Beyond Degrees
Degrees (\(360^\circ\)) are arbitrary; they were invented by humans based on the calendar year. Physics uses Radians, which are based on the circle itself. One radian is the angle where the "arc length" is equal to the "radius."
Conversion: \(\pi \text{ radians} = 180^\circ\)
Common Conversions
- \(90^\circ = \pi/2\)
- \(180^\circ = \pi\)
- \(270^\circ = 3\pi/2\)
- \(360^\circ = 2\pi\)
Worked Examples
Example 1: Degrees to Radians
Convert \(60^\circ\) to radians.
- \(60 \cdot (\frac{\pi}{180}) = \frac{60\pi}{180} = \frac{\pi}{3}\).
Example 2: Radians to Degrees
Convert \(\frac{\pi}{4}\) to degrees.
- \(\frac{\pi}{4} \cdot (\frac{180}{\pi}) = \frac{180}{4} = 45^\circ\).
The Bridge to Quantum Mechanics
In Quantum Mechanics, we never use degrees. The Schrödinger Equation only works in Radians. Why? Because the derivative of \(\sin(x)\) is only \(\cos(x)\) if \(x\) is in radians. If you use degrees, a messy constant (\(\pi/180\)) appears in every single equation. Radians are the "Natural Language" of the universe because they relate the geometry of the circle directly to the values of the functions. Every time you see \(\pi\) in a quantum formula, it's there because we are working in radians.