7.1 重新审视右三角三角三角三角测量-interactive

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  重新审视右三角三角三角三角体

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  Lesson Objectives
  ::经验教训目标

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  • Solve triangles using trigonometric ratios and the .
    ::使用三角比和三角比来解决三角形。
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  • Prove polynomial identities and use them to describe numerical relationships.
    ::验证多面性身份, 并用这些身份描述数字关系 。

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  Introduction: The Train Problem
  ::导言:火车培训问题

  

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  A local bridge that holds the tracks for a large, high-speed passenger train is too low, and trucks keep getting stuck under the bridge. City officials are exploring ways to raise the clearance of the bridge. One option is to raise the height of the bridge. However, one issue with this option is that increasing the angle of the tracks can be problematic for a high-speed train. Trains are incredibly heavy, and steel wheels on a steel track do not have much inherent friction - these factors explain why a train takes a long time between applying the brakes and coming to a stop, and also explain why it's difficult for a train to go up inclined surfaces . For a train to travel uphill, even 2 °, takes a lot of fuel and is highly inefficient. With a few notable exceptions such as a Portuguese railway incline (the steepest in the world) that climbs approximately 13.8°, trains very rarely run on inclines greater than 2°.
  ::连接大型高速客运列车轨迹的当地桥梁太低, 卡车一直被困在桥下。 市政官员正在探索提高桥干线的方法。 一种选择是提高桥的高度。 但是, 其中一个选择是提高铁路的高度。 但是, 这个选择的一个问题是, 提高铁路的角对高速列车来说可能是一个问题。 火车非常重, 钢轨上的钢轮没有多少固有的摩擦 — — 这些因素解释了为什么火车在使用刹车和停车之间需要很长的时间, 也解释了为什么火车很难爬上倾斜的地表。 上山的火车, 需要大量燃料, 并且效率很高。 除了少数显著的例外, 比如葡萄牙铁路( 世界上最陡峭的) 直线( 世界上最陡峭的) 大约13.8 度, 火车很少在超过 2 度的直线上运行 。

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  The bridge is currently flat, with an angle of 0°. If the city officials want to raise the height of the bridge by 3 feet while limiting the incline to 2°, how far back from the bridge will the incline start and how much replacement track will be needed?
  ::这座桥目前是平的,角度为0°。 如果城市官员想将桥的高度提高3英尺,同时将斜度限制在2°,那么离桥的距离会有多远?

   

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  Activity 1: Pythagorean Theorem
  ::活动1:毕达哥里定理

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  The train bridge problem can be solved using t rigonometric ratios. Trigonometric ratios come from the idea of similarity and the Pythagorean Theorem:
  ::火车桥问题可以通过三角比来解决。 三角比来自相似性和毕达哥里安理论的概念:

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  $$\begin{align*}a^2 + b^2 = c^2\end{align*}$$
  ::a2+b2=c2 = c2 =

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  If one triangle is similar to another, they will have proportional sides and equal angles. Thus, all similar right triangles will have the same ratios between corresponding sides. Two common examples of this are 30 ° : 60° : 90° and 45° : 45° : 90° triangles.
  ::如果一个三角形与另一个三角形相似,它们将具有成比例的边角和相等的角。 因此,所有类似的右三角形在对应的边边上将具有相同的比例。 其中两个常见的例子是30°:60°:90°和45°:45°:90°三角形。

  

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  While the ratios in other right triangles do not work out as nicely as they do in the ones above, these ratios can be used to fill in missing information about a given triangle. Use the interactive below to explore how this can be done.
  ::虽然其他右三角的比重不如以上三角的比重好,但这些比重可用于填充某一三角的缺失信息。 使用下面的交互作用来探索如何做到这一点。

   

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  Activity 2: Extending the Pythagorean Theorem
  ::活动2:扩展毕达哥里定理

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  T he Pythagorean Theorem is a tool that can be used to derive new algebraic relationships. For example, this theorem can be used to write the following relationship:
  ::Pytagorean Theorem 是一个可用于生成新代数关系的工具。 例如, 此理论可用于写入以下关系 :

