1.20 大于360度的角的三角函数

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  大于360度角的角的三角函数

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  While out at the local amusement park with friends, you take a ride on the Go Karts. You ride around a circular track in the carts three and a half times, and then stop at a "pit stop" to rest. While waiting for your Go Kart to get more fuel, you are talking with your friends about the ride. You know that one way of measuring how far something has gone around a circle (or the trig values associated with it) is to use angles. However, you've gone more than one complete circle around the track.
  ::与朋友一起在本地游乐场玩耍时, 您可以搭乘Go Karts。 您在马车上绕着圆形车道骑了3次半, 然后停在“ 便车站” 休息。 在等待您的 Go Kart 获得更多燃料的时候, 您正在和朋友谈论这趟车。 您知道, 测量环绕( 或与其相关的三角值) 所走过的路的一个方法就是使用角度。 然而, 您在车道上已经跨过一个完整的圈子了 。

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  Is it still possible to find out what the values of are for the change in angle you've made?
  ::是否仍有可能找出 你改变角度的值是多少?

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  Angles Greater Than 360°
  ::大于360度的角

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  Consider the angle $\begin{array}{r}{390}^{\circ }\end{array}$ . As you learned previously, you can think of this angle as a full 360 degree rotation, plus an additional 30 degrees. Therefore $\begin{array}{r}{390}^{\circ }\end{array}$ is coterminal with $\begin{array}{r}{30}^{\circ }\end{array}$ . As you saw above with negative angles, this means that $\begin{array}{r}{390}^{\circ }\end{array}$ has the same ordered pair as $\begin{array}{r}{30}^{\circ }\end{array}$ , and so it has the same trig values. For example,
  ::考虑角度 390 。 正如您以前学到的, 您可以将这个角度视为完全360度旋转, 加上另外30度。 因此 390 与 30 是共同的。 正如您在上面用负角度看到的那样, 这意味着 390 与 30 有相同的定购对, 因此它具有相同的三角值。 例如,

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  $$\begin{array}{r}\cos {390}^{\circ }=\cos {30}^{\circ }=\frac{\sqrt{3}}{2}\end{array}$$

  
  ::390°C 30°C32°C

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  In general, if an angle whose measure is greater than
  ::一般而言,如果某一角度的测量量大于

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  $\begin{array}{r}{360}^{\circ }\end{array}$  has a reference angle of  $\begin{array}{r}{30}^{\circ }\end{array}$ ,  $\begin{array}{r}{45}^{\circ }\end{array}$ , or  $\begin{array}{r}{60}^{\circ }\end{array}$ , or if it is a quadrantal angle , we can find its ordered pair, and so we can find the values of any of the trig functions of the angle. Again, determine the reference angle first.
  ::360 ,

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  Let's look at some problems involving angles greater than  $\begin{array}{r}{360}^{\circ }\end{array}$ . 
  ::让我们来看看一些问题 涉及的角度 大于360。

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  Find the value of the following expressions:
  ::查找下列表达式的值:

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  1.  $\begin{array}{r}\sin {420}^{\circ }\end{array}$
  ::1. 罪行420 _________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

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  $\begin{array}{r}\sin {420}^{\circ }=\frac{\sqrt{3}}{2}\end{array}$
  ::42032

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  $\begin{array}{r}{420}^{\circ }\end{array}$ is a full rotation of 360 degrees, plus an additional 60 degrees. Therefore the angle is coterminal with $\begin{array}{r}{60}^{\circ }\end{array}$ , and so it shares the same ordered pair, $\begin{array}{r}\left(\frac{1}{2},\frac{\sqrt{3}}{2}\right)\end{array}$ . The sine value is the $\begin{array}{r}y-\end{array}$ coordinate.
  ::420是360度的完全旋转, 加上60度。 因此角是60度的交点, 所以它分享相同的一对( 12, 32) 。 正弦值是 Y - 坐标 。

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  2.  $\begin{array}{r}\tan {840}^{\circ }\end{array}$
  ::2. tan840_____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

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  $\begin{array}{r}\tan {840}^{\circ }=-\sqrt{3}\end{array}$
  ::{\fn黑体\fs22\bord1\shad0\3aHBE\4aH00\fscx67\fscy66\2cHFFFFFF\3cH808080}8403 {\fn黑体\fs22\bord1\shad0\3aHBE\4aH00\fscx67\fscy66\2cHFFFFFF\3cH808080}8403

