7.4 逆三角比-interactive

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  逆三角比

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  Trigonometry - Solving for Missing Angles
  ::三角测量 - 解决缺失角

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  The trigonometric ratios can all be defined in terms of ratios of the sides of a right triangle. Since all right triangles obey , as long as the angles of two right triangles are the same, their sides are proportional. Because of this, the ratio of one side to another will always be the same. 
  ::三角比可以按照右三角的两边比率来定义。 由于所有右三角都服从 , 只要两个右三角的角相同, 三角比是成比例的 。 因此, 一方对另一方的比例总是一样 。

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  • The sine of an angle in a right triangle is the ratio of the side opposite the angle to the hypotenuse .
    ::右三角形角的正弦值是角对面的侧面与下限的比。
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  • The cosine of an angle in a right triangle is the ratio of the side adjacent to the angle to the hypotenuse.
    ::右三角形角的余弦值是角与下角的相邻边比。
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  • The tangent of an angle in a right triangle is the ratio of the side opposite the angle to the side adjacent to the angle.
    ::右三角形角的正切值是角对面方与角对面方对面方与角对面方之比。

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  You can use the trigonometric ratios to find missing sides of right triangles when given at least one side length and one angle measure.
  ::您可以使用三角比来找到右三角形的缺失边, 当给定至少一个边长和一个角测量值时 。

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  Consider the right triangle below.
  ::考虑下方的右三角形。

  

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  From this triangle,  you know how to determine all three trigonometric ratios for both $\begin{array}{r}\mathrm{\angle }L\end{array}$ and $\begin{array}{r}\mathrm{\angle }N\end{array}$ .  You can write  $\begin{array}{r}\sin \theta =\frac{\text{opposite}}{\text{hypotenuse}}=\frac{7}{25}.\end{array}$
  ::从这个三角形上, 你知道如何确定所有三个三角比, 包括L和N。 您可以写 sin oppositehypotenuse=725 。

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  But this doesn't help much  with finding the measure of the angle $\begin{array}{r}\theta \end{array}$ .
  ::但这对寻找角度的度量没有多大帮助。

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  Inverse Trig Ratios: Arcsin, Arccos, Arctan
  ::逆三角比率: Arcsin, Arcccos, Arctan

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  T o determine the angle measure when at least two sides of a right triangle are known, we use the inverse trigonometric ratios. The particular ratio that should be used depends on which two sides are known. For example, if you know the hypotenuse and the side opposite the angle in question, you could use the inverse sine ratio. If you know the side opposite and the side adjacent to the angle in question, the inverse tangent ratio is used.
  ::当已知右三角至少两边时,我们使用反三角比来确定角度。应该使用的特定比取决于已知的两边。例如,如果您知道该角的下限和对面,您可以使用反正弦比。如果您知道该角的对面和对面,则使用反正弦比。如果您知道该角的对面和对面,则使用反正弦比。

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  • The inverse sine of a ratio gives the angle in a right triangle whose sine is the given ratio . Inverse sine is also called arcsine and is labeled $\begin{array}{r}{\sin }^{-1}\end{array}$ or arcsin.
    ::比例的逆正弦给出了右三角角的角, 右三角角的正弦是给定比例。 逆正弦也被称为弧素, 并标有 sin - 1 或 arcsin 。
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  • The inverse cosine of a ratio gives the angle in a right triangle whose cosine is the given ratio . Inverse cosine is also called arccosine and is labeled $\begin{array}{r}{\cos }^{-1}\end{array}$ or arccos.
    ::比例的反余弦给出了右三角的角, 右三角的余弦是给定的比例。 反余弦也被称为弧辛, 并被标为cos-1 或arcccos 。
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  • The inverse tangent of a ratio gives the angle in a right triangle whose tangent is the given ratio. Inverse tangent is also called arctangent and is labeled $\begin{array}{r}{\tan }^{-1}\end{array}$ or arctan.
    ::比例的反正切值给出了右三角的角, 右三角的正切值是给定的比例。 反正正切值也被称为正切值, 并被标为 tan-1 或 arctan 。