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  $$\begin{align*}{(x^2-y^2)}^{2}+{(2xy)}^{2}={(x^2+y^2)}^{2}\end{align*}$$
  :x2-y2)2+(2xy)2=(x2+y2)2

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  S implifying $\begin{align*}(x^2-y^2)^2\end{align*}$ will result in $\begin{align*}x^4-2x^2y^2+y^4.\end{align*}$ T he expression $\begin{align*}(2xy)^2\end{align*}$ can be rewritten as $\begin{align*}4x^2y^2.\end{align*}$ Combining these expressions on the right side of the equation will produce $\begin{align*}x^4-2x^2y^2+y^4+4x^2y^2,\end{align*}$ which simplifies to $\begin{align*}x^4+2x^2y^2+y^4.\end{align*}$ The expression represents a perfect square trinomial and can be factored to $\begin{align*}(x^2+y^2)^2.\end{align*}$ Double-check this result by multiplying out $\begin{align*}(x^2+y^2)^2.\end{align*}$ T his property is used to state that the square of the sum of two values is equal to the square of the difference plus the square of twice the product of the values. Can you think of another way to prove this property?
  ::简化 (x2- y2) 2 将导致 x4 - 2x2x2y2+y4。 表达式 (2xy) 2 可以重写为 4x2y2 。 将方程式右侧的这些表达式合并将产生 x4 - 2x2y2+y4+4x2+4x2y2, 简化为 x4+2x2- y2+y4。 该表达式代表一个完美的平方三角方形, 并且可以乘以 (x2+y2) 2 。 双倍校验此结果, 乘以乘以乘以( x2+y2) 2 。 此属性用于说明两个值之和的平方等于差的平方, 加上值的两倍的产值。 您能想到另一种方法来证明此属性吗 ?

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  Another property that can be derived from the Pythagorean Theorem is the formula for the graph of a circle.
  ::从 Pythagorena 理论可以得出的另一个属性是圆形图形的公式。

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  Use the interactive below to derive this formula.
  ::使用下面的交互式数据得出此公式 。

   

   

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  Discussion Questions:
  ::讨论问题:

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  1. Name 6 input-output pairs from the function $\begin{align*}x^2+y^2=25.\end{align*}$
     ::从 函数x2+y2=25 中列出 6 对输入输出对 。
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  2. What shape would you expect the equation $\begin{align*}y = \sqrt{r^2 - x^2}\end{align*}$ to take when graphed. Will it form a function?
     
     ::当图形化时, 您期望 yr2 - x2 方程式会采取什么样的形状。 它会形成函数吗 ?

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  Activity 3: Sine, Cosine, and Tangent
  ::活动3:松、和唐

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  Before calculators, mathematicians would use tables to keep track of all the ratios for right triangles. The first trigonometric table dates back to about 1800 BC and is written on an ancient Babylonian tablet known as Plympton 322, named after the discoverer. Over time, what began as a simple table of right triangle ratios was expanded in India, Greece, and many other cultures. Currently, tables are no longer used; instead, mathematicians use calculators that take advantage of infinite series to produce the values that would be found in the table.
  ::在计算之前,数学家会使用表格来跟踪右三角形的所有比率。 第一个三角测量表可追溯到公元前1800年左右,并写在以发现者命名的古代巴比伦平板上,称为Plympton 322。随着时间推移,印度、希腊和许多其他文化都扩大了以右三角比的简单表。目前,表格已不再使用;相反,数学家使用利用无限序列的计算器来生成表中发现的值。

