5.1 单位圆圈

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  联合圆圈

  The unit circle is a circle of radius one, centered at the origin, that summarizes all the 30-60-90 and 45-45-90 triangle relationships that exist.  When memorized, it is extremely useful for evaluating expressions like $\begin{align*}\cos (135^\circ)\end{align*}$  or $\begin{align*}\sin \left(-\frac{5 \pi}{3}\right)\end{align*}$ .  It also helps to produce the parent graphs of . 

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  How can you use the unit circle to evaluate $\begin{align*}\cos (135^\circ)\end{align*}$  and $\begin{align*}\sin \left(-\frac{5 \pi}{3}\right)\end{align*}$ ? 
  ::您如何使用此单位圆来评估 cs( 135) 和 sin( 5) ?

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  The Unit Circle
  ::联合圆圈

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  You already know how to translate between degrees and radians and the triangle ratios for 30-60-90 and 45-45-90 right triangles.  In order to be ready to completely fill in and memorize a unit circle, two triangles need to be worked out.  Start by finding the side lengths of a 30-60-90 triangle and a 45-45-90 triangle each with hypotenuse equal to 1.
  ::您已经知道如何翻译度和弧度, 以及30- 60- 90 和 45- 45- 45- 90 右三角形的三角比。 为了准备完全填充和记忆一个单位圆, 需要找到两个三角形。 首先找到一个 30- 60- 90 三角形和一个 45- 45- 90 三角形的侧长, 每个三角形的下限等于 1 。

  $\begin{align*}30^\circ\end{align*}$	$\begin{align*}60^\circ\end{align*}$	$\begin{align*}90^\circ\end{align*}$
  $\begin{align*}x\end{align*}$	$\begin{align*}x \sqrt{3}\end{align*}$	$\begin{align*}2x\end{align*}$
  $\begin{align*}\frac{1}{2}\end{align*}$	$\begin{align*}\frac{\sqrt{3}}{2}\end{align*}$	1

  $\begin{align*}45^\circ\end{align*}$	$\begin{align*}45^\circ\end{align*}$	$\begin{align*}90^\circ\end{align*}$
  $\begin{align*}x\end{align*}$	$\begin{align*}x\end{align*}$	$\begin{align*}x \sqrt{2}\end{align*}$
  $\begin{align*}\frac{\sqrt{2}}{2}\end{align*}$	$\begin{align*}\frac{\sqrt{2}}{2}\end{align*}$	1

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  This is enough information to fill out the important points in the first quadrant of the unit circle.  The values of the $\begin{align*}x\end{align*}$  and $\begin{align*}y\end{align*}$  coordinates for each of the key points are shown below.  Remember that the $\begin{align*}x\end{align*}$  and $\begin{align*}y\end{align*}$  coordinates come from the lengths of the legs of the special right triangles, as shown specifically for the $\begin{align*}30^\circ\end{align*}$  angle.  Always remember to measure the angle from the positive portion of the $\begin{align*}x\end{align*}$ -axis.
  ::此信息足以填充单位圆第一个象限中的重要点。 以下显示每个关键点的 x 和 y 坐标值。 记住, x 和 y 坐标来自特殊右三角的腿长, 具体显示为 30 角 。 记住要从 x 轴的正部分测量角度 。

  

  
  

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  Knowing the first quadrant well is the key to knowing the entire unit circle.  Every other point on the unit circle can be found using logic and this quadrant, so there is no need to memorize the whole circle.
  ::了解第一个象限井是了解整个单位圆的关键。单位圆的每一个点都可以用逻辑和这个象限找到,所以不需要记住整个圆。

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  To use your knowledge of the first quadrant of the unit circle to identify the angles and important points of the second quadrant, notice that the heights are mirrored and equal which correspond to the  $\begin{align*}y\end{align*}$ values.  The  $\begin{align*}x\end{align*}$ values are all negative.
  ::要使用您对单位圆第一个象数的知识来识别第二个象数的角和重要点,请注意高度是反射的,与y值相等。 x 值均为负值。

  

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  There is a pattern in the heights of the points in the first quadrant that can help you remember the points. 
  ::第一个象限点的高度有一个模式,可以帮助您记住这些点。

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  Notice that the heights of the points in the first quadrant are the $\begin{align*}y\end{align*}$ -coordinates:  $\begin{align*}0, \frac{1}{2}, \frac{\sqrt{2}}{2}, \frac{\sqrt{3}}{2}, 1\end{align*}$
  ::注意第一个象数点的高度是 Y 坐标: 0,12, 22, 32,1

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  When rewritten, the pattern becomes clear: $\begin{align*}\frac{0}{2}, \frac{\sqrt{1}}{2}, \frac{\sqrt{2}}{2}, \frac{\sqrt{3}}{2}, \frac{\sqrt{4}}{2}\end{align*}$ .
  ::改写后,图案变得清晰:02, 12, 22, 32, 42。