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  $\begin{array}{r}{840}^{\circ }\end{array}$ is two full rotations, or 720 degrees, plus an additional 120 degrees:
  ::840为两个完全旋转,即720度,加上另外120度:

  $$\begin{array}{r}840=360+360+120\end{array}$$

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  Therefore $\begin{array}{r}{840}^{\circ }\end{array}$ is coterminal with $\begin{array}{r}{120}^{\circ }\end{array}$ , so the ordered pair is $\begin{array}{r}\left(-\frac{1}{2},\frac{\sqrt{3}}{2}\right)\end{array}$ . The tangent value can be found by the following:
  ::因此,840(840)与120(120)相交,因此订购的一对是(-12.32)。

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  $$\begin{array}{r}\tan {840}^{\circ }=\tan {120}^{\circ }=\frac{y}{x}=\frac{\frac{\sqrt{3}}{2}}{-\frac{1}{2}}=\frac{\sqrt{3}}{2}\times -\frac{2}{1}=-\sqrt{3}\end{array}$$

  
  ::$840$1200$120$1,120$1,200$1,200$1,200$3$3$3$3$3$3$3$3$3$1$3$3$3$3$1$1$1$1$1$1$1$1$1$1$1$1$1$1$1$1$1$1$1$1$1$1$1$1$1$1$1$1$1$1$1$1$1$1$1$1$1$1$1$1$1$1$1$1$1$1$1$1$1$3$1$1$1$1$1$1$1$1$3$3$1$1$1$3$1$1$1$3$1$1$1$1$1$1$1$1$1$1$1$1$3$1$1$1$1$1$1$1$1$1$1$1$1$1$1$1$1$1$1$1$1$1$1$1$1$1$1$1$1$1$1$1$1$1$3

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  3.  $\begin{array}{r}\cos {540}^{\circ }\end{array}$
  ::3 cos 540

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  $\begin{array}{r}\cos {540}^{\circ }=-1\end{array}$
  ::=5401

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  $\begin{array}{r}{540}^{\circ }\end{array}$ is a full rotation of 360 degrees, plus an additional 180 degrees. Therefore the angle is coterminal with $\begin{array}{r}{180}^{\circ }\end{array}$ , and the ordered pair is (-1, 0). So the cosine value is -1.
  ::540 是一个360度的完全旋转, 加上180度。 因此角度是 180 的交点, 订购的对数是 (-1, 0) 。 所以余弦值是 - 1 。

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  Examples
  ::实例

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  Example 1
  ::例1

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  Earlier, you were asked if it is still possible to find out what the values of sine and cosine are for the change in angle. 
  ::早些时候,有人问您是否仍然能够找出 弦和弦的价值观是什么 改变角度。

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  Since you've gone around the track 3.5 times, the total angle you've traveled is $\begin{array}{r}{360}^{\circ }\times 3.5={1260}^{\circ }\end{array}$ . However, as you learned in this unit, this is equivalent to $\begin{array}{r}{180}^{\circ }\end{array}$ . So you can use that value in your computations:
  ::自从你绕过轨道3.5倍, 你所经过的总角度是 3603.5=1260。 然而, 正如你在这个单位所学到的, 这相当于 180。 所以, 您可以在计算时使用这个数值 :

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  $$\begin{array}{r}\sin {1260}^{\circ }=\sin {180}^{\circ }=0\\ \cos {1260}^{\circ }=\cos {180}^{\circ }=-1\end{array}$$

  
  ::1260 180 0cos 1260 ocos 180 $1

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  Example 2
  ::例2

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  Find the value of the expression: $\begin{array}{r}\sin {570}^{\circ }\end{array}$
  ::查找表达式的值: sin570

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  Since $\begin{array}{r}{570}^{\circ }\end{array}$ has the same terminal side as $\begin{array}{r}{210}^{\circ }\end{array}$ , $\begin{array}{r}\sin {570}^{\circ }=\sin {210}^{\circ }=\frac{\frac{-1}{2}}{1}=\frac{-1}{2}\end{array}$
  ::自570起 与210起 570起 1121起 12

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  Example 3
  ::例3

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  Find the value of the expression: $\begin{array}{r}\cos {675}^{\circ }\end{array}$
  ::查找表达式的值: cos675

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   Since $\begin{array}{r}{675}^{\circ }\end{array}$ has the same terminal side as $\begin{array}{r}{315}^{\circ }\end{array}$ , $\begin{array}{r}\cos {675}^{\circ }=\cos {315}^{\circ }=\frac{\frac{\sqrt{2}}{2}}{1}=\frac{\sqrt{2}}{2}\end{array}$
  ::由于675的终端面与315相同, cos675cos315221=22

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  Example 4
  ::例4

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  Find the value of the expression: $\begin{array}{r}\sin {480}^{\circ }\end{array}$
  ::查找表达式的值: sin480

