7.3 补充角的松树和脊弦-interactive

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• 选择节补充角的松弦和余弦

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  补充角的松弦和余弦

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  Two angles are said to be complementary if their sum is $\begin{array}{r}{90}^{\text{o}}.\end{array}$   Therefore,   $\begin{array}{r}\theta \end{array}$ and $\begin{array}{r}({90}^{\text{o}}-\theta )\end{array}$ are complementary angles. (Note that in a right triangle like the one below,  $\begin{array}{r}\theta \end{array}$  will be an acute angle.)
  ::如果其总和为90o,则两个角度是互补的。因此,和(90o)是互补的角度。 (注意,在右三角形中,如下三角形,将是一个急性角度。 )

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  Recall the definitions of :
  ::回顾以下定义:

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  • The  sine  of an angle in a right triangle is the ratio of the hypotenuse to the side opposite the angle.
    ::右三角形角的正弦值是角对面的下限与侧边之比。
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  • The  cosine of an angle in a right triangle is the ratio of the hypotenuse to the side adjacent  the angle. 
    ::右三角形角的余弦值是角的侧侧的下限比。

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  The Complementary Angle Theorem
  ::补充角定理

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  Consider  right angle triangle $\begin{array}{r}PQR,\end{array}$  like the one in the image above.
  ::考虑右角三角PQR, 和上方图像中的一样 。

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  •   As with any triangle, t he sum of the measures of the three angles is $\begin{array}{r}{180}^{\text{o}}.\end{array}$  This means that $\begin{array}{r}m\mathrm{\angle }P+m\mathrm{\angle }Q+m\mathrm{\angle }R={180}^{\text{o}}.\end{array}$
    ::与任何三角形一样,三个角度的测量总和为180o。这意味着 mP+mmmR=180o。
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  •  Angle $\begin{array}{r}Q\end{array}$  is a right angle, so $\begin{array}{r}m\mathrm{\angle }Q={90}^{\text{o}}.\end{array}$
    ::角 Q 是一个右角度, 所以 m 90o 。
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  •   $\begin{array}{r}{180}^{\text{o}}-m\mathrm{\angle }Q={90}^{\text{o}},\end{array}$  therefore, $\begin{array}{r}m\mathrm{\angle }P+m\mathrm{\angle }R={90}^{\text{o}}.\end{array}$
    ::因此,180-m90o, mP+mR=90o。
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  •   $\begin{array}{r}\mathrm{\angle }P\end{array}$  and $\begin{array}{r}\mathrm{\angle }R\end{array}$  are complementary.
    ::“P”和“R”是相辅相成的。

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  For the reference angle $\begin{array}{r}\theta :\end{array}$
  ::对于参考角 :

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    $\begin{array}{r}\sin \theta =\frac{QR}{PR},\end{array}$  and  $\begin{array}{r}\cos \theta =\frac{PQ}{PR}\end{array}$
  ::和COSQPR 和COSQPR

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  For the reference angle $\begin{array}{r}({90}^{\text{o}}-\theta ):\end{array}$
  ::用于参考角( 90o) :

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    $\begin{array}{r}\sin ({90}^{\text{o}}-\theta )=\frac{PQ}{PR},\end{array}$  and  $\begin{array}{r}\cos ({90}^{\text{o}}-\theta )=\frac{QR}{PR}\end{array}$
  ::=PQPR, 和cos(90o)RPR

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  Taken together:
  ::综合起来:

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    $\begin{array}{r}\sin ({90}^{\text{o}}-\theta )=\cos \theta ,\end{array}$ and $\begin{array}{r}\cos ({90}^{\text{o}}-\theta )=\sin \theta \end{array}$
  :90o)=cos, 和cos(90o)=sin

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  In other words, the sine of any acute angle is equal to the cosine of its complement, and the cosine of any acute angle is equal to the sine of its complement.
  ::换句话说,任何急性角的正弦等于其补充的余弦,任何急性角的余弦等于其补充的正弦。

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  In a right triangle , if $\begin{array}{r}\cos ({40}^{\text{o}}+x)=\sin {30}^{\text{o}}\end{array}$ , then what is the value of $\begin{array}{r}x\end{array}$ ?
  ::在右三角形中,如果cos(40o+x)=sin30o,x的值是多少?

