side and it is the side of the triangle, along with the hypotenuse, that forms the reference angle.
  ::在两个三角形中, 下角是右角对面的。 在第一个三角形中, 红色标记的角被称为引用角, 用来命名三角形的腿。 引用角的度量可以给定, 也可以计算其量度。 不论情况如何, 问题或问题中都会提到一个急角。 引用角的对面被称作对面的对面。 第三个边被称作相邻边, 是三角形的对面, 加上下角, 构成引用角的对面 。

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  In the second triangle the angle marked in red is called the reference angle and it is used to name the legs of this triangle. The side across from the reference angle is called the opposite side . The third side is called the adjacent side and it is the side of the triangle, along with the hypotenuse, that forms the reference angle.
  ::在第二个三角形中,以红色标记的角被称为参考角,用来命名这个三角形的腿。引用角的对面称为对面。第三边称为相邻的侧面,是三角形的侧面,加上下角,构成参考角的就是三角形的侧面。

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  The location of the hypotenuse never changes but the location of the opposite and adjacent sides depends upon which acute angle is the reference angle.
  ::下限的位置从未改变,但相向和相邻两侧的位置取决于哪个急性角是参考角。

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  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:

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  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)

  

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  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 hypotenuse is known as the s ine ratio . The sine ratio is related to the reference angle of the right triangle, not to the right angle.
  ::右三角形两侧长度之比称为三角比。引用角对面的侧长度与下限长度之比被称为正弦比。正弦比与右三角形的参考角有关,而不是右角。

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  The sine ratio for the first triangle can be written in words as:
  ::第一个三角形的正弦比例可以以下列文字写入:

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  sine of  $\begin{align*}\angle A = \frac{\text{side opposite }\angle A}{\text{hypotenuse}}\end{align*}$  Or in a shortened form as sine of  $\begin{align*}\angle A = \frac{\text{opposite}}{\text{hypotenuse}}\end{align*}$
  ::A的正弦值 = 相对的 亚聚苯或以缩写形式 A = 对面的 亚聚苯的正弦值

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  The sine ratio for the first triangle can be written in symbols as:
  ::第一个三角形的正弦比可以以下列符号写成:

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

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  The sine ratio for the second triangle can be written in words as:
  ::第二个三角形的正弦比例可以以下列文字写入:

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  sine of $\begin{align*}\angle F = \frac{\text{side opposite } \angle F}{\text{hypotenuse}}\end{align*}$  Or in a shortened form as sine of  $\begin{align*}\angle F = \frac{\text{opposite}}{\text{hypotenuse}}\end{align*}$
  ::F=对面的Fhypotenuse 或作为F=对面hypotenuse 的正弦的缩写形式

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  The sine ratio for the first triangle can be written in symbols as:
  ::第一个三角形的正弦比可以以下列符号写成:

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  $\begin{align*}\sin F = \frac{DE}{EF}\end{align*}$
  ::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

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  Let’s determine the value for the sine ratio of each of the acute angles using the following right triangle. Express the sine 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弦比值。 以文字和符号表示正弦比。 然后, 使用相应边上的值, 将符号替换为数字, 先以小数表示比率, 然后以小数四舍五入到最接近的一万位( 小数后四位数)。

  

  

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  For the first triangle:
  ::第一个三角形 :

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  First, with the respect to the reference angle $\begin{align*}A\end{align*}$ , name the sides of the triangle.
  ::首先,关于参考角度A,请列出三角形的侧面。

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  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 。

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  Next, write the sine ratio for $\begin{align*}\angle A\end{align*}$  in all of the required forms.
  ::下一步,用所有要求的表格填入 A 的正弦比例 。

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  Words:
  ::单词 :

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  sine of  $\begin{align*}\angle A = \frac{\text{opposite}}{\text{hypotenuse}}\end{align*}$
  ::A=对面日光化的正弦值

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  Symbols: 
  ::符号 :

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

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  Fraction:
  ::分数 :

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  $\begin{align*}\sin A = \frac{12}{13}\end{align*}$
  ::A=1213

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  Decimal:
  ::十进制 :

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  $\begin{align*}\sin A = 0.9231\end{align*}$
  ::A= 0.9231

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  The second triangle is the first triangle with $\begin{align*}\angle B\end{align*}$  as the reference angle. 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.
  ::第二个三角形是第一个三角形, 以 B 为参考角度。 注意对面和相邻边的位置已经从 A 为参考角度时的位置改变 。

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  First, with the respect to the reference angle $\begin{align*}B\end{align*}$ , name the sides of the triangle.
  ::首先,关于参考角度B,请列出三角形的侧面。

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  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 。

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  Next, write the sine ratio for $\begin{align*}\angle B\end{align*}$  in all of the required forms.
  ::下一步,用所有要求的表格填入 {B} 的正弦比例 。

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  Words:
  ::单词 :

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  sine of  $\begin{align*}\angle B = \frac{\text{opposite}}{\text{hypotenuse}}\end{align*}$
  ::B=对面日光利用的正弦值

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  Symbols:
  ::符号 :

