Using degree find the period of each function then graph write the table of value.
1. \( y=2 \sin \frac{x}{3} \) use \( 60^{\circ} \) Interval
2. \( y=4 \cos 3 x \), use \( 60^{\circ} \) Interval
3. \( y=\frac{1}{2} \tan \frac{x}{3} \)
4. \( y=\frac{1}{2} \sec x \)
5. \( y=2 \operatorname{crc} 2 x \)
6. \( y=2 \) cos
\( 2 x \)
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Answer:
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Step-by-step explanation:
Certainly! Let's go through each function one by one.
### 1. \( y = 2 \sin \frac{x}{3} \)
**Period:**
The period of \( y = A \sin(Bx) \) is \( \frac{360}{|B|} \) degrees.
For this function, \( B = \frac{1}{3} \), so the period is \( \frac{360}{\frac{1}{3}} = 1080 \) degrees.
**Graph:**
I can't provide a graph here, but you can use any graphing tool or calculator to plot points.
**Table of Values:**
\[ \begin{array}{|c|c|} \hline
x (\text{degrees}) & y \\ \hline
0 & 0 \\ \hline
60 & 2 \sin(20) \\ \hline
120 & 2 \sin(40) \\ \hline
180 & 0 \\ \hline
240 & -2 \sin(20) \\ \hline
300 & -2 \sin(40) \\ \hline
360 & 0 \\ \hline
\end{array} \]
### 2. \( y = 4 \cos 3x \)
**Period:**
The period of \( y = A \cos(Bx) \) is \( \frac{360}{|B|} \) degrees.
For this function, \( B = 3 \), so the period is \( \frac{360}{3} = 120 \) degrees.
**Graph:**
Again, use a graphing tool or calculator for the graph.
**Table of Values:**
\[ \begin{array}{|c|c|} \hline
x (\text{degrees}) & y \\ \hline
0 & 4 \\ \hline
60 & 4 \cos(180) \\ \hline
120 & 0 \\ \hline
180 & -4 \\ \hline
240 & -4 \cos(180) \\ \hline
300 & 0 \\ \hline
360 & 4 \\ \hline
\end{array} \]
### 3. \( y = \frac{1}{2} \tan \frac{x}{3} \)
**Period:**
The period of \( y = A \tan(Bx) \) is \( \frac{180}{|B|} \) degrees.
For this function, \( B = \frac{1}{3} \), so the period is \( \frac{180}{\frac{1}{3}} = 540 \) degrees.
**Graph:**
Graph this function using a graphing tool.
**Table of Values:**
\[ \begin{array}{|c|c|} \hline
x (\text{degrees}) & y \\ \hline
0 & 0 \\ \hline
60 & \frac{1}{2} \tan(20) \\ \hline
120 & \frac{1}{2} \tan(40) \\ \hline
180 & 0 \\ \hline
240 & -\frac{1}{2} \tan(20) \\ \hline
300 & -\frac{1}{2} \tan(40) \\ \hline
360 & 0 \\ \hline
\end{array} \]
### 4. \( y = \frac{1}{2} \sec x \)
**Period:**
The period of \( y = A \sec(Bx) \) is \( \frac{360}{|B|} \) degrees.
For this function, \( B = 1 \), so the period is \( \frac{360}{1} = 360 \) degrees.
**Graph:**
Graph this function using a graphing tool.
**Table of Values:**
\[ \begin{array}{|c|c|} \hline
x (\text{degrees}) & y \\ \hline
0 & \frac{1}{2} \sec(0) \\ \hline
60 & \frac{1}{2} \sec(60) \\ \hline
120 & 0 \\ \hline
180 & -\frac{1}{2} \sec(180) \\ \hline
240 & -\frac{1}{2} \sec(240) \\ \hline
300 & 0 \\ \hline
360 & \frac{1}{2} \sec(360) \\ \hline
\end{array} \]
### 5. \( y = 2 \csc 2x \)
**Period:**
The period of \( y = A \csc(Bx) \) is \( \frac{180}{|B|} \) degrees.
For this function, \( B = 2 \), so the period is \( \frac{180}{2} = 90 \) degrees.
**Graph:**
Graph this function using a graphing tool.
**Table of Values:**
\[ \begin{array}{|c|c|} \hline
x (\text{degrees}) & y \\ \hline
0 & 2 \csc(0) \\ \hline
60 & 2 \csc(120) \\ \hline
120 & 0 \\ \hline
180 & -2 \csc(180) \\ \hline
240 & -2 \csc(240) \\ \hline
300 & 0 \\ \hline
360 & 2 \csc(360) \\ \hline
\end{array} \]
### 6. \( y = 2 \cos 2x \)
**Period:**
The period of \( y = A \cos(Bx) \) is \( \frac{360}{|B|} \) degrees.
For this function, \( B = 2 \), so the period is \( \frac{360}{2} = 180 \) degrees.
**Graph:**
Graph this function using a graphing tool.
**Table of Values:**
\[ \begin{array}{|c|c|} \hline
x (\text{degrees}) & y \\ \hline
0 & 2 \cos(0) \\ \hline
60 & 2 \cos(120) \\ \hline
120 & 0 \\ \hline
180 & -2 \cos(180) \\ \hline
240 & -2 \cos(240) \\ \hline
300 & 0 \\ \hline
360 & 2 \cos(360) \\ \hline
\end{array} \]
Please note that these tables provide values for specific angles within each interval. Adjust the intervals as needed for more comprehensive tables.