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  The value of $\begin{array}{r}x\end{array}$ is .
  ::x 值为 。

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  Complementary Angle Theorem Proof
  ::补充角定理校验

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  Consider the right triangle below. Note  the sine and cosine of angles $\begin{array}{r}A\end{array}$ and $\begin{array}{r}B\end{array}$ in terms of sides  $\begin{array}{r}a,b,\end{array}$  and $\begin{array}{r}c.\end{array}$
  ::考虑下方的右三角形。 注意角度A和角度B的正弦和连弦 a、 b 和 c。

  

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  $\begin{array}{r}\sin A=\frac{a}{c},\cos A=\frac{b}{c}\phantom{\underset{X}{\overset{X}{\sum }}}\\ \sin B=\frac{b}{c},\cos B=\frac{a}{c}\phantom{\underset{X}{\overset{X}{\sum }}}\end{array}$
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  N otice that $\begin{array}{r}\sin A=\cos B,\end{array}$   and $\begin{array}{r}\sin B=\cos A\end{array}$  
  ::注意罪与罪与罪

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  In the same triangle, how is $\begin{array}{r}\mathrm{\angle }A\end{array}$ related to $\begin{array}{r}\mathrm{\angle }B\end{array}$ ?
  ::在同一三角形上,ZA和B有什么关系?

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  •  The sum of the measures of the three angles in a triangle is $\begin{array}{r}{180}^{\text{o}}.\end{array}$  This means that $\begin{array}{r}m\mathrm{\angle }A+m\mathrm{\angle }B+m\mathrm{\angle }C={180}^{\text{o}}.\end{array}$  
    ::三角形中三个角度的测量总和为 180o。 这意味着 mA+mB+mC=180o 。
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  •   $\begin{array}{r}\mathrm{\angle }C\end{array}$ is a right angle so $\begin{array}{r}m\mathrm{\angle }C={90}^{\circ }\end{array}$ . Therefore, $\begin{array}{r}m\mathrm{\angle }A+m\mathrm{\angle }B={180}^{\text{o}}\text{- }{90}^{\text{o}}={90}^{\text{o}}.\end{array}$
    ::C是一个右角, 所以 mC=90 。 因此, mA+mB=180o- 90o=90o。
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  •  Angles $\begin{array}{r}A\end{array}$ and $\begin{array}{r}B\end{array}$ are complementary angles because their sum is $\begin{array}{r}{90}^{\text{o}}.\end{array}$
    ::A和B是互补角度 因为它们的总和是90o
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  •  From above:  $\begin{array}{r}\sin A=\cos B\end{array}$ and $\begin{array}{r}\sin B=\cos A\end{array}$ .
    ::上面写着:sinA=cosB 和sinB=cosA。

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  You just verified the Complementary Angle Theorem :
  ::您刚刚验证了补充角度定理 :

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  The sine of an angle is equal to the cosine of its complementary angle, and t he cosine of an angle is equal to the sine of its complementary angle.  
  ::一个角的正弦等于其互补角的余弦,一个角的余弦等于其互补角的正弦。

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  Find $\begin{array}{r}\sin {80}^{\text{o}}\end{array}$  and $\begin{array}{r}\cos {10}^{\text{o}}.\end{array}$  Explain the result.
  ::找到sin80o和cos10o 解释一下结果

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  $\begin{array}{r}\sin {80}^{\text{o}}\approx 0.985\end{array}$ and $\begin{array}{r}\cos {10}^{\text{o}}\approx 0.985\end{array}$ .
  $\begin{array}{r}\sin {80}^{\text{o}}=\cos {10}^{\text{o}}\end{array}$ because $\begin{array}{r}{80}^{\text{o}}\end{array}$ and $\begin{array}{r}{10}^{\text{o}}\end{array}$ are complementary angle measures.
  $\begin{array}{r}\sin {80}^{\text{o}}=\cos {10}^{\text{o}}=\frac{BC}{AB}\end{array}$ are the ratios of the same sides of a right triangle, as shown below.
  ::80o 0.985 和 cos 10o 0.985. sin 80o =cos 10o 因为80o 和 10o 的补充角度度量。sin 80o =cos 10o = BCAB 是右三角形的同一边的比,如下所示。

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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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  $\begin{array}{r}\mathrm{△}ABC\end{array}$ is a right triangle with $\begin{array}{r}m\mathrm{\angle }C={90}^{\text{o}}\end{array}$ and $\begin{array}{r}\sin A=k\end{array}$ . What is $\begin{array}{r}\cos B\end{array}$ ?
  ::ABC是一个右三角形,与 mC=90oand sinA=k。 什么是cosB?

