Section: Sine 和 Consine 图形图 | 14.1 Sine和Consine图案

Section outline

Your mission, should you choose to accept it, as Agent Trigonometry is to graph the function $\begin{aligned} y = 2 \cos x \end{aligned}$ . What are the minimum and maximum of your graph?
::您的任务, 您是否愿意接受它, 作为Trigonology Agent 来绘制 y= 2cosx 的函数。您的图表的最小值和最大值是多少 ?

Graphing Sine and Cosine
::Sine 和 Consine 图形图

In this concept, we will take the unit circle and graph it on the Cartesian plane.
::在这个概念中,我们将用单位圆来图解这个单位圆,然后用笛卡尔飞机来图解它。

To do this, we are going to “unravel” the unit circle. Recall that for the unit circle the coordinates are $\begin{aligned} (\cos θ, \sin θ) \end{aligned}$ where $\begin{aligned} θ \end{aligned}$ is the central angle. To graph, $\begin{aligned} y = \sin x \end{aligned}$ rewrite the coordinates as $\begin{aligned} (x, \sin x) \end{aligned}$ where $\begin{aligned} x \end{aligned}$ is the central angle, in radians. Below we expanded the sine coordinates for $\begin{aligned} \frac{3 π}{4} \end{aligned}$ .
::为此,我们将“ unravel” 单位圆。提醒注意对于单位圆来说, 坐标是 ° 的中心角( cos, sin) 。要图形, y=sinx 将坐标重写为 x (x, sinx) 的中心角, 以弧度表示。在下面, 我们将正弦坐标扩大为 3}4 。

Notice that the curve ranges from 1 to -1. The maximum value is 1, which is at $\begin{aligned} x = \frac{π}{2} \end{aligned}$ . The minimum value is -1 at $\begin{aligned} x = \frac{3 π}{2} \end{aligned}$ . This “height” of the sine function is called the . The amplitude is the absolute value of average between the highest and lowest points on the curve.
::注意曲线从 1 到 - 1 。最大值为 1 , 在 x @ @ 2 。最小值为 - 1 , 在 x=3% 2 。正弦函数的“ 高度” 称为。振幅是曲线上最高点和最低点之间的平均值的绝对值。

Now, look at the domain. It seems that, if we had continued the curve, it would repeat. This means that the sine curve is periodic . Look back at the unit circle, the sine value changes until it reaches $\begin{aligned} 2 π \end{aligned}$ . After $\begin{aligned} 2 π \end{aligned}$ , the sine values repeat. Therefore, the curve above will repeat every $\begin{aligned} 2 π \end{aligned}$ units, making the period $\begin{aligned} 2 π \end{aligned}$ . The domain is all real numbers.
::现在,看看这个域。如果我们继续曲线, 它会重复。这意味着正弦曲线是周期性的。回顾单位圆, 正弦值会变化, 直到它达到 2Q。在 2Q 之后, 正弦值会重复。因此, 上面的曲线会重复每个 2Q 单位, 使时间段为 2Q 。域是全部真实的数字。

Similarly, when we expand the cosine curve, $\begin{aligned} y = \cos x \end{aligned}$ , from the unit circle, we have:
::同样,当我们从单位圆中扩展余弦曲线y=cosx时,我们有:

Notice that the range is also between 1 and -1 and the domain will be all real numbers. The cosine curve is also periodic, with a period of $\begin{aligned} 2 π \end{aligned}$ . If we draw the graph past $\begin{aligned} 2 π \end{aligned}$ , it would look like:
::注意范围也介于 1 和 - 1 之间, 域将全部为真实数字。余弦曲线也是定期的, 时长为 2 。如果我们将图画过 2, 它看起来像 :

Comparing $\begin{aligned} y = \sin x \end{aligned}$ and $\begin{aligned} y = \cos x \end{aligned}$ (below), we see that the curves are almost identical, except that the sine curve starts at $\begin{aligned} y = 0 \end{aligned}$ and the cosine curve starts at $\begin{aligned} y = 1 \end{aligned}$ .
::比较 y=sinx 和 y=cosx (以下),我们看到曲线几乎完全相同,但正弦曲线以 y=0 开头,余弦曲线以 y=1开头。