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  Since $\begin{array}{r}{480}^{\circ }\end{array}$ has the same terminal side as $\begin{array}{r}{120}^{\circ }\end{array}$ , $\begin{array}{r}\sin {480}^{\circ }=\sin {120}^{\circ }=\frac{\frac{\sqrt{3}}{2}}{1}=\frac{\sqrt{3}}{2}\end{array}$
  ::自480起, 与120起相同, 480起,120起,321起,32起。

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  Review
  ::回顾

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  Find the value of each expression.
  ::查找每个表达式的值 。

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  1. $\begin{array}{r}\sin {405}^{\circ }\end{array}$
     ::-405 -405 -405
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  2. $\begin{array}{r}\cos {810}^{\circ }\end{array}$
     ::810
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  3. $\begin{array}{r}\tan {630}^{\circ }\end{array}$
     ::{\fn黑体\fs22\bord1\shad0\3aHBE\4aH00\fscx630
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  4. $\begin{array}{r}\cot {900}^{\circ }\end{array}$
     ::科特 900
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  5. $\begin{array}{r}\csc {495}^{\circ }\end{array}$
     :csc495) (csc495) (csc495)
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  6. $\begin{array}{r}\sec {510}^{\circ }\end{array}$
     ::510111111111111111111111111111111111111111111111111111111111111111111111111111111111111111
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  7. $\begin{array}{r}\cos {585}^{\circ }\end{array}$
     ::585################################################################################################################################################################################################################################################################################################################################################################################################
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  8. $\begin{array}{r}\sin {600}^{\circ }\end{array}$
     ::来 来 来 来 来 来 来 来 来 来 来 来 来 来 来 来 来 来 来 来 来 来 来 来 来 来 来 来 来 来 来 来 来 来 来 来 来 来 来 来 来 来 来 来 来 来 来 来 来 来 来 来 来 来 来 来 来
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  9. $\begin{array}{r}\cot {495}^{\circ }\end{array}$
     ::科特495
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  10. $\begin{array}{r}\tan {405}^{\circ }\end{array}$
      ::{\fn黑体\fs22\bord1\shad0\3aHBE\4aH00\fscx67\fscy66\2cHFFFFFF\3cH808080}我... {\fn黑体\fs22\bord1\shad0\3aHBE\4aH00\fscx67\fscy66\2cHFFFFFF\3cH808080}我... {\fn黑体\fs22\bord1\shad0\3aHBE\4aH00\fscx67\fscy66\2cHFFFFFF\3cH808080}我... {\fn黑体\fs22\bord1\shad0\3aHBE\4aH00\fscx67\fscy66\2cHFFFFFF\3cH808080}
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  11. $\begin{array}{r}\cos {630}^{\circ }\end{array}$
      ::630
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  12. $\begin{array}{r}\sec {810}^{\circ }\end{array}$
      ::81011111111111111111111111111111111111111111111111111111111111111111111111111111111111111
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  13. $\begin{array}{r}\csc {900}^{\circ }\end{array}$
      :csc900) (csc900) (csc900) (c) (c) (c) (c) (c) (c) (a) (c) (c) (c) (c)) (c) (c) (c) (c)
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  14. $\begin{array}{r}\tan {600}^{\circ }\end{array}$
      ::{\fn黑体\fs22\bord1\shad0\3aHBE\4aH00\fscx67\fscy66\2cHFFFFFF\3cH808080}现在... {\fn黑体\fs22\bord1\shad0\3aHBE\4aH00\fscx67\fscy66\2cHFFFFFF\3cH808080}现在... {\fn黑体\fs22\bord1\shad0\3aHBE\4aH00\fscx67\fscy66\2cHFFFFFF\3cH808080}现在...
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  15. $\begin{array}{r}\sin {585}^{\circ }\end{array}$
      ::问题:585 ________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________
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  16. $\begin{array}{r}\tan {510}^{\circ }\end{array}$
      ::{\fn黑体\fs22\bord1\shad0\3aHBE\4aH00\fscx67\fscy66\2cHFFFFFF\3cH808080}我... {\fn黑体\fs22\bord1\shad0\3aHBE\4aH00\fscx67\fscy66\2cHFFFFFF\3cH808080}我... {\fn黑体\fs22\bord1\shad0\3aHBE\4aH00\fscx67\fscy66\2cHFFFFFF\3cH808080}
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  17. Explain how to evaluate a trigonometric function for an angle greater than $\begin{array}{r}{360}^{\circ }\end{array}$ .
      ::解释如何对大于 360 角的三角函数进行评估 。

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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 。