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

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  Fraction:
  ::分数 :

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  $\begin{align*}\sin B = \frac{5}{13}\end{align*}$
  ::B= 513

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  Decimal:
  ::十进制 :

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  $\begin{align*}\sin B = 0.3846\end{align*}$
  ::B=0.3846)\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\

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  If $\begin{align*}\sin B = 0.3846\end{align*}$ , then the measure of $\begin{align*}\angle B\end{align*}$  can be found using the TI-84 calculator.
  ::如果sinB=0.3846,那么使用TI-84计算器可以找到 B 的测量值。

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  Remember you measure angles in units called degrees. Press $\begin{align*}\boxed{\text{mode}}\end{align*}$  and look at the screen.
  ::记住,您用称为度的单位测量角度。 按住模式, 并查看屏幕 。

  

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  In the fourth row of the screen, DEGREE is not highlighted.
  ::在屏幕第四行,DEGREE没有突出显示。

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  Follow the Key Press History below to put the calculator in DEGREE mode.
  ::遵循下面的密钥新闻史将计算器设置为 DEGREE 模式 。

  

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  In the fourth row of the screen, DEGREE is now highlighted.
  ::在屏幕第四行,现在突出显示的是DEGREE。

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  Next, follow the Key Press History below to find the measure of $\begin{align*}\angle B\end{align*}$ .
  ::接下来,跟随下面的主要新闻史 来找到ZZB的量度。

  

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  Then, look at the screen of the calculator:
  ::然后,看看计算器的屏幕:

  

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  Round the measure of the angle to the nearest tenth. 
  ::圆形角的量度为最接近的十分之一。

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

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  The answer is 22.6°
  ::答案是22.6摄氏度

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  Examples
  ::实例

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  Example 1
  ::例1

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  Earlier, you were given a problem about  Jacob and the angle of his ladder. Jacob needs to make sure that the ladder makes an angle of less than 75° with the ground? Jacob will have to use the sine ratio to figure this out.
  ::早些时候,雅各和他梯子角度的问题就给了你。雅各需要确保梯子与地面的角小于75度。雅各必须用正弦比来解决这个问题。

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  First, draw and label a right triangle with the information given.
  ::首先,用给定的信息绘制右三角形并贴上标签。

  

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  Next, name the sides of the triangle with reference to $\begin{align*}\angle A\end{align*}$ .
  ::下一位是三角形的两侧名称 参考 A 。

  

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  Next, express the ratio $\begin{align*}\sin A\end{align*}$  using words.
  ::接下来,用文字来表示 sinA 的比例 。

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

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  Next, express the ratio using symbols.
  ::接下来,用符号表示比例。

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

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  Next, express the ratio as a fraction using the corresponding values of the sides.
  ::接下来,用两侧的相应数值将比率表示成一个分数。

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  $\begin{align*}\sin A = \frac{2.5}{6.5}\end{align*}$
  ::A=2.56.5

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  Next, express the ratio as a decimal. 
  ::接下来,以小数点表示该比率。

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  $\begin{align*}\sin A = 0.3846\end{align*}$
  ::A=0.3846)\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\

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  Next, use the $\begin{align*}\sin^{-1}\end{align*}$  function 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 sin-1 函数来查找 A 的量度 。

  

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  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。

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  $\begin{align*}\angle A = 22.6^\circ\end{align*}$
  ::A=22.6

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  The answer is 22.6°.
  ::答案是22.6摄氏度

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  It is safe for Jacob to go up the ladder.
  ::雅各上梯子是安全的

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  Example 2
  ::例2

  

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  Use the sine ratio to calculate the measure of $\begin{align*}\angle A\end{align*}$  to the nearest tenth.
  ::使用正弦比计算 A 到最近的十分之一的度量。

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  First, using $\begin{align*}\angle A\end{align*}$  name the sides of the triangle.
  ::首先,使用 A 命名三角形的侧面。

  

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  Next, write the sine ratio in words.
  ::接下来,用文字写下正弦比例。

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

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

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

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  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。

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  $\begin{align*}\sin A = \frac{8}{14}\end{align*}$
  ::A=814个

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  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。

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  $\begin{align*}\sin A = 0.5714\end{align*}$
  ::A=0.5714

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  Then, use the inverse sine function $\begin{align*}(\sin^{-1})\end{align*}$  on the TI calculator to find the measure of $\begin{align*}\angle A\end{align*}$ .
  ::然后在 TI 计算器上使用反正正弦函数( sin-1) 来找到 A 的量度 。

  

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  Then, look at the screen and round the measure of the angle to the nearest tenth.
  ::然后,看看屏幕 并绕过角度的度量 到最近的十分之一。

  

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  Notice, it is not necessary to close the bracket after the decimal has been entered into the calculator. 
  ::注意, 在计算器输入小数点后无需关闭括号 。

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  $\begin{align*}\angle A = 34.9^\circ\end{align*}$
  ::A=34.9

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  The answer is 34.9°
  ::答案是34.9摄氏度

