联合圆圈

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Lesson Objectives
::经验教训目标

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• Explain how the unit circle in the coordinate plane enables the extension of trigonometric functions to all real numbers, interpreted as degrees and radian measures of angles traversed counterclockwise around the unit circle.
  ::解释一下坐标平面上的单位圆如何使三角函数延伸至所有实际数字,被解释为在单位圆周围逆时针穿行角度的度和弧度测量。

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Introduction: Deriving the Unit Circle
::一. 导言:形成 " 团结圈 "

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You may recognize $\begin{align*}30^\circ - 60^\circ - 90^\circ\end{align*}$  and  $\begin{align*}45^\circ - 45^\circ - 90^\circ\end{align*}$ triangles from prior studies .
::您可以识别之前学习的 30 6090 和 45 4590 三角形 。

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In a   $\begin{align*}30^\circ - 60^\circ - 90^\circ\end{align*}$   triangle, the sides are in the ratio   $\begin{align*}1:\sqrt{3}:2,\end{align*}$   and in an isosceles   $\begin{align*}45^\circ - 45^\circ - 90^\circ\end{align*}$  triangle, the sides are in the ratio   $\begin{align*}1:1:\sqrt{2}.\end{align*}$  If the hypotenuse for each of these triangles was 1 unit,   t he dimensions of the triangles would look like the two below.
::在 30 60 90 三角形中,两边在1: 3: 2的比例中, 在 4 45 90 三角形中, 两边在 1: 1: 2. 如果每个三角形的下限为 1 单位, 三角形的尺寸会像下面两个三角形的尺寸 。

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Using these triangles, you can easily evaluate sine, cosine, and tangent for each of the angle measures. Since the hypotenuse always equals 1, the sine of the angle will always be equal to the opposite.
::使用这些三角形,您可以很容易地对每个角度量的正弦、连弦和正切值进行评估。由于次元总是等于 1, 角的正弦总是等于相反的 。

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$$\begin{align*}\sin = \frac{\text{opp}}{\text{hyp}} &= \frac{\text{opp}}{{1}}=\text{opp}\\ \ \\ \sin 45^\circ = \frac{\sqrt{2}}{2} \qquad \quad \sin 60^\circ &= \frac{\sqrt{3}}{2} \qquad \quad \sin 30^\circ = \frac{1}{2} \\ \end{align*}$$
::=opphyp=opp1=oppsin45=22sin60=32sin30=12

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A right triangle can be constructed inside a circle to represent the location of any point along the circle. This triangle can be constructed by placing one vertex on the origin, one vertex on the circle, and running the vertex with the right angle along the x-axis.
::右三角可以在圆内构造一个圆形, 以代表圆上任何点的位置。 此三角形可以通过在圆上放置一个顶点来构造, 在圆上放置一个顶点, 并在X轴上以右角运行顶点 。

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Use the  ratios in the special triangles above and the interactive to explore how trigonometric ratios relate to the points on a circle.
::使用上文特殊三角形中的比率和交互性来探讨三角比与圆上点的关系。

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Discussion Question: What issues would you have finding the of a triangle when $\begin{align*}m \angle \theta > 90º?\end{align*}$
::讨论问题:您在m90o时会发现三角形的问题是什么?

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Activity 1: Expanding Sine and Cosine
::活动1:扩大松和

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In the interactive a bove, special right triangles were used to identify trigonometric ratios of the angles 30°, 60°, 90°. It was also speculated that trigonometric ratios could be used for any angle between 0° and 90°.
::在互动的上文中,使用了特别右三角形来确定角30°、60°、90°的三角比,还推测角0°至90°的任何角都可以使用三角比。

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Given that angles may be negative and/or extend beyond the range of 0° to 90°, what would the sine or the cosine of these angles look like?
::鉴于角度可能是负的和/或超出0°至90°的范围,这些角度的正弦或余弦是什么样子?

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Use the interactive  below to observe what happens to sine and cosine as the angle approaches 0° and 90°.
::在角度接近0°和90°时,使用下面的交互式观察正弦和余弦发生的情况。

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Discussion Question: Use information from the questions above to derive sin(90°) and cos(90°).
::讨论问题:利用上述问题的资料得出罪(90°)和罪(90°)。

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Activity 2: Going B eyond the First Quadrant
::活动2:超越第一个四方

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Thus far, the following information has been derived from the first quadrant  in the circle that you have been building. 
::到目前为止,以下信息来自您正在建造的圆圈中的第一个象限。