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  Note that in each case the “-1” is to indicate inverse , and is not an exponent. Hence the expression $\begin{array}{r}{\tan }^{-1}\theta \end{array}$ is not the same as $\begin{array}{r}\frac{1}{\tan \theta }\end{array}$ .
  ::请注意,在每种情况下,“-1”是指反向表示,而不是表率,因此,“tan-1”一词与“-1”不相同。

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  To find the measure of an angle using an inverse trigonometric ratio, you will need to use your calculator. Most scientific and graphing calculators have buttons that look like $\begin{array}{r}[{\sin }^{-1}],[{\cos }^{-1}]\end{array}$ , and $\begin{array}{r}[{\tan }^{-1}]\end{array}$ . You will want to make sure that your calculator is in degree mode so that the angle measure that the calculator produces is in degrees.
  ::要使用反三角比来找到角度的度量, 您需要使用您的计算器。 大多数科学和图形计算器都有类似 [sin-1] 、 [cos-1] 、 [cos-1] 和 [tan-1] 的按钮。 您需要确保您的计算器处于度度模式, 以便计算器产生的角度量以度表示 。

  Trigonometric ratios            Inverse trigonometric ratios

  $\begin{array}{r}\sin (\theta )=\frac{\text{opposite}}{\text{hypotenuse}}\end{array}$ $\begin{array}{r}\to \end{array}$ $\begin{array}{r}{\sin }^{-1}\left(\frac{\text{opposite}}{\text{hypotenuse}}\right)=\theta \end{array}$

  $\begin{array}{r}\cos (\theta )=\frac{\text{adjacent}}{\text{hypotenuse}}\end{array}$ $\begin{array}{r}\to \end{array}$ $\begin{array}{r}{\cos }^{-1}\left(\frac{\text{adjacent}}{\text{hypotenuse}}\right)=\theta \end{array}$

  $\begin{array}{r}\tan (\theta )=\frac{\text{opposite}}{\text{adjacent}}\end{array}$        $\begin{array}{r}\to \end{array}$ $\begin{array}{r}{\tan }^{-1}\left(\frac{\text{opposite}}{\text{adjacent}}\right)=\theta \end{array}$

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  CK-12 PLIX Interactive
  ::CK-12 PLIX 交互式互动

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  Inverse Trigonometric Ratios
  ::逆三角比

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  Solving for an Angle
  ::角的溶解

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  1. Solve for $\begin{array}{r}\theta \end{array}$ .
  ::1. 解决_______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

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  Since  you  know the length of the side opposite angle ' $\begin{array}{r}\theta \end{array}$ ' is 10, and the hypotenuse is 22, we can use the inverse sine ratio to determine the angle measure of ' $\begin{array}{r}\theta \end{array}$ '.
  ::既然你知道对角“”的长度是10, 而下限是22, 我们可以使用逆正弦比来确定“”的角度量。

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  $\begin{array}{rl}\sin \theta & =\frac{\text{opposite}}{\text{hypotenuse}}=\frac{BC}{AB}=\frac{10}{22}\\ \theta & ={\sin }^{-1}\left(\frac{10}{22}\right)\\ \theta & \approx {27.04}^{\text{o}}\end{array}$
  ::=BCAB=1022sin -1(1022)\\\27.04o

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  2. Find $\begin{array}{r}m\mathrm{\angle }B\end{array}$ . 
  ::2. 寻找 mB.

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  Since we know the length of the side opposite   $\begin{array}{r}\mathrm{\angle }B\end{array}$ is 15 and the side adjacent is 8, we can use the inverse tangent ratio to determine the measure of $\begin{array}{r}\mathrm{\angle }B.\end{array}$
  
  ::由于我们知道B区对面的长度是15,而相邻的距离是8,我们可以使用反正正切比来确定B区的度量。

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  3. Find $\begin{array}{r}m\mathrm{\angle }A\end{array}$ . 
  ::3. 寻找maA。

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  You only need to know two sides of the triangle in order to find the measure of one of the angles. Since you are given all three sides, you can choose which two sides you want to use.
  ::您只需要知道三角形的两边就可以找到一个角度的度量。 既然您得到了所有三个角度的度量, 您可以选择您想要使用的两边 。