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  Example 3
  ::例3

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  For the following right triangle, express $\begin{align*}\sin B\end{align*}$  in words, symbols, as a fraction and as a decimal.
  ::对于以下的右三角形,以文字、符号、小数和小数表示sinB。

  

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  First, name the sides of the triangle using the reference angle $\begin{align*}B\end{align*}$ .
  ::首先,使用参考角度 B 命名三角形的边。

  

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  Next, express the ratio $\begin{align*}\sin B\end{align*}$  using words.
  ::接下来,用文字来表达比例 sin_B 。

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

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  Next, express the ratio using symbols.
  ::接下来,用符号表示比例。

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

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  Next, express the ratio as a fraction using the corresponding values of the sides.
  ::接下来,用两侧的相应数值将比率表示成一个分数。

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  $\begin{align*}\sin B = \frac{8.5}{10.5}\end{align*}$
  ::B=8.510.5

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  Then, express the ratio as a decimal to the ten thousandth. 
  ::然后,用小数点和十万分之一表示这个比率。

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  $\begin{align*}\sin B = 0.8095\end{align*}$
  ::B=0.8095

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  The answer is 0.8095.
  ::答案是0.8095。

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  Example 4
  ::例4

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  If $\begin{align*}\sin A = 0.3872\end{align*}$ , what is the measure of $\begin{align*}\angle A\end{align*}$  to the nearest tenth.
  ::如果sinA=0.3872,那么A的量度是多少? 至最近的十分之一。

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  First, use the TI-calculator and follow the key press history.
  ::首先,使用TI计算器,并遵循主要新闻史。

  

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  Next, look at the screen of the calculator.
  ::接下来,看看计算器的屏幕。

  

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  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。

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  $\begin{align*}\angle A = 22.8^\circ\end{align*}$
  ::A=22.8

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  The answer is 22.8°.
  ::答案是22.8摄氏度

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  Example 5
  ::例5

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  Determine the measure of $\begin{align*}\angle B\end{align*}$  in the following triangle.
  ::在以下三角形中确定 B 的度量。

  

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

  

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  Next, express the ratio $\begin{align*}\sin A\end{align*}$  using words.
  ::接下来,用文字来表示 sinA 的比例 。

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

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  Next, express the ratio using symbols.
  ::接下来,用符号表示比例。

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

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  Next, express the ratio as a fraction using the corresponding values of the sides.
  ::接下来,用两侧的相应数值将比率表示成一个分数。

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  $\begin{align*}\sin A = \frac{6}{12}\end{align*}$
  ::A=612号

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  Next, express the ratio as a decimal.
  ::接下来,以小数点表示该比率。

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  $\begin{align*}\sin A = 0.5\end{align*}$
  ::A=0.5

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  Next, use the $\begin{align*}\sin^{-1}\end{align*}$  function 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 sin-1 函数来查找 A 的量度 。

  

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  Then, write the measure of $\begin{align*}\angle A\end{align*}$ . 
  ::然后写下A的量度

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  $\begin{align*}\angle A = 30^\circ\end{align*}$
  ::A=30

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  The answer is 30°.
  ::答案是30摄氏度

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  Review
  ::回顾

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  Solve each problem.
  ::解决每一个问题。

  

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  1. What is the sine of $\begin{align*}\angle G\end{align*}$ ?
  ::1. 什么是G的必然条件?

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  2. What is the sine of $\begin{align*}\angle H\end{align*}$ ?
  ::2. 什么是`H'的必然?

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  3. Can you find the sine of $\begin{align*}\angle A\end{align*}$ ?
  ::3. 你能找到"A"的正弦吗?

  

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  4. What is the sine of $\begin{align*}\angle R\end{align*}$ ?
  ::4. 什么是`R'的必然?

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  5. What is the sine of $\begin{align*}\angle S\end{align*}$ ?
  ::5. 什么是必要条件?

  

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  6. What is the sine of $\begin{align*}\angle A\end{align*}$ ?
  ::6. 什么是`A'的必然?

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  7. What is the sine of $\begin{align*}\angle B\end{align*}$ ?
  ::7. 什么是B的必要条件?

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  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?

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  Answer each question true or false.
  ::回答每个问题是否真实或虚假。

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  9. You can use the Pythagorean Theorem to find the length of a missing side in a right triangle.
  ::9. 您可以使用毕达哥里安神话在右三角形中找到缺失方的长度。

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  10. A right triangle must have a 90 degree angle.
  ::10. 右三角必须有90度角。

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  11. The sine ratio is the hypotenuse over the opposite side.
  ::11. 必要比例是对面的下限。

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  12. If you know only side length, then you can figure out all the side lengths.
  ::12. 如果你只知道副长度,那么你就可以找出所有副长度。

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  13. Sine ratio has to do with side lengths.
  ::13. 线比与侧长有关。

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  14. The hypotenuse is always opposite the right angle.
  ::14. 低压作用总是与正确的角度相反。

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  15. You must be given all three side lengths to figure out the sine ratio.
  ::15. 要确定正弦比例,必须给您所有三个侧长度。

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  Review (Answers)
  ::回顾(答复)

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  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 。