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  Trigonometric functions like $\begin{align*}\sin{x}\end{align*}$ and $\begin{align*}\cos{x}\end{align*}$ can be used to solve for missing sides and angles in right triangles. When $\begin{align*}\sin{(15°)}\end{align*}$ is entered into a calculator, you are asking for the side ratio that produces a 15° angle. The calculator will give the ratio 0.2588, which is not meant to be an actual length. To get the length of one side, use this ratio and a given side length to determine what the proportional missing side length would be . You are essentially using the base of a similar right triangle with a hypotenuse of 1 and scaling it to the proper size.
  ::exinx 和 osx 等三重函数可用于解析右三角形中缺失的边和角。 当将罪( 15°) 输入计算器时, 您正在询问产生 15 ° 角的边比。 计算器将给出0. 2588 的比, 而不是实际长度 。 要获取一边的长度, 请使用此比和给定的边长来确定对应的边长。 您实际上是使用一个类似右三角形的基数, 其下限为 1 , 并缩放到合适的大小 。

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  Use the interactive below to visualize how trigonometry can be seen in a ladder leaning on a building.
  ::利用下面的交互作用来想象三角测量 如何在靠着建筑物的梯子上 能看到三角测量。

   

  The t rigonometric ratios describe the side ratios relative to a given angle. In the right triangle below, opp stands for the side opposite the angle $\begin{align*}\theta\end{align*}$ , hyp stands for the hypotenuse of the right triangle, and adj stands for the side adjacent to the angle $\begin{align*}\theta\end{align*}$ .

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  $$\begin{align*}\sin \theta = \frac{opp}{hyp} \end{align*}$$
  ::一九九九一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一二一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一一

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  $$\begin{align*}\cos \theta = \frac{adj}{hyp} \end{align*}$$
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  $$\begin{align*}\tan \theta = \frac{opp}{adj}\end{align*}$$
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  Example
  ::示例示例示例示例

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  G iven the following triangle, solve for side $\begin{align*}x\end{align*}$ .
  ::根据以下三角形, 解决侧面 x 。

  

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  You can solve for side $\begin{align*}x\end{align*}$ using either $\begin{align*}\angle A\end{align*}$ or $\begin{align*}\angle C\end{align*}$ . The solution below shows how to find x using $\begin{align*}\angle C,\end{align*}$ but you can double-check using $\begin{align*}\angle A.\end{align*}$ S ide x is opposite , you know the length of the hypotenuse is z = 20, and that $\begin{align*}\angle C\end{align*}$ = 35º, so you can set up and solve the following equation for x . Use this information to w rite the following equation:
  ::您可以使用 A 或 C 来解析侧面 x 。 下面的解决方案显示如何使用 C 找到 x, 但是您可以使用 A 进行重复检查。 侧面 x 是对的, 您知道下限的长度是 z = 20 , 而 C = 35o, 您可以为 x 设置和解析以下方程式。 使用此信息来写入以下方程式 :

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  $$\begin{align*}\sin ({35}^{o}) = \frac{x}{20}\\ \end{align*}$$
  ::Sin( 35o) =x20

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  To solve this, multiply both sides by 20 to isolate x.
  ::要解决这个问题,将两边乘以20,分离x。

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  $$\begin{align*}20 \times (\sin ({35}^{o})) &= (\frac{x}{20}) \times 20\\\\ 20 \cdot \sin(35) &= x\end{align*}$$
  ::20xx( 辛( 35o)) = (x20) xx20x2020xx( 35) sin( 35) =x

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  T he expression $\begin{align*}\sin(35),\end{align*}$ means to insert the ratio of the opposite side to the hypotenuse that will produce a 35º angle. Multiplying by 20 will result in the desired scaled side length value.
  ::表达式罪( 35) , 意思是插入相对面与下限之比, 从而产生35o 角度。 乘以 20 将产生所期望的侧长缩放值 。

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  Answer: $\begin{align*}x\approx 11.47 \end{align*}$ units
  ::答复:x11.47个单位

   

   

   

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  Discussion Question (critical thinking): What are some other ways to increase the clearance of the bridge?
  ::讨论问题(批判性思维):还有什么其他方法可以提高桥梁的通航率?