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  $\begin{array}{r}\mathrm{\angle }A\end{array}$ and $\begin{array}{r}\mathrm{\angle }B\end{array}$ are complementary because they are the two non-right angles of a right triangle. This means that $\begin{array}{r}\sin A=\cos B\end{array}$ and $\begin{array}{r}\sin B=\cos A\end{array}$ . If $\begin{array}{r}\sin A=k\end{array}$ , then $\begin{array}{r}\cos B=k\end{array}$ as well.
  ::“A”和“B”是互补的,因为它们是右三角的两个非右角度。这意味着sin“A”=“Cos”和“sin”=“B”=“Cos”A。如果sin“A”和“K”是互补的。如果sin“A”和“K”,那么“Cos”也是。

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  Example 2
  ::例2

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  If $\begin{array}{r}\sin {30}^{\text{o}}=\frac{1}{2},\cos \underset{\_}{???}=\frac{1}{2}?\end{array}$
  ::如果罪证是30o=12,好吗?12?

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  The sine and cosine of complementary angles are equal.
  ::互补角度的内在和共性是平等的。

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  $\begin{array}{r}{90}^{\text{o}}\text{- }{30}^{\text{o}}={60}^{\text{o}},\end{array}$ so $\begin{array}{r}{60}^{\text{o}}\end{array}$   is complementary to $\begin{array}{r}{30}^{\text{o}}.\end{array}$
  ::90 - 30o=60o,so 60o是对30o的补充。

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  Therefore, $\begin{array}{r}\cos {60}^{\circ }=\frac{1}{2}.\end{array}$
  ::因此,cos6012。

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  Example 3
  ::例3

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  Consider the right triangle below. Find $\begin{array}{r}\tan A\end{array}$ and $\begin{array}{r}\tan B\end{array}$ .
  ::考虑下方的右三角形。 找到 tanA 和 tanB 。

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  $\begin{array}{r}\tan A=\frac{BC}{AC}=\frac{a}{b}\end{array}$ and $\begin{array}{r}\tan B=\frac{AC}{BC}=\frac{b}{a}\end{array}$ .
  ::TANA=BCAC=AB和TANB=ACBC=BA。

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  The tangent of any angle is opposite over adjacent so $\begin{array}{r}\tan B=\frac{AC}{BC}\end{array}$ and $\begin{array}{r}\tan A=\frac{BC}{AC}\end{array}$ .
  ::任何角的正切值在相邻的对面,所以TanB=ACBC和TanA=BCAC。

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  These are inverses of each other, therefore the tangents of complementary angles are inverses of each other.
  ::这些是互相反射的, 因此互补角度的相切是互相反射的。

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  In general, the tangents of complementary angles are reciprocals .
  ::一般来说,互补角度的相切性是互惠的。

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

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  Sine, Cosine and Tangent of Complementary Angles
  ::补充角的松弦、脊弦和切敏

  Summary

  播放段落 Two angles are said to be complementary if their sum is 90 o . Therefore, $\begin{array}{r}\theta \end{array}$  and $\begin{array}{r}\left({90}^{\circ }-\theta \right)\end{array}$  are complementary angles. ::据说,如果其总和为90o,则两个角度是互补的。因此, 和(90)是互补的角度。 播放段落 $\begin{array}{r}\sin (\theta )=\frac{\text{opp}}{\text{hyp}}\end{array}$   ::sin = opphyp = opphyp = sin = opphyp 播放段落 $\begin{array}{r}\cos (\theta )=\frac{\text{adj}}{\text{hyp}}\end{array}$   ::cos =adjhyp

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  Review
  ::审查审查审查审查

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  1. How are the two non-right angles in a right triangle related? Explain.
  ::1. 右三角形的两个非右角度如何相关?

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  2. How are the sine and cosine of complementary angles related? Explain.
  ::2. 互补角度的正弦和连弦如何相关?

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  3. How are the tangents of complementary angles related? Explain.
  ::3. 互补角度的相切关系如何?

