松和余弦比率

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Here you will learn how to use sine and cosine ratios to find missing sides of right triangles.
::在这里您将学习如何使用正弦和正弦比来找到右三角形的缺失边。

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Two right triangles with one pair of non-right congruent angles are similar by $\begin{array}{r}AA\sim \end{array}$ . This means the ratio between the side lengths of the first triangle must be congruent to the ratio between the corresponding side lengths of the second triangle.
::两个右三角形与一对非右对齐角度相似。 这意味着第一个三角形的侧长比必须与第二个三角形的对应侧长比相匹配。

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 For example, in the image below, $\begin{array}{r}\frac{a}{c}=\frac{d}{f}:\end{array}$
::例如,在下面的图像中, ac=df:

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Because there are three pairs of sides for any triangle, there are three relevant ratios for a given angle.
::因为任何三角形的边有三对对, 某一角度有三对相关比例 。

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1. The tangent ( ) of an angle gives the ratio $\begin{array}{r}\frac{\text{opposite leg}}{\text{adjacent leg}}\end{array}$   $\begin{array}{r}\phantom{[}\end{array}$  
   ::角的正切值() 表示腿部对面的比 [
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2. The sine ( ) of an angle gives the ratio $\begin{array}{r}\frac{\text{opposite leg}}{\text{hypotenuse}}\end{array}$   $\begin{array}{r}\phantom{[}\end{array}$    
   ::角的正弦 () 表示腿部对面的比 [
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3. The cosine ( ) of an angle gives the ratio $\begin{array}{r}\frac{\text{adjacent leg}}{\text{hypotenuse}}\end{array}$   $\begin{array}{r}\phantom{[}\end{array}$  
   ::角的余弦 () 给出了相邻的支流比 [

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SohCahToa?
::索加托亚?

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SOH-CAH-TOA stands for:  S ine equals O pposite over H ypotenuse, C osine equals A djacent over H ypotenuse, and T angent equals O pposite over A djacent. This mnemonic may help you remember which sides are associated with which trigonometric ratio.
::SOH-CAH-TOA 表示 : Sine 等于 ypotenuse 的对面, Cosine 等于 ypotenuse 的对面, Cosine 等于 ypotenuse 的对面, Tangent 等于 aypotenuse 的对面。 这个中音可以帮助您记住与三角比相关的两边 。

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These are the basic trigonometric ratios. The y  are called trigonometric ratios because they apply to triangles, and trigonometry is the study of triangles.
::这些是基本的三角比。它们被称为三角比,因为它们适用于三角,三角是三角的研究。

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Just as your scientific or graphing calculator has tangent programmed into it, it also has sine and cosine programmed into it. This means that you can use your calculator to determine the ratio between the lengths of the sides for any angle within a right triangle.
::正如您的科学或图形计算器已编入正切, 它也编入正弦和正弦。 这意味着您可以使用您的计算器来确定右三角内任何角的侧边长度比 。

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Can the value of $\begin{array}{r}\sin A\end{array}$ or $\begin{array}{r}\cos A\end{array}$ exceed 1?
::罪状或罪状是否值超过1?

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The value of $\begin{array}{r}\sin A\end{array}$ or $\begin{array}{r}\cos A\end{array}$ exceed 1.
::罪状A或cosáA的值超过1。

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

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Signs of Trigonometric Functions
::三角函数的标志

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Solving for Unknown Values
::解决未知值

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Solve for missing side length  $\begin{array}{r}x\end{array}$  in the triangle below. Use your calculator to find the sine ratio and cosine ratio for a $\begin{array}{r}{27}^{\circ }\end{array}$ angle. 
::解决下面三角形中缺失的侧长 x 。 使用您的计算器来找到一个 27 角的正弦比和余弦比 。

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$\begin{array}{r}\sin ({27}^{\circ })\approx 0.454\text{ and}\cos ({27}^{\circ })\approx \mathrm{0.891.}\end{array}$
::{\fn黑体\fs22\bord1\shad0\3aHBE\4aH00\fscx67\fscy66\2cHFFFFFF\3cH808080}什么? {\fn黑体\fs22\bord1\shad0\3aHBE\4aH00\fscx67\fscy66\2cHFFFFFF\3cH808080}什么? {\fn黑体\fs22\bord1\shad0\3aHBE\4aH00\fscx67\fscy66\2cHFFFFFF\3cH808080}什么? {\fn黑体\fs22\bord1\shad0\3aHBE\4aH00\fscx67\fscy66\2cHFFFFFF\3cH808080}什么? {\fn黑体\fs22\bord1\shad0\3aHBE\4aH00\fscx67\fscy66\2cHFFFFFF\3cH808080}什么?