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  Here we are choosing the length of the side adjacent to  $\begin{array}{r}\mathrm{\angle }A\end{array}$ and the hypotenuse , we can use the inverse cosine ratio to determine the measure of $\begin{array}{r}\mathrm{\angle }A\end{array}$ .
  ::在这里,我们选择了 A 附近的边的长度 和下丘, 我们可以使用逆余弦比 来决定 A 的量度 。

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  Examples
  ::实例实例实例实例

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  Example 1
  ::例1

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  The maximum slope of a wheelchair ramp is 1:12. For a wheelchair ramp made with these specifications, what angle does the ramp make with the flat ground?
  ::轮椅坡道的最大斜坡是1:12。 对于以这些规格制成的轮椅坡道,坡道与平坦地面的角是什么?

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  Draw a picture to represent this situation.
  ::绘制图片来代表这种情况 。

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  Since  you know the length of the side opposite angle $\begin{array}{r}\theta \end{array}$ is 1, and the side adjacent is 12, you can use the inverse tangent ratio to determine the angle measure of $\begin{array}{r}\theta \end{array}$ .
  ::由于您知道对角的长度 为 1, 相邻的一面为 12, 您可以使用反正正切比来确定 的角度量 。

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  $\begin{array}{rl}\tan \theta & =\frac{\text{opposite}}{\text{adjacent}}=\frac{1}{12}\\ \theta & ={\tan }^{-1}\frac{1}{12}\\ \theta & \approx {4.76}^{\text{o}}\end{array}$
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  The wheelchair ramp makes approximately a $\begin{array}{r}{4.76}^{\text{o}}\end{array}$ angle with the ground.
  ::轮椅坡道与地面的角大约为4.76o。

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  Example 2
  ::例2

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  Use the triangle below to find
  ::使用下方三角形查找

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  1. $\begin{array}{r}m\mathrm{\angle }A\end{array}$
     ::ma
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  2. $\begin{array}{r}m\mathrm{\angle }B\end{array}$
     ::mB
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  3. $\begin{array}{r}AB\end{array}$
     ::AB

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  i) Since we know the length of the side opposite   $\begin{array}{r}\mathrm{\angle }A\end{array}$ is 3.5 and the side adjacent is 6.2, we can use the inverse tangent ratio to determine the measure of $\begin{array}{r}\mathrm{\angle }A\end{array}$ .
  :一) 由于我们知道QA对面的长度是3.5,而相邻的一侧是6.2,我们可以使用反正正切比来确定QA的度量。

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  $\begin{array}{rl}\tan A& =\frac{\text{opposite}}{\text{adjacent}}=\frac{BC}{AC}=\frac{3.5}{6.2}\\ m\mathrm{\angle }A& ={\tan }^{-1}\frac{3.5}{6.2}\\ m\mathrm{\angle }A& \approx {29.45}^{\text{o}}\end{array}$
  ::a=BCAC=3.56200m_A=tan-13.56200m_A2945o

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  ii) $\begin{array}{r}\mathrm{\angle }A\end{array}$ and $\begin{array}{r}\mathrm{\angle }B\end{array}$ are complementary:
  :二) A和B是相辅相成的:

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  $\begin{array}{rl}m\mathrm{\angle }B& ={90}^{\text{o}}-m\mathrm{\angle }A\\ & ={90}^{\text{o}}-{29.45}^{\text{o}}\\ & ={60.55}^{\text{o}}\end{array}$
  ::mB=90o-mA=90o-2945o=60.55o

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  You could also use inverse tangent to find $\begin{array}{r}m\mathrm{\angle }B\end{array}$ .
  ::您也可以使用反正切来找到 mB 。

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  iii) With all angle measures and two sides, you could use sine, cosine, or the Pythagorean Theorem to find $\begin{array}{r}AB\end{array}$ . Using the Pythagorean Theorem:
  :三) 利用所有角度措施和两面,可以使用正弦、cosine或毕达哥里安理论来找到AB。