If we shift either curve $\begin{aligned} \frac{π}{2} \end{aligned}$ units to the left or right, they will overlap. Any horizontal shift of a trigonometric function is called a phase shift .
::如果我们将曲线 2 单位向左或向右移动, 它们就会重叠。三角函数的任何水平移动都被称为相移。

Let's identify the highlighted points on $\begin{aligned} y = \sin x \end{aligned}$ and $\begin{aligned} y = \cos x \end{aligned}$ below.
::让我们确定下面的 y=sinx 和y=cosx 上的亮点。

For each point, think about what the sine or cosine value is at those values. For point $\begin{aligned} A \end{aligned}$ , $\begin{aligned} \sin \frac{π}{4} = \frac{\sqrt{2}}{2} \end{aligned}$ , therefore the point is $\begin{aligned} (\frac{π}{4}, \frac{\sqrt{2}}{2}) \end{aligned}$ . For point $\begin{aligned} B \end{aligned}$ , we have to work backwards because it is not exactly on a vertical line, but it is on a horizontal one. When is $\begin{aligned} \sin x = - \frac{1}{2} \end{aligned}$ ? When $\begin{aligned} x = \frac{7 π}{6} \end{aligned}$ or $\begin{aligned} \frac{11 π}{6} \end{aligned}$ . By looking at point $\begin{aligned} B \end{aligned}$ ’s location, we know it is the second option. Therefore, the point is $\begin{aligned} (\frac{11 π}{6}, \frac{1}{2}) \end{aligned}$ .
::对于每一个点,请想一想这些值的正弦值或余弦值。对于 A 点, sin4=22, 因此点是( 4, 22) 。对于 B 点, 我们必须倒着工作, 因为它不完全在垂直线上, 而是在水平线上。当sinx12 时? 当 x= 76 或 116 时。通过查看 B 点的位置, 我们知道这是第二个选项。因此, 点是 (116, 12) 。因此, 点是 (116, 12) 。

For the cosine curve, point $\begin{aligned} C \end{aligned}$ is the same as point $\begin{aligned} A \end{aligned}$ because the for $\begin{aligned} \frac{π}{4} \end{aligned}$ is the same. As for point $\begin{aligned} D \end{aligned}$ , we use the same logic as we did for point $\begin{aligned} B \end{aligned}$ . When does $\begin{aligned} \cos x = - \frac{1}{2} \end{aligned}$ ? When $\begin{aligned} x = \frac{2 π}{3} \end{aligned}$ or $\begin{aligned} \frac{4 π}{3} \end{aligned}$ . Again, looking at the location of point $\begin{aligned} D \end{aligned}$ , we know it is the second option. The point is $\begin{aligned} (\frac{4 π}{3}, \frac{1}{2}) \end{aligned}$ .
::对于余弦曲线, C点与 A 点相同, 因为 + 4 相同。至于 D 点, 我们使用的逻辑与 B 点相同。当 = x = 12 时? 当 x= 2 3 或 4 3 时。同样, 查看 D 点的位置, 我们知道这是第二个选项。点是 (4Q3 3 12 ) 。

Amplitude
::振幅

In addition to graphing $\begin{aligned} y = \sin x \end{aligned}$ and $\begin{aligned} y = \cos x \end{aligned}$ , we can stretch the graphs by placing a number in front of the sine or cosine, such as $\begin{aligned} y = a \sin x \end{aligned}$ or $\begin{aligned} y = a \cos x \end{aligned}$ . $\begin{aligned} | a | \end{aligned}$ is the amplitude of the curve.
::除了绘制 y=sinx 和 y=cosx 外, 我们可以在正弦或余弦前放置数字, 如 y=asinx 或 y=acosx 。 a 表示曲线的振幅。

Let's graph $\begin{aligned} y = 3 \sin x \end{aligned}$ over two periods.
::让我们在两个周期内绘制 y=3sinx 的图表。