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  Let $\begin{array}{r}A\end{array}$ and $\begin{array}{r}B\end{array}$ be the two non-right angles in a right triangle.
  ::让A和B在右三角形中成为两个非右角度。

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  4. If $\begin{array}{r}\tan A=\frac{1}{2}\end{array}$ , what is $\begin{array}{r}\tan B\end{array}$ ?
  ::4. 如果TanA=12,什么是TanB?

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  5. If $\begin{array}{r}\sin A=\frac{7}{10}\end{array}$ , what is $\begin{array}{r}\cos B\end{array}$ ?
  ::5. 如果sinA=710,什么是CosB?

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  6. If $\begin{array}{r}\cos A=\frac{1}{4}\end{array}$ what is $\begin{array}{r}\sin B\end{array}$ ?
  ::6. 如果CosA=14 什么是罪? B?

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  7. If $\begin{array}{r}\sin A=\frac{3}{5},\end{array}$ then  $\begin{array}{r}\cos B=\frac{3}{5}.\end{array}$ True or False?
  ::7. 如果sinA=35,那么cosB=35。

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  8. Simplify $\begin{array}{r}\frac{\sin A+\cos B}{2}\end{array}$ .
  ::8. 简化sin*A+cos*B2。

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  9. If $\begin{array}{r}\tan A=\frac{2}{3}\end{array}$ what is $\begin{array}{r}\tan B\end{array}$ ?
  ::9. 如果TanA=23,什么是TanB?

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  10. If $\begin{array}{r}\tan B=\frac{1}{5}\end{array}$ , what is $\begin{array}{r}\tan A\end{array}$ ? Which angle is bigger, $\begin{array}{r}\mathrm{\angle }A\end{array}$ or $\begin{array}{r}\mathrm{\angle }B\end{array}$ ?
  ::10. 如果TanB=15,什么是TanA?哪个角度更大,A或B?

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

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  11. $\begin{array}{r}\cos {30}^{\text{o}}=\sin \theta \end{array}$
  ::11,cos30o=sin

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  12. $\begin{array}{r}\sin {75}^{\text{o}}=\cos \theta \end{array}$
  ::12 sin 75o=cos = 12sin 75o=cos = 12sin 75o=cos =cos = 12sin 75o=cos =cos

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  13. $\begin{array}{r}\cos {52}^{\text{o}}=\sin \theta \end{array}$
  ::13,cos52o=sin

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  14. $\begin{array}{r}\sin {18}^{\text{o}}=\cos \theta \end{array}$
  ::14罪 18o=cos

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  15. $\begin{array}{r}\cos {49}^{\text{o}}=\sin \theta \end{array}$
  ::15岁,49o=sin 15岁,49o=sin 15岁,49o=sin

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  16. What is the slope of an increasing line that intersects the $\begin{array}{r}x\text{-}\end{array}$ axis at an acute angle $\begin{array}{r}\theta ?\end{array}$  What is the slope of any line parallel to this line? What is the slope of any line perpendicular to this line? Why?
  ::16. 将X轴交错于急性角的日益扩大的线的斜坡是什么? 与该线平行的任何线的斜坡是什么? 任何与该线垂直的线的斜坡是什么? 为什么?

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  17. Sine is opposite over hypotenuse , and cosine is adjacent over hypotenuse. What is the sine of angle divided by the cosine of the same angle? What do you observe?
  ::17. 松是相对的,高于低温,而松是相邻的。 角的正弦由同一角的余弦分割是什么?你看到了什么?

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  18. In this lesson you explored the sine, cosine and tangent of complements. Use a calculator to explore the sine, cosine and tangent of complements. What do you observe?
  ::18. 在这一教训中,你探讨了补丁的正弦、正弦和正弦。用计算器来探讨补丁的正弦、正弦和正弦。你观察什么?

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  19. Do the patterns you observed for sine, cosine and tangent of complements hold true for $\begin{array}{r}{0}^{\text{o}}\end{array}$ and $\begin{array}{r}{90}^{\text{o}}?\end{array}$  Use a calculator to explore further. Why or why not?
  ::19. 你观察到的正弦、正弦、正弦和正弦补丁的图案是否对0oand 90o是真实的?使用计算器进一步探索。为什么或为什么没有?

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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 。