Let's go through the steps for the given problem involving a $30^\circ$ angle:

Part a: Drawing the Angle in Standard Position

In standard position, the vertex of the angle is at the origin of the coordinate plane, and the initial side lies along the positive x-axis. The terminal side of the angle is drawn counterclockwise from the x-axis for positive angles.

For $30^\circ$, the terminal side would be slightly above the x-axis, in the first quadrant, because the angle is less than $90^\circ$.

Part b: Converting to Radian Measure

To convert degrees to radians, use the formula: $\text{Radians} = \frac{\pi}{180^\circ} \times \text{Degrees}$

For $30^\circ$: $\text{Radians} = \frac{\pi}{180^\circ} \times 30^\circ = \frac{\pi}{6}$ Thus, $30^\circ = \frac{\pi}{6}$ radians.

Part c: Reference Angle in Degrees and Radians

The reference angle is the smallest angle that the terminal side makes with the x-axis. For angles in the first quadrant (which $30^\circ$ is), the reference angle is simply the angle itself.

• In degrees: The reference angle is $30^\circ$.
• In radians: The reference angle is $\frac{\pi}{6}$.

Summary:

• a. The angle is drawn with its vertex at the origin, initial side along the positive x-axis, and terminal side 30 degrees above the x-axis in the first quadrant.
• b. $30^\circ = \frac{\pi}{6}$ radians.
• c. The reference angle is $30^\circ$ or $\frac{\pi}{6}$ radians.

Would you like me to go into more details about any of these steps?

Here are five questions that build upon this concept:

1. How would you find the reference angle for an angle in the second quadrant?
2. What is the reference angle of $120^\circ$ in radians?
3. How would you convert $45^\circ$ to radians?
4. What is the radian measure of $90^\circ$?
5. How do you identify the quadrant of an angle based on its radian measure?

Tip: To easily convert degrees to radians, remember that $180^\circ = \pi$ radians!