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Look to see how the marked sides are related to the $\begin{array}{r}{27}^{\circ }\end{array}$ angle.
::看看标记的两面 与27角的关系如何

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The length of the adjacent leg to the angle is 11 cm and the length of the hypotenuse of the triangle is $\begin{array}{r}x.\end{array}$
::角的相邻腿长度为11厘米,三角形的下限长度为x。

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When working with the adjacent side and the hypotenuse, you should use the cosine ratio. 
::当与相邻的侧面和下限一起工作时, 您应该使用余弦比 。

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Note that 12.346 is an approximate answer because you rounded the value of $\begin{array}{r}\cos ({27}^{\circ })\end{array}$ whereas $\begin{array}{r}\frac{11}{\cos ({27}^{\circ })}\end{array}$ would be an exact answer.
::请注意,12.346是一个大致答案,因为您四舍五入 Cos(27) 的值,而 11cos(27) 的值将是一个准确答案。

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Calculating Sine and Cosine Functions
::计算正弦和余弦函数

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1. Find $\begin{array}{r}\sin \theta \end{array}$   and $\begin{array}{r}\cos \theta \end{array}$   in the triangle below:
::1. 在以下三角形中查找sin和cos:

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Relative to angle $\begin{array}{r}\theta :\end{array}$  3 is the opposite leg and 4 is the adjacent leg .
::相对于角度 : 3 是相反的腿, 4 是相邻的腿。

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In order to find the sine and cosine ratios, you also need to know the hypotenuse of the triangle. Use to find the hypotenuse:
::要找到正弦和余弦比率, 您也需要知道三角形的下限值。 使用以查找下限值 :

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Now, write the sine and cosine ratios:
::现在,写下正弦和余弦比率:

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$\begin{array}{rl}\sin \theta & =\frac{\text{opposite leg}}{\text{hypotenuse}}=\frac{MN}{LM}=\frac{3}{5}\\ \cos \theta & =\frac{\text{adjacent leg}}{\text{hypotenuse}}=\frac{LN}{LM}=\frac{4}{5}\end{array}$
::

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2. In a right triangle, if $\begin{array}{r}\sin A=\frac{4}{5}\end{array}$ , what is the value of $\begin{array}{r}\cos A\end{array}$ ?
::2. 在右三角,如果sinA=45,CosA的价值是什么?

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The value of $\begin{array}{r}\cos A\end{array}$ is equal to .
::COSA 值等于 。

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Examples
::实例实例实例实例

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Use your calculator to find the sine and cosine ratios for a $\begin{array}{r}{39}^{\circ }\end{array}$ angle.
::使用计算器找到39角的正弦和余弦比率。

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$\begin{array}{r}\sin ({39}^{\circ })\approx 0.629\end{array}$ and $\begin{array}{r}\cos ({39}^{\circ })\approx 0.777\end{array}$ .
::-=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=-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:

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Example 2
::例2

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

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Look to see how the sides that are marked are related to the $\begin{array}{r}{39}^{\circ }\end{array}$ angle. The length of the opposite leg from the angle is 17 cm and the length of the hypotenuse of the triangle is $\begin{array}{r}x\end{array}$ . When working with the opposite side and the hypotenuse, you should use the sine ratio.
::查看标记的边与 39 angle 的关系。 角度对面的腿长度为 17 cm,三角形的下限长度为 x。 在与对面和下限一起工作时, 您应该使用正弦比例 。

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Example 3
::例3

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Find $\begin{array}{r}\sin \theta \end{array}$ and $\begin{array}{r}\cos \theta \end{array}$ .
::找到罪与罪

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Relative to angle $\begin{array}{r}\theta \end{array}$ , 4 is the opposite leg and 2 is the adjacent leg. In order to find the sine and cosine ratios you also need to know the hypotenuse of the triangle. Use to find the hypotenuse:
::相对于角度 , 4 是相反的腿, 2 是相邻的腿。 为了找到正弦和正弦比例, 您也需要知道三角形的下拉值 。 用于查找下拉值 :

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$\begin{array}{rl}{BC}^2+{AB}^2& ={AC}^2\\ 4^2+2^2& =h^2\\ 20& =h^2\\ h& =2\sqrt{5}\end{array}$
::BC2+AB2=AC242+22=h220=h2h=25