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  The three points in the middle are the most often mixed up.  This pattern illustrates how they increase in size from small $\begin{align*}\frac{1}{2}\end{align*}$ , to medium $\begin{align*}\frac{\sqrt{2}}{2}\end{align*}$ , to large $\begin{align*}\frac{\sqrt{3}}{2}\end{align*}$ .  When you fill in the unit circle, look for the heights that are small, medium and large and this will tell you were each value should go.  Notice that the heights for these five points in the second quadrant are also $\begin{align*}0, \frac{1}{2}, \frac{\sqrt{2}}{2}, \frac{\sqrt{3}}{2}, 1\end{align*}$ .
  ::中间的三点是最常见的混合点。 这个模式显示它们大小是如何从小12点到中22点增加到大32点的。 当您填满单数圆时, 寻找小、 中、 大高度, 这样会显示您每个值都应该去。 请注意, 第二个象限的这五个点的高度也是 0, 12, 22, 32, 1 。

  

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  This technique also works for the widths.  This can make memorizing the 16 points of the unit circle a matter of logic and the pattern: $\begin{align*}\frac{0}{2}, \frac{\sqrt{1}}{2}, \frac{\sqrt{2}}{2}, \frac{\sqrt{3}}{2}, \frac{\sqrt{4}}{2}\end{align*}$ .
  ::这一技术也用于宽度,可以使单位圆的16点被记住,这是逻辑和模式问题:02, 12, 22, 32, 42。

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  One last item to note is that are sets of angles such as 90º, 450 º , and -270 º that start at the positive x-axis and end at the same terminal side. Since coterminal angles end at identical points along the unit circle, trigonometric expressions involving coterminal angles are equivalent:  $\begin{align*}\sin{(90)}=\sin{(450)}=\sin{(-270)}\end{align*}$ . 
  ::最后要注意的一个项目是90o、450o和-270o等一组角度,它们从正x轴开始,到同一终点端结束。由于共同终点角度在单位圆的相同点结束,涉及共同终点角的三角度表达式相等于:sin(90)=sin(450)=sin(270)。

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  Examples
  ::实例

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  Example 1
  ::例1

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  Earlier, you were asked how you can use the unit circle to evaluate  $\begin{align*}\cos{(135)}\end{align*}$  and  $\begin{align*}\sin \left(-\frac{5 \pi}{3}\right)\end{align*}$ . The  $\begin{align*}x\end{align*}$ value of a point along the unit circle corresponds to the cosine of the angle.  The  $\begin{align*}y\end{align*}$ value of a point corresponds to the sine of the angle.  When the angles and points are memorized, simply recall the  $\begin{align*}x\end{align*}$ or  $\begin{align*}y\end{align*}$ coordinate. If the construction of the unit circle is understood, it becomes easier to determine the coordinates.
  ::早些时候,有人询问您如何使用单位圆来评估 cos(135) 和 sin(-5) 。 单位圆周围的点的 x 值与角的余弦相对应。 点的 y 值对应角的正弦。 当角和点被记住时, 只需记住 x 或 y 坐标。 如果能够理解单位圆的构造, 则更容易确定坐标 。

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  When evaluating $\begin{align*}\cos (135^\circ)\end{align*}$  your thought process should be something like this:
  ::当评价 cos(135) 时, 您的思考过程应该是这样 :

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  You know $\begin{align*}135^\circ\end{align*}$  goes with the point $\begin{align*}\left(-\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right)\end{align*}$  and cosine is the  $\begin{align*}x\end{align*}$ portion.  So, $\begin{align*}\cos (135^\circ)=-\frac{\sqrt{2}}{2}\end{align*}$ .
  ::你知道135 与点 (22,22) 和cosine 是 x 部分。 所以, coms( 135) 22 。

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  When evaluating $\begin{align*}\sin \left(-\frac{5 \pi}{3}\right)\end{align*}$  your thought process should be something like this:
  ::当评价罪(- 5) 时, 您的思考过程应该是这样的 :

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  You know  $\begin{align*}-\frac{5 \pi}{3}\end{align*}$ goes with the point $\begin{align*}\left(\frac{1}{2},-\frac{\sqrt{3}}{2}\right)\end{align*}$  and sine is the  $\begin{align*}y\end{align*}$ portion.  So, $\begin{align*}\sin \left(-\frac{5 \pi}{3}\right)=-\frac{\sqrt{3}}{2}\end{align*}$
  ::你知道- 53与点( 12, 32) 和正弦是 y 部分。 所以, 罪恶 (- 53) \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\

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  Example 2
  ::例2

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  Evaluate $\begin{align*}\cos 60^\circ\end{align*}$  using the unit circle and . What is the connection between the  $\begin{align*}x\end{align*}$ coordinate of the point and the cosine of the angle? 
  ::使用单位圆和 。 点的 x 坐标和角的余弦之间有什么联系 ?