Start with the basic sine curve. Recall that one period of the parent graph, $\begin{aligned} y = \sin x \end{aligned}$ , is $\begin{aligned} 2 π \end{aligned}$ . Therefore, two periods will be $\begin{aligned} 4 π \end{aligned}$ . The 3 indicates that the range will now be from 3 to -3 and the curve will be stretched so that the maximum is 3 and the minimum is -3. The red curve is $\begin{aligned} y = 3 \sin x \end{aligned}$ .
::以基本正弦曲线开始。提醒注意, 父图形的一个时段, y=sinx, 是 2。因此, 两个时段将是 4。 3 表示范围现在为 3 到 - 3, 曲线将被拉伸, 最大值为 3, 最小值为 - 3 。红曲线是 y= 3sinx 。

Notice that the $\begin{aligned} x \end{aligned}$ -intercepts are the same as the parent graph. Typically, when we graph a trigonometric function, we always show two full periods of the function to indicate that it does repeat.
::请注意 X 界面与父图形相同。通常, 当我们绘制三角函数时, 我们总是显示两个完整的函数周期, 以显示它重复。

Now, let's graph $\begin{aligned} y = \frac{1}{2} \cos x \end{aligned}$ over two periods.
::现在,让我们用Y=12cos=x来图两个时期。

Now, the amplitude will be $\begin{aligned} \frac{1}{2} \end{aligned}$ and the function will be “smooshed” rather than stretched.
::现在,振幅将是12,功能将是“消沉”而不是拉伸。

Finally, let's graph $\begin{aligned} y = - \sin x \end{aligned}$ over two periods.
::最后,让我们用两段时间来图表ysinx。

The last two problems dealt with changing $\begin{aligned} a \end{aligned}$ and $\begin{aligned} a \end{aligned}$ was positive. Now, $\begin{aligned} a \end{aligned}$ is negative. Just like with other functions, when the leading coefficient is negative, the function is reflected over the $\begin{aligned} x \end{aligned}$ -axis. $\begin{aligned} y = - \sin x \end{aligned}$ is in red.
::后两个问题涉及改变一个是正。现在, 一个是负的。就像其他函数一样, 当主要系数为负时, 该函数会反映在 x 轴上。 ysinx 是红色的。

Examples
::实例

Example 1
::例1

Earlier, you were asked to find the minimum and maximum of the graph $\begin{aligned} y = 2 \cos x \end{aligned}$ .
::早些时候,你被要求找到 y=2cosx 的最小和最大图形。

The 2 in front of the cosine function indicates that the range will now be from 2 to –2 and the curve will be stretched so that the maximum is 2 and the minimum is –2.
::余弦函数前面的 2 表示范围现在从 2 到 - 2, 曲线将被拉伸, 最大值为 2 , 最小值为 - 2 。

Example 2
::例2

Is the point $\begin{aligned} (\frac{5 π}{6}, \frac{1}{2}) \end{aligned}$ on $\begin{aligned} y = \sin x \end{aligned}$ ? How do you know?
::y=sinx是点(56,12)吗?你怎么知道?

Substitute in the point for $\begin{aligned} x \end{aligned}$ and $\begin{aligned} y \end{aligned}$ and see if the equation holds true.
::以 x 和 y 的点替代, 看看方程是否正确。

$\begin{aligned} \frac{1}{2} = \sin (\frac{5 π}{6}) \end{aligned}$
::12=sin( 56)

This is true, so $\begin{aligned} (\frac{5 π}{6}, \frac{1}{2}) \end{aligned}$ is on the graph.
::这是事实,所以图上是(5-6-12)图。

Graph the following functions for two full periods.
::以下图示两个完整期间的函数。

Example 3
::例3

$\begin{aligned} y = 6 \cos x \end{aligned}$
::y=6xx y=6xx

Stretch the cosine curve so that the maximum is 6 and the minimum is -6.
::伸展余弦曲线,使最大值为6,最低值为-6。

Example 4
::例4

$\begin{aligned} y = - 3 \cos x \end{aligned}$
::y 3cos x y 3cos x

The graph is reflected over the $\begin{aligned} x \end{aligned}$ -axis and stretched so that the amplitude is 3.
::该图在 X 轴上方反射,并拉伸,使振幅为 3。