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T his circle lies at the heart of trigonometry. When the radius of the circle is 1 unit, it  is referred to as the unit circle. The   unit circle   is a circle of radius one, centered at the origin, that summarizes all the  right   triangle relationships that exist. As you saw in the introduction, the points along the circle are determined by the sine and cosine of the angle. T he   $\begin{align*}x\end{align*}$ and   $\begin{align*}y\end{align*}$ coordinates of the points come from the lengths of the legs of the right triangles. In the introduction, you were asked what would happen if you had to find the sine or cosine of an  angle greater than 90°. While it is not possible to construct a right triangle with an obtuse angle, a right triangle can still be used to determine the location of a point in this quadrant. Keep in mind that the $\begin{align*}x\end{align*}$ and $\begin{align*}y\end{align*}$  coordinates of this point represent the cosine and sine, respectively. This approach can be used to find the sine and cosine of an angle between 90° and 180°.
::此圆位于三角测量的中心。 当圆的半径为 1 个单位时, 它被称为单位圆。 单位圆是一个半径的圆, 以起始点为中心, 概括所有右三角关系。 正如您在导言中看到的那样, 圆的点是由角的正弦和正弦决定的。 点的x 和 y 坐标来自右三角的腿长。 在导言中, 有人询问, 如果您必须找到一个大于 90 度角的正弦或正弦值, 将会发生什么 。 虽然无法用模糊角度构造右三角, 但仍可以使用右三角来确定此方形中某个点的位置 。 请注意, 此点的x 和 y 坐标分别代表正弦和正弦的长度 。 此方法可用于查找 90 度到 180 之间的角的正弦和正弦 。

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Example
::示例示例示例示例

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Find sin(120°) and cos(120°).
::查找罪( 120°) 和 cos( 120°) 。

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B egin by constructing a right triangle. A rotation of 120° will land in the second quadrant. 
::开始建造右三角形。 120°的旋转将在第二个象限降落 。

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T he sine of the given angle will have the same value as the y-coordinate of the point on the circle. To find this point, construct a right triangle that runs along the negative x-values.
::给定角的正弦值将与圆圆上点的 Y 坐标值相同。要找到此点,请在负 x 值上构造一个右三角形。

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To find the y-value of a 120° rotation, simply find sin(60°), which you should recognize from a bove as being  $\begin{align*}\frac{\sqrt{3}}{2}.\end{align*}$  To find the cosine, use the same triangle to find the $\begin{align*}x\text{-}\end{align*}$ coordinate of the point along the unit circle. The $\begin{align*}x\text{-}\end{align*}$ coordinate can be determined using cos(60°), which you know from above is  $\begin{align*}\frac{1}{2}.\end{align*}$  However, since this $\begin{align*}x\text{-}\end{align*}$ value resides to the left of 0, it will be negative.
::要找到一个 120 ° 旋转的 Y 值, 只需找到罪( 60 °) , 您应该从上面识别为 32 。 要找到余弦, 请使用同一个三角来查找单位圆周围点的 x 坐标 。 x 坐标可以使用 COs( 60 °) 确定, 您从上面知道的是 12 。 但是, 由于这个 X 值位于 0 左侧, 它将是负的 。

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Answer:   $\begin{align*}\sin{(120^{\circ})} = \frac{\sqrt{3}}{2} \text{ and } \cos{(120^{\circ})} = \:–\frac{1}{2}\end{align*}$  
::答复:罪(120) 32和cos(120) 12

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When connecting a right triangle to the point on the circle with a side running perpendicular to the $\begin{align*}x\text{-}\end{align*}$ axis, the angle created is called a reference angle .  A reference angle is an acute angle formed by the terminal side of an angle and the $\begin{align*}x\text{-}\end{align*}$ axis. Use the interactive below to practice identifying the remaining special triangle points in the second quadrant.
::当将右三角形连接到圆的圆角上与直角为X轴的侧面时,所创建的角被称为参考角。参考角是一个角的末端和X轴形成的急性角。使用下面的交互作用来练习识别第二个象限中剩余的特殊三角点。

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Discussion Question: How can you use the special angle points in quadrant 1 to quickly find the values in quadrant 2?
::讨论问题:您如何使用象限 1 中的特殊角度点快速找到象限 2 中的值 ?

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Activity 3: Finishing the Unit Circle
::活动3:完成联合圆圈

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Use what you know about reference angles to complete the unit circle.  Remember that the signs of each coordinate will depend on whether or not the $\begin{align*}x\end{align*}$  -axis or $\begin{align*}y\end{align*}$  -axis is negative in that quadrant.
::使用您所知道的关于引用角度来完成单位圆。 记住, 每个坐标的标记将取决于 X - 轴或 y - 轴是否在该象限中为负值 。

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Use the interactive below to finish the unit circle. You can then use the unit circle to quickly identify the sine and cosine of angles that form special right triangles.  
::使用下面的交互作用来完成单位圆。然后,您可以使用单位圆来快速识别形成特殊右三角形的角的正弦和余弦。

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Discussion Question: What patterns do you notice in the $\begin{align*}x\end{align*}$ and $\begin{align*}y\end{align*}$  coordinates of the unit circle? 
::讨论问题:单位圆的x和y座标中你注意到什么模式?

