For the angle

7π/6
Graph the angle in standard position. Show both the initial and terminal sides.

To graph the angle \( \frac{7\pi}{6} \) in standard position, we need to understand the following steps:

1. **Convert the angle to degrees (optional):**  
   Since \( \frac{7\pi}{6} \) is already in radians, we don't need to convert it to degrees to plot it on the unit circle. However, for better understanding, we can convert it to degrees.  
   \[
   \frac{7\pi}{6} \times \frac{180^\circ}{\pi} = 210^\circ
   \]
   The angle of \( \frac{7\pi}{6} \) radians is equivalent to 210 degrees.

2. **Locate the angle on the unit circle:**  
   The angle \( \frac{7\pi}{6} \) is more than \( \pi \) (which is \(180^\circ\)), so it is in the third quadrant.  
   To find the terminal side, start at the positive \( x \)-axis (initial side) and rotate counterclockwise around the origin.

3. **Draw the initial side:**  
   The initial side of the angle in standard position starts at the origin and extends along the positive \( x \)-axis.

4. **Draw the terminal side:**  
   To find the terminal side for \( \frac{7\pi}{6} \), rotate counterclockwise from the initial side. Since \( \frac{7\pi}{6} \) is in the third quadrant (between \( \pi \) and \( \frac{3\pi}{2} \)), the terminal side will end up in the third quadrant.

### Graph of the Angle \( \frac{7\pi}{6} \)

To illustrate this, we can visualize a coordinate plane with the initial side along the positive \( x \)-axis and the terminal side passing through the third quadrant.

Let's draw the graph:

- **Initial Side:** Along the positive \( x \)-axis.
- **Terminal Side:** Extends into the third quadrant at an angle of \( \frac{7\pi}{6} \) from the positive \( x \)-axis.

Here is the step-by-step approach to draw the graph:

1. **Draw the coordinate axes**: Mark the \(x\) (horizontal) and \(y\) (vertical) axes.
2. **Mark the unit circle**: Draw a circle centered at the origin (0,0) with a radius of 1.
3. **Draw the initial side**: This is the positive \(x\)-axis line.
4. **Determine the terminal side**:
   - Since \( \frac{7\pi}{6} = \pi + \frac{\pi}{6} \), rotate \(180^\circ\) (or \( \pi \) radians) and then an additional \(30^\circ\) (or \( \frac{\pi}{6} \) radians).
   - The terminal side should be located \(30^\circ\) past \(180^\circ\) (which is the negative \(x\)-axis).

### Visual Representation of the Angle
To visualize the angle \( \frac{7\pi}{6} \):

1. Draw the initial side along the positive \(x\)-axis.
2. Rotate counterclockwise to reach 210 degrees or \(7\pi/6\).
3. Mark this terminal side in the third quadrant.

### Conclusion

You should now have a graph with:
- The initial side along the positive \(x\)-axis.
- The terminal side in the third quadrant, indicating a counterclockwise rotation of \(210^\circ\) or \( \frac{7\pi}{6} \) radians.

Would you like me to provide more details on graphing angles or any related topics? Here are some related questions to consider:

1. What are the coordinates of the terminal point on the unit circle for the angle \( \frac{7\pi}{6} \)?
2. How would the graph look for a negative angle such as \(-\frac{7\pi}{6}\)?
3. What is the reference angle for \( \frac{7\pi}{6} \)?
4. How do we calculate the sine and cosine of the angle \( \frac{7\pi}{6} \)?
5. What are coterminal angles, and can you find one for \( \frac{7\pi}{6} \)?

**Tip:** When plotting angles in standard position, always start from the positive \(x\)-axis and rotate counterclockwise for positive angles.

Explore how to graph the angle 7π/6 in standard position with detailed steps and visual representation. This problem covers trigonometric concepts, including radians and degrees conversion, unit circle interpretation, and angle positioning on the coordinate plane. Suitable for students in Grades 10-12.

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