Example 5
::例5

$\begin{aligned} y = \frac{3}{2} \sin x \end{aligned}$
::y=32sinx

The fraction is equivalent to 1.5, making 1.5 the amplitude.
::分数等于1.5,使1.5成振幅。

Review
::回顾

Determine the exact value of each point on $\begin{aligned} y = \sin x \end{aligned}$ or $\begin{aligned} y = \cos x \end{aligned}$ .
::确定 y=sinx 或 y=cosx 上的每个点的准确值。

List all the points in the interval $\begin{aligned} [0, 4 π] \end{aligned}$ where $\begin{aligned} \sin x = \cos x \end{aligned}$ . Use the graph from #1 to help you.
::列出 sinx=cosx 的间距 [0,4] 中的所有点。使用 # 1 的图表来帮助您。

Draw from $\begin{aligned} y = \sin x \end{aligned}$ from $\begin{aligned} [0, 2 π] \end{aligned}$ . Find $\begin{aligned} f (\frac{π}{3}) \end{aligned}$ and $\begin{aligned} f (\frac{5 π}{3}) \end{aligned}$ . Plot these values on the curve.
::从 y=sinx 从 [ 0, 2] 中绘制。查找 f 3 和 f( 53)。在曲线上绘制这些值。

For questions 4-12, graph the sine or cosine curve over two periods.
::对于问题4-12,请用图形显示两个时期的正弦或余弦曲线。

$\begin{aligned} y = 2 \sin x \end{aligned}$
::y=2sinx

$\begin{aligned} y = - 5 \cos x \end{aligned}$
::y 5cos x y 5cosx y 5cosx

$\begin{aligned} y = \frac{1}{4} \cos x \end{aligned}$
::y=14cosx

$\begin{aligned} y = - \frac{2}{3} \sin x \end{aligned}$
::y y 23sin x y 23sin_x

$\begin{aligned} y = 4 \sin x \end{aligned}$
::y=4sinx

$\begin{aligned} y = - 1.5 \cos x \end{aligned}$
::y$1.5cosx( y$1.5cosx)

$\begin{aligned} y = \frac{5}{3} \cos x \end{aligned}$
::y=53cosx

$\begin{aligned} y = 10 \sin x \end{aligned}$
::y=10sinx

$\begin{aligned} y = - 7.2 \sin x \end{aligned}$
::y7.2sinx

Graph $\begin{aligned} y = \sin x \end{aligned}$ and $\begin{aligned} y = \cos x \end{aligned}$ on the same set of axes. How many units would you have to shift the sine curve (to the left or right) so that it perfectly overlaps the cosine curve?
::图形 y=sinx 和 y=cosx 在同一组轴上。您需要将正弦曲线( 向左或向右) 移动多少个单位才能完全重合余弦曲线 ?

Graph $\begin{aligned} y = \sin x \end{aligned}$ and $\begin{aligned} y = - \cos x \end{aligned}$ on the same set of axes. How many units would you have to shift the sine curve (to the left or right) so that it perfectly overlaps $\begin{aligned} y = - \cos x \end{aligned}$ ?
::相同轴上的 y=sinx 和 ycosx 图形。您需要将正弦曲线( 向左或向右) 移动多少个单位才能完全重合 ycosx ?

Write the equation for each sine or cosine curve below. $\begin{aligned} a > 0 \end{aligned}$ for both questions.
::为下方的正弦或余弦曲线写出方程。 a>0 为两个问题写出方程。

Review (Answers)
::回顾(答复)

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。

Section outline

Graphing Sine and Cosine ::Sine 和 Consine 图形图

Amplitude ::振幅

Examples ::实例

Example 1 ::例1

Example 2 ::例2

Example 3 ::例3

Example 4 ::例4

Example 5 ::例5

Review ::回顾

Review (Answers) ::回顾(答复)

Graphing Sine and Cosine
::Sine 和 Consine 图形图

Amplitude
::振幅

Examples
::实例

Example 1
::例1

Example 2
::例2

Example 3
::例3

Example 4
::例4

Example 5
::例5

Review
::回顾

Review (Answers)
::回顾(答复)