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Extension: Reference Angles in Four Quadrants
::扩展名: 四二次曲线中的参考角

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Use the interactive below to visually explore how reference angles work in each of the four quadrants.
::利用下面的交互作用,对四个四重角中的每个角的参考角度如何工作进行视觉探索。

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Activity 4: Coterminal Angles
::活动4:共同终点角

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Co terminal angles are sets of angles that have the same initial  and terminal sides. Since end at identical points along the unit circle, trigonometric expressions involving coterminal angles are equivalent.
::直角是具有相同初始边和末端边的一组角度。由于在单位圆一带相同点结束,涉及共界角的三角表达式是等同的。

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Example
::示例示例示例示例

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Use the unit circle to evaluate  $\begin{align*}\cos{\left(\frac{11\pi}{4}\right)}.\end{align*}$
::使用单位圆来评价 cos( 11_ 4) 。

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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.  Since $\begin{align*}\frac{11\pi}{4}\end{align*}$  is greater than $\begin{align*}2\pi,\end{align*}$  you know that  $\begin{align*}\frac{11\pi}{4}\end{align*}$ isn't on the unit circle that we have explored thus far. To evaluate $\begin{align*}\cos{\left(\frac{11\pi}{4}\right)},\end{align*}$ find the equivalent angle on the unit circle. S ubtract  $\begin{align*}2\pi\end{align*}$ from the angle until you get a value on the unit circle. This effectively rotates the angle 360° clockwise or counterclockwise until you get an angle between 0° and 360°.
::单位圆周围一个点的 X 值与角的余弦对应。 一个点的 Y 值与角的正弦对应。 由于 11 4 大于 2 , 您知道 11 4 不在我们迄今所探索的单位圆上。 要对单位圆进行评估, 请在单位圆上找到相等的角度 。 从角度上减去 2 , 直到您在单位圆上找到一个值。 这有效地旋转了 360 o 时或逆时针的角度, 直到您得到 0 o 和 360 o 之间的角 。

$$\begin{align*}\frac{11\pi}{4} - 2\pi = \frac{11\pi}{4} - \frac{8\pi}{4} = \frac{3\pi}{4}\end{align*}$$

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Since  $\begin{align*}\frac{3\pi}{4}\end{align*}$  is on the unit circle,  $\begin{align*}\cos\left(\frac{11\pi}{4}\right)\end{align*}$  is equivalent to  $\begin{align*}\cos\left(\frac{3\pi}{4}\right).\end{align*}$   The rotation   $\begin{align*}\frac{3\pi}{4}\end{align*}$  goes with the point $\begin{align*}(-\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}),\end{align*}$  and the cosine is in the $\begin{align*}x\text{-}\end{align*}$  position.  
::由于3×4在单位圆上,cos(11×4)等于cos(3×4),旋转3×4与点(22,22)相同,余弦处于x位置。

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Example
::示例示例示例示例

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Use the unit circle to evaluate  $\begin{align*}\sin \left(-45^o\right).\end{align*}$  
::使用单位圆来评价罪(- 45o) 。

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The angle -45° is less than 0°, so add increments of 360° until you have an angle on the unit circle.
::角度 - 45 度小于 0 度,所以增加360 度的递增,直到您在单位圆上有一个角度。

$$\begin{align*}-45^\circ + 360^\circ = 315^\circ\end{align*}$$

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The angle 315° is on the unit circle and goes to the point $\begin{align*}\left(\frac{\sqrt{2}}{2}, \frac{-\sqrt{2}}{2}\right).\end{align*}$  S ine is in the y-position of the coordinate: $\begin{align*}\frac{-\sqrt{2}}{2}.\end{align*}$
::角度 315°在单位圆上,直达点(22,22)。Sine位于坐标的Y位置:22。

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Answer:   $\begin{align*}\sin(-45^\circ) = \frac{-\sqrt{2}}{2}\end{align*}$
::答复:罪(-45) 22

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Use the interactive below to practice using the unit circle to find the sine and cosine of the given coterminal angles.
::使用下面的交互作用练习使用单位圆以找到给定共同终点角的正弦和正弦。

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Discussion Question:   The domain of the sine function for a unit circle is all x's for which  you can find sin(x). What values are in the domain of sin(x)? The  range of the sine function for a unit circle is the set of all possible outputs . What values are in the range of sin(x)? 
::讨论问题: 单位圆正弦函数的域是您能找到罪(x) 的全部 x 的域。 罪(x) 的域是哪些值? 单位圆正弦函数的范围是所有可能的输出。 罪(x) 的域是哪些值?

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Wrap-Up: Review Questions
::总结:审查问题

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This video explores how to determine trigonometric function values using the unit circle.
::此视频探索如何使用单位圆来确定三角函数值。

Summary

播放段落 The unit circle is a circle of radius one, centered at the origin, that summarizes all the right triangle trigonometric ratios that exist.  ::单位圆是半径一的圆圈, 以原点为中心, 总结所有右三角三角间现有三角比 。 播放段落 The x and y coordinates of a given point on the unit circle correspond to the cosine and sine respectively of the angle that is formed. ::单位圆上一个给定点的 x 和 y 坐标分别对应所形成角的正弦和正弦。 播放段落 A reference angle is an acute angle formed by the terminal side of an angle and the x-axis. ::参考角是一个角度和 X 轴末端形成的急性角。

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Extension: Medieval Castle Defense
::延长期:中世纪城堡防御

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Use the interactive below for more practice with the unit circle.
::使用下面的交互方式对单位圆做更多的练习。