Breaking the Triangle
Up until now, angles were restricted to being inside a triangle (0 to 90 degrees). To understand physics, we need to allow angles to be anything—even negative or over 360 degrees. We do this by placing a circle of radius 1 (the Unit Circle) on the Cartesian plane.
Degrees and Quadrants
- \(0^\circ\): Positive x-axis.
- \(90^\circ\): Positive y-axis.
- \(180^\circ\): Negative x-axis.
- \(270^\circ\): Negative y-axis.
Worked Examples
Example 1: Coterminal Angles
Find an angle between 0 and 360 that is at the same spot as \(450^\circ\).
- Subtract one full circle: \(450 - 360 = 90^\circ\).
- Result: \(90^\circ\).
Example 2: Negative Rotation
Where is \(-30^\circ\)?
- Negative means move clockwise.
- Move 30 degrees down from the x-axis. This is the same spot as \(330^\circ\).
- This is in Quadrant IV.
The Bridge to Quantum Mechanics
Quantum particles have a property called Phase. Think of the phase as a clock hand spinning around the unit circle. As a particle moves through space, its "quantum clock" spins. If two particles arrive at the same spot and their "clocks" are pointing the same way (0 degrees), they add together. If one clock is at 0 and the other is at 180, they cancel out. This rotation around a circle is the fundamental reason behind Quantum Interference.