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  The point on the unit circle for $\begin{align*}60^\circ\end{align*}$  is $\begin{align*}\left(\frac{1}{2}, \frac{\sqrt{3}}{2}\right)\end{align*}$  and the point is one unit from the origin.  This can be represented as a 30-60-90 triangle.
  ::60单位圆的点为(1232),点是原点的一个单位,可以表示为30-60-90三角形。

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  Since cosine is adjacent over hypotenuse, cosine turns out to be exactly the  $\begin{align*}x\end{align*}$ coordinate $\begin{align*}\frac{1}{2}\end{align*}$ .
  ::由于连弦在低温上空相邻,因此,连弦正好是X座标12。

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  Example 3
  ::例3

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  Using knowledge of the first quadrant of the unit circle, identify the angles and important points of the third quadrant. 
  ::使用单位圆第一个象限的知识,确定第三个象限的角和重要点。

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  Both the  $\begin{align*}x\end{align*}$ values and  $\begin{align*}y\end{align*}$ values are negative and their respective coordinates correspond to those of the other quadrants.
  ::x 值和 y 值均为负值,它们各自的坐标与其他四分位数对应。

  

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  Example 4
  ::例4

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  For each of the six trigonometric functions , identify the quadrants where they are positive and the quadrants where they are negative. 
  ::6个三角函数中的每个功能,标明正的四分位数和负的四分位数。

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  In quadrant I, the hypotenuse, adjacent and opposite side are all positive.  Thus all 6 trigonometric functions are positive. 
  ::在象限一中,下限、相邻和对面的侧面都是正数。因此,所有6个三角函数都是正数。

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  In quadrant II the hypotenuse and opposite sides are positive and the adjacent side is negative.  This means that every trigonometric expression involving an adjacent side is negative.  Sine and its reciprocal cosecant are the only two trigonometric functions that do not refer to the adjacent side which makes them the only positive ones. 
  ::在象限二中,下限和对面是正的,而相邻是负的。这意味着涉及相邻一面的三角表达式都是负的。 辛尼及其对等的共生体是唯一两个不指相邻一面的三角函数,因此它们是唯一正的三角函数。

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  In quadrant III only the hypotenuse is positive.  Thus the only trigonometric functions that are positive are tangent and its reciprocal cotangent because these functions refer to both adjacent and opposite sides which will both be negative. 
  ::在象限三中,只有下限是正数。 因此,唯一正数的三角函数是正数及其正数,因为这些函数指相邻和对面的两面,两者都是负数。

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  In quadrant IV the hypotenuse and the adjacent sides are positive while the opposite side is negative.  This means that only cosine and its reciprocal secant are positive. 
  ::在四等离子体中,下限和相邻两侧是正数,而对面是负数,这意味着只有余弦和对等分离是正数。

  

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  A mnemonic device to remember which trigonometric functions are positive and which trigonometric functions are negative is “ All S tudents T ake C alculus.”  All refers to all the trigonometric functions are positive in quadrant I.  The letter S refers to sine and its reciprocal cosecant that are positive in quadrant II.  The letter T refers to tangent and its reciprocal cotangent that are positive in quadrant III.  The letter C refers to cosine and its reciprocal secant that are positive in quadrant IV.
  ::用于记住三角函数为正数,三角函数为负数的负数设备是“所有学生采取计算法。” 所有的三角函数都指象限一中的正数。 S 字母指正数二中的正数,S 指正数二中的正数和正数。 T 字母指正数及其正数三中的正数。 C 字母指象限四中的正数及其正数。 C 字母指正数四中的正数和正数四。

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  Example 5
  ::例5

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  Evaluate the following trigonometric expressions using the unit circle. 
  ::使用单位圆评价以下三角表达式。

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  1. $\begin{align*}\sin \frac{\pi}{2}\end{align*}$
     ::罪2

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  $\begin{align*}\sin \frac{\pi}{2}=1\end{align*}$
  ::sin2=1

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  2. $\begin{align*}\cos 210^\circ\end{align*}$
     ::COs210

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  $\begin{align*}\cos 210^\circ=-\frac{\sqrt{3}}{2}\end{align*}$
  ::CO210 32

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  3. $\begin{align*}\tan 315^\circ\end{align*}$
     ::tan315______________________________________________________________________________________________________________________________________________________________________________________________________________

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  $\begin{align*}\tan 315^\circ=-1\end{align*}$
  ::tan3151

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  4. $\begin{align*}\cot 270^\circ\end{align*}$
     ::科特270________________________________________________________________________________________________________________________________________________________________________________________________________

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  $\begin{align*}\cot 270^\circ=0\end{align*}$
  ::科特2700

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  5. $\begin{align*}\sec \frac{11 \pi}{6}\end{align*}$
     ::秒11=6