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  $\begin{array}{rl}{AB}^2& ={BC}^2+{AC}^2\\ {AB}^2& ={3.5}^2+{6.2}^2\\ {AB}^2& =12.25+38.44\\ {AB}^2& =50.69\\ AB& \approx 7.12\end{array}$
  ::AB2=BC2+BC2+AC2AB2=3.52+6.22AB2=12.25+38.44AB2=50.69AB_7.12

  Summary

  播放段落 The inverse trigonometric ratios : 播放段落 $\begin{array}{r}\arcsin ={\sin }^{-1}\left(\frac{\text{opp}}{\text{hyp}}\right)=\theta \end{array}$   ::arcsin = sin - 1 (opphyp) 播放段落 $\begin{array}{r}\arccos ={\cos }^{-1}\left(\frac{\text{adj}}{\text{hyp}}\right)=\theta \end{array}$   ::arccos=cos- 1 (adjhyp) 播放段落 $\begin{array}{r}\arctan ={\tan }^{-1}\left(\frac{\text{opp}}{\text{adj}}\right)=\theta \end{array}$   ::arctan=tan-1(oppadj) ::逆三角比: arcsin=sin-1(opphyp)arccos=cos-1(adjhyp)arctan=tan-1(oppadj)______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

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  Review
  ::审查审查审查审查

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  1. What is the difference between $\begin{array}{r}{\sin }^{-1}\end{array}$ and $\begin{array}{r}\sin \end{array}$ ?
  ::1. 罪与罪之间有什么区别?

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  2. When do you use regular trigonometric ratios and when do you use inverse trigonometric ratios?
  ::2. 何时使用正常三角比率,何时使用逆三角比率?

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  In questions 3 - 9 solve for $\begin{array}{r}\theta \end{array}$ . Round your answers to the nearest hundredths place.
  ::问题3 - 9 解答 。 将您的答复转至最近的一百个地方 。

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  10. How could you have found $\begin{array}{r}\theta \end{array}$ in #9 without using an inverse trigonometric ratio?
  ::10. 在9号中,没有使用逆三角比,你怎么会发现?

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  In questions 11 - 13, find all missing information (sides and angles) for each triangle. Round your answers to the nearest hundredths place.
  ::在问题11 - 13中, 查找每个三角形的所有缺失信息( 边和角度) 。 将您的答复转至最近的一百个位置 。

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  14. How are inverse cosine and inverse sine of the same ratio related?
  ::14. 同一比率的反余弦和反正弦如何相关?

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  15. How are inverse tangents of $\begin{array}{r}a\end{array}$ and $\begin{array}{r}\frac{1}{a}\end{array}$ related?
  ::15. A和1a的反正相切关系如何?

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  16. Consider the graphs of the following lines.  How does the slope change from line to line? How does this impact the graphs of the lines? Is it true that if we double the slope, the angle of intersection with the x-axis also doubles? Why or why not?
  ::16. 考虑下几行的图表:斜坡如何从线向线?这如何影响线的图?如果我们将斜坡翻一番,那么与x轴相交的角也翻一番是真的吗?为什么或为什么不是?

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  $\begin{array}{r}\begin{array}{l}y=\frac{1}{4}x\\ \\ y=\frac{1}{2}x\\ \\ y=x\\ \\ y=2x\\ \\ y=4x\\ \\ y=8x\end{array}\end{array}$
  ::y=14xy=12xy=xy=2xy=4xy=8x

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  17. Consider a line with a slope of 100. Rounding to two decimal places, what is the angle of intersection of this line with the x-axis? Is there any limit to how steep a line can get? Describe the steepness in terms of slope and in terms of the angle with the x-axis.
  ::17. 以100的斜坡乘以四舍五入到小数点后两个位数,该直线与 X 轴相交的角是什么?一条直线的斜坡和X 轴角的陡度是否有限制?描述一下斜坡和角与 X 轴的陡度。

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  18. Use a calculator to compute $\begin{array}{r}\tan {90}^{\text{o}}\end{array}$ . What do you observe? Explain this result.
  ::18. 使用计算器计算 tan 90o。您观测到什么? 请解释此结果 。

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  Review (Answers)
  ::审查(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 。