To determine the quadrant in which an angle of $-270^\circ$ falls, we follow these steps:

1. Understand the Standard Position: An angle in standard position has its vertex at the origin and its initial side along the positive x-axis. Positive angles are measured counterclockwise, while negative angles are measured clockwise.

2. Convert the Angle to a Positive Coterminal Angle: To find the coterminal angle (an equivalent angle between $0^\circ$ and $360^\circ$): $-270^\circ + 360^\circ = 90^\circ$

3. Determine the Quadrant: The coterminal angle $90^\circ$ is on the positive y-axis, which technically is on the boundary between the first and second quadrants. However, for practical purposes, we often consider it as part of the first quadrant.

Thus, the angle $-270^\circ$ falls along the boundary of the first quadrant, specifically on the positive y-axis.

Would you like a detailed explanation or have any questions about this?

Here are some related questions:

1. How do you find coterminal angles for any given angle?
2. What are the characteristics of each quadrant in the Cartesian plane?
3. How do positive and negative angle measurements differ in standard position?
4. Can an angle lie exactly on an axis, and how is it categorized?
5. How do you convert degrees to radians, and vice versa?
6. What are the main trigonometric functions and their relationships in each quadrant?
7. How do you determine the reference angle for any given angle?
8. What is the significance of the unit circle in trigonometry?

Tip: To quickly determine the quadrant of an angle, subtract or add $360^\circ$ repeatedly until the angle is within the range of $0^\circ$ to $360^\circ$, then analyze its position accordingly.