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  $\begin{align*}\sec \frac{11 \pi}{6}=\frac{1}{\cos \frac{11 \pi}{6}}=\frac{2}{\sqrt{3}}=\frac{2 \sqrt{3}}{3}\end{align*}$
  ::秒11=1cos11=6=23=233

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  6. $\begin{align*}\csc - \frac{5 \pi}{4}\end{align*}$
     ::csc-54

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  $\begin{align*}\csc-\frac{5 \pi}{4}=\frac{1}{\sin-\frac{5 \pi}{4}}=-\frac{2}{\sqrt{2}}=-\sqrt{2}\end{align*}$
  ::csc-54=1sin-5422222

  Summary

  播放段落 The unit circle is a circle of radius one, centered at the origin, summarizing 30-60-90 and 45-45-90 triangle relationships. ::单位圆为半径一圆,以原为中心,总结了30-60-90和45-45-90三角关系。 播放段落 The entire unit circle can be determined using logic and the first quadrant, as other quadrants have mirrored and equal heights. ::整个单位圆圈可以使用逻辑和第一个象限来确定,因为其他象限已经反射和等高。 播放段落 A pattern in the coordinates can be used to help memorize the order: $\begin{align*}\frac{\sqrt{0}}{2}, \frac{\sqrt{1}}{2}, \frac{\sqrt{2}}{2}, \frac{\sqrt{3}}{2}, \frac{\sqrt{4}}{2}.\end{align*}$   ::坐标上的图案可以用来帮助记住顺序: 02, 12, 22, 32, 42。 播放段落 Coterminal angles , such as 90º, 450º, and -270º, end at identical points along the unit circle, making trigonometric expressions involving them equivalent. ::直角,如90o、450o和-270o,以单位圆一带相同的点结束,使三角表达式与之等同。

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  Review
  ::回顾

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  Name the angle between  $\begin{align*}0^\circ\end{align*}$ and  $\begin{align*}360^\circ\end{align*}$ that is coterminal with…
  ::命名 0 和 360 之间的角度, 该角度与...

  1. $\begin{align*}-20^\circ\end{align*}$

  2. $\begin{align*}475^\circ\end{align*}$

  3. $\begin{align*}-220^\circ\end{align*}$

  4. $\begin{align*}690^\circ\end{align*}$

  5. $\begin{align*}-45^\circ\end{align*}$

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  Use your knowledge of the unit circle to help determine whether each of the following trigonometric expressions is positive or negative.
  ::使用您对单位圆的了解来帮助确定以下三角表达式是否为正或负。

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  6.  $\begin{align*}\tan 143^\circ\end{align*}$
  ::6. tan143_____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

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  7. $\begin{align*}\cos \frac{\pi}{3}\end{align*}$
  ::7 COs%3 , 7 COs%3

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  8. $\begin{align*}\sin 362^\circ\end{align*}$
  ::第8362号罪

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  9. $\begin{align*}\csc \frac{3 \pi}{4}\end{align*}$
  ::9. csc34

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  Use your knowledge of the unit circle to evaluate each of the following trigonometric expressions.
  ::使用您对单位圆的了解来评价下列三角表达式。

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  10.  $\begin{align*}\cos 120^\circ\end{align*}$
  ::10. COS120__________________________________________________________________________________________________________________________

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  11. $\begin{align*}\sec \frac{\pi}{3}\end{align*}$
  ::11 秒=3

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  12. $\begin{align*}\tan 225^\circ\end{align*}$
  ::12. 坦 坦 225 __________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

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  13. $\begin{align*}\cot 120^\circ\end{align*}$
  ::13. COT120______________________________________________________________________________________________________________________________________________________________________________________

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  14. $\begin{align*}\sin \frac{11 \pi}{6}\end{align*}$
  ::14. 罪行11 6

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  15. $\begin{align*}\csc 240^\circ\end{align*}$
  ::15 csc240

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  16. Find $\begin{align*}\sin \theta\end{align*}$  and $\begin{align*}\tan \theta\end{align*}$  if $\begin{align*}\cos \theta=\frac{\sqrt{3}}{2}\end{align*}$  and $\begin{align*}\cot \theta > 0\end{align*}$ .
  ::16. 找到sin和tan,如果cos32和cot0。

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  17. Find  $\begin{align*}\cos \theta\end{align*}$ and  $\begin{align*}\tan \theta\end{align*}$ if $\begin{align*}\sin \theta =-\frac{1}{2}\end{align*}$  and $\begin{align*}\sec \theta < 0\end{align*}$ .
  ::17. 如果罪行12 和秒0 找到Cos 和 tan。

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  18. Draw the complete unit circle (all four quadrants) and label the important points.
  ::18. 绘制完整的单位圆(全部四个四方圆)并标出重要点。

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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 。