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.