Part 1: Sketching the Angle

To start, let's understand the angle $-420^\circ$ and where its terminal side lies.

A negative angle means that we rotate clockwise from the positive x-axis. Since angles are measured counterclockwise from the positive x-axis in standard position, rotating clockwise would mean we subtract from $360^\circ$.

Let's break it down:

• $-420^\circ$ is larger than $-360^\circ$, so we first subtract $360^\circ$ to bring the angle into a more manageable range:

  $-420^\circ + 360^\circ = -60^\circ$

Thus, the terminal side of the angle $-420^\circ$ is equivalent to $-60^\circ$, which lies in the fourth quadrant (clockwise from the positive x-axis). The answer is Quadrant 4.

• Correct Answer for Part 1: C. An angle with its initial side on the positive x-axis and its vertex at the origin has its terminal side in Quadrant 4.

Part 2: Reference Angle in Degrees

The reference angle is the smallest positive angle between the terminal side of the angle and the x-axis. To find it:

1. For a negative angle, we find the difference between $0^\circ$ and the angle's position within the quadrant.

2. Since $-60^\circ$ lies in the fourth quadrant, the reference angle is the absolute value of the difference between $-60^\circ$ and $-360^\circ$ (the closest x-axis line), which gives:

   $\text{Reference angle} = 360^\circ - 60^\circ = 60^\circ$

• Reference Angle in Degrees: 60°

Part 3: Reference Angle in Radians

To convert the reference angle from degrees to radians, we use the fact that $180^\circ = \pi$ radians. So:

$\text{Reference angle in radians} = 60^\circ \times \frac{\pi}{180^\circ} = \frac{\pi}{3}$

• Reference Angle in Radians: $\frac{\pi}{3}$

Summary of Answers:

• Part 1: The correct graph is C. (Quadrant 4)
• Part 2: The reference angle in degrees is 60°.
• Part 3: The reference angle in radians is $\frac{\pi}{3}$.

Would you like more details or further questions on this topic?

Additional Questions:

1. What happens if we rotate a positive angle, say $420^\circ$, instead of a negative one?
2. How do you find the reference angle for angles greater than $360^\circ$?
3. How is the reference angle different for angles in different quadrants?
4. What is the general formula to calculate the reference angle for any given angle in standard position?
5. Can you calculate the reference angle for a negative angle in radians, like $- \frac{5\pi}{6}$?

Tip:

Always remember that the reference angle is the smallest positive angle between the terminal side and the nearest x-axis.