   

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  Extension: Trigonometric Relationships
  ::扩展名: 三角关系

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  Use the interactive below to explore the relationships between sine, cosine, and tangent.
  ::使用下文互动来探索正弦、正弦和正弦之间的关系。

   

   

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  Activity 4: Inverse Trig Ratios
  ::活动4:逆三角比率

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  Example
  ::示例示例示例示例

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  What is the measure of angle X in the diagram below?
  ::下图中角度 X 的度量是多少 ?

  

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  I n this problem, you know the length of two sides and need to find the missing angle. To accomplish this, use inverse trigonometric functions , which do the opposite of regular trig functions. When you enter $\begin{align*}\sin x\end{align*}$ into a calculator, you are asking for a ratio that forms an angle. When you enter the inverse function into a calculator, essentially asking for the angle formed by the ratio.
  ::在此问题上, 您知道两边的长度, 并且需要找到缺失的角度。 要完成此任务, 请使用反向三角函数, 这与普通三角函数相反 。 当您在计算器中输入 sinx 时, 您正在要求一个形成角度的比值。 当您在计算器中输入反向函数时, 基本上要求由比例构成的角 。

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  • $\begin{align*}sin^{-1}(x)\end{align*}$ , or $\begin{align*}\arcsin{(x)}\end{align*}$ , does the opposite of the sine function.
    ::sin- 1(x) , 或 arcsin(x) , 与正弦函数相反 。
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  • $\begin{align*}cos^{-1}(x)\end{align*}$ , or $\begin{align*}\arccos{(x)}\end{align*}$ , does the opposite of the cosine function.
    ::COs- 1(x), 或 arccos(x) , 与余弦函数相反 。
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  • $\begin{align*}tan^{-1}(x)\end{align*}$ , or $\begin{align*}\arctan{(x)}\end{align*}$ , does the opposite of the tangent function.
    ::tan- 1(x) , 或 arctan(x) , 与正切函数相反 。

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  In this problem, you know the length of the adjacent side and the hypotenuse, so you can set up the following equation using the cosine function.
  ::在此问题上, 您知道相邻侧和下限的长度, 这样您就可以使用余弦函数设置以下方程 。

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  $$\begin{align*}\cos(X)=\frac{10}{15}\end{align*}$$
  ::CO(X)=1015

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  To get the $\begin{align*}x\end{align*}$ alone, take the inverse cosine function to both sides. A function will be canceled when the inverse function is applied.
  ::要单独获得 x, 请将反余弦函数移到两侧。 当应用反弦函数时, 函数将被取消 。

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  $$\begin{align*}{\cos}^{-1}(\cos(X))={\cos}^{-1}\left(\frac{10}{15}\right)\end{align*}$$
  ::COs- 1 (cos( X)) =cos- 1( 1015)

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  The function $\begin{align*}{\cos}^{-1}(\cos(x))\end{align*}$ is essentially asking, "what is the angle that will form the ratio formed by the angle x?" The answer is x because the questions cancel each other out.
  ::函数 cos-1(cos(x)) 实质上是问 , “ 将形成角度 x 所构成的比率的角度是什么? ” 答案是 x, 因为问题相互勾销 。

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  $$\begin{align*}X&=cos^{-1}\left(\frac{10}{15}\right)\\\\ X&\approx 48.19\end{align*}$$
  
  ::X=cos- 1( 1015X) 48.19

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  Answer: $\begin{align*}m \angle X \approx 48.19°\end{align*}$
  ::答复:mX48.19°

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  Although $\begin{align*}cos^{-1}(x)\end{align*}$ is the inverse of cosine (or $\begin{align*}\arccos\end{align*}$ ), this is different from $\begin{align*}cos({x})^{-1}.\end{align*}$ The function $\begin{align*}cos(x)^{-1}\end{align*}$ is a reciprocal function and means $\begin{align*}\frac{1}{\cos{x}}.\end{align*}$ This function is called the secant, and is represented using the abbreviation $\begin{align*}\sec{x}.\end{align*}$
  ::虽然 Cos- 1 (x) 是 cosine (或 arcccos) 的反差, 但与 os( x) - 1 不同。 函数 cos( x) - 1 是一种对等函数, 意味着 1cosx 。 此函数被称为 secont, 使用缩写 secx 表示 。