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Now, write the sine and cosine ratios:
::现在,写下正弦和余弦比率:

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$\begin{array}{rl}\sin \theta & =\frac{\text{opposite leg}}{\text{hypotenuse}}=\frac{BC}{AC}=\frac{4}{2\sqrt{5}}=\frac{2}{\sqrt{5}}=\frac{2\sqrt{5}}{5}\\ \cos \theta & =\frac{\text{adjacent leg}}{\text{hypotenuse}}=\frac{AB}{AC}=\frac{2}{2\sqrt{5}}=\frac{1}{\sqrt{5}}=\frac{\sqrt{5}}{5}\end{array}$
::=BCAC=425=2555cos=2555cos *bAC=225=15=55

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Note that in the last step of each calculation the denominator was rationalized.  Generally speaking, you should always rationalize the denominator of a fractional answer.
::请注意, 在每次计算的最后一步中, 分母被合理化了 。 一般说来, 您应该总是将分解答案的分母合理化 。

Summary

播放段落 SOH-CAH-TOA stands for: Sine equals Opposite over Hypotenuse, Cosine equals Adjacent over Hypotenuse, and Tangent equals Opposite over Adjacent.  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 播放段落 $\begin{array}{r}\tan (\theta )=\frac{\text{opp}}{\text{adj}}\end{array}$   ::tan =oppadj ::SOH-CAH-TOA 表示 : Sine 等于 ypotenuse 的对面, Cosine 等于 ypotenuse 的对面, Tangent 等于 ypotenuse 的对面。 sin = opphyp cos = adjhyp tan = oppadj

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For questions 1 through 6, use the triangle below. Find each exact value.
::对于问题1至6, 请使用下面的三角形, 找到每个精确值 。

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1. $\begin{array}{r}\sin E\end{array}$
::1. 罪 义 义

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2. $\begin{array}{r}\cos E\end{array}$
::2. COsE

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3. $\begin{array}{r}\tan E\end{array}$
::3. 铁环

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4. $\begin{array}{r}\sin F\end{array}$
::4 sin *F 4sin *F 4sin *F

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5. $\begin{array}{r}\cos F\end{array}$
::5 COSF

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6. $\begin{array}{r}\tan F\end{array}$
::6. 坦 坦 法

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Identify whether sine, cosine, or tangent ratio is used to solve the problem. Then, solve for $\begin{array}{r}x\end{array}$ and round your answer to the nearest hundredths place.
::确定是否使用正弦、 连弦或正弦比来解决问题。 然后, 解答 x , 并绕过您答案到最近的百位 。

7.

8.

9.

10.

11.

12.

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Use the triangle below for #13-#15.
::使用下面的三角形进行 # 13- # 15 。

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13. Find $\begin{array}{r}m\mathrm{\angle }C\end{array}$ .
::13. 寻找 mC。

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14. Draw an altitude from $\begin{array}{r}\mathrm{\angle }B\end{array}$ to divide the triangle into two right triangles. Use trigonometry to find the lengths of the sides of each of these right triangles. Round your answers to the nearest thousandths place.
::14. 从 B 绘制高度, 将三角形分为两个右三角形。 使用三角测量法查找这些右三角形两侧的长度。 将答案转至最近的千位位置 。

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15. Using the values you found in the previous question, find the perimeter of $\begin{array}{r}\mathrm{\Delta }ABC\end{array}$ and round your answer to nearest whole inch.
::15. 使用您在前一个问题中发现的值,找到 ABCand 的周界,将您的答案绕到最近的整英寸。

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16. What do the trigonometric ratios have to do with similar triangles?
::16. 三角比与类似的三角有何关系?

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17. The curvature of the earth makes it impossible to see beyond the horizon, but as an observer ascends, the distance to the horizon increases. The radius of the earth is 3,959 miles. How high must an individual ascend in order to see a horizon line corresponding to a $\begin{array}{r}{1}^{\circ }\end{array}$ rotation around the center of the earth? Round your answer to five decimal places. (Note: sketch is not to scale.)
::17. 地球的弯曲使得不可能超越地平线,但随着观察者升起,地平线的距离会增加,地球的半径为3,959英里,个人必须升到多高才能看到一个与地球中心周围的 1- 旋转相适应的地平线? 将答案转至小数点后五位。 (注:草图不是缩放的。 )