  Function	Abbreviation	Definition
  Cosecant	$\begin{align*}\csc{\theta }\end{align*}$	$\begin{align*}\frac{1}{\sin{\theta }}\end{align*}$
  Secant	$\begin{align*}\sec{\theta }\end{align*}$	$\begin{align*}\frac{1}{\cos{\theta }}\end{align*}$
  Cotangent	$\begin{align*}\cot{\theta }\end{align*}$	$\begin{align*}\frac{1}{\tan{\theta }}\end{align*}$

   

   

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  Discussion Question: Will $\begin{align*}\sin(\csc(x)) = x?\end{align*}$ Why or why not?
  ::讨论问题:犯罪(csc(x)=x?为什么或为什么不是?

  Summary

  播放段落 The Pythagorean Theorem shows a relationship between the three sides of a right triangle. $$\begin{align*}a^2 + b^2 = c^2\end{align*}$$ ::Pythatagorean 理论显示右三角形的三边之间的关系。 a2+b2=c2 播放段落 Two common examples of similar right triangles are $\begin{align*}30^{\circ} : 60^{\circ} : 90^{\circ}\end{align*}$   and $\begin{align*}45^{\circ} : 45^{\circ} : 90^{\circ}.\end{align*}$   ::类似右三角的常见例子有:30:60:90;45:45:90。 播放段落 Trigonometric functions relate the ratio of two sides of a right triangle to an angle of the triangle. 播放段落 $\begin{align*}\sin(\theta) = \frac{\text{opp}}{\text{hyp}}\end{align*}$   ::sin = opphop 播放段落 $\begin{align*}\cos(\theta) = \frac{\text{adj}}{\text{hyp}}\end{align*}$   ::cos =adjhyp 播放段落 $\begin{align*}\tan(\theta) = \frac{\text{opp}}{\text{adj}}\end{align*}$   ::tan = oppadj ::三角函数将右三角形两侧与三角形角的比值联系起来。sin =opphyp cos =adjhyp tan =oppadj 播放段落 Inverse trigonometric functions will “undo” their corresponding trig function. For example, while taking the sine of an angle will give output a ratio, taking the inverse sine of the ratio will output the angle. ::逆三角函数将“ 反” 对应的三角函数 。 例如, 使用角度的正弦值将给输出一个比例, 使用比率的正弦值将输出角 。 播放段落 Reciprocal trigonometric functions have their own names 播放段落 Cosecant: $\begin{align*}\csc(\theta) = \frac{1}{\sin(\theta)}\end{align*}$   ::Cosecant: csc = 1sin 播放段落 Secant: $\begin{align*}\sec(\theta) = \frac{1}{\cos(\theta)}\end{align*}$   ::seccant: 秒 = 1cos 播放段落 Cotangent: $\begin{align*}\cot(\theta) = \frac{1}{\tan(\theta)}\end{align*}$   ::相切性: cot = 1tan ::对等三角函数自有名称 Cosecant: csc =1sin Secant: sec =1cos 共切度: cot =1tan

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  Wrap-Up: Review Questions
  ::总结:审查问题

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  T he video below provides a good overview of right triangle trigonometry and demonstrates some examples of how to solve related problems.
  ::以下影片很好地概述了右三角三角三角测量,并举例说明了如何解决相关问题的一些例子。

   

  ythagorean-theorem" quiz-url="https://www.ck12.org/assessment/ui/embed.html?test/view/5f5784c9da229adeea6a4097&collectionHandle=trigonometry&collectionCreatorID=3&conceptCollectionHandle=trigonometry-:ythagorean-theorem&mode=lite" test-id="5f5784c9da229adeea6a4097">

   

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  Extension: Knitting a Right Triangle
  ::扩展名: 右三角形编织

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  Use the interactive below for more practice with the trigonometric ratios.
  ::在三角比方面,使用下面的交互方法进行更多的实践。