Speeding Up the Cycle
When the angle inside is \(2x\) or \(3x\), the wave repeats more often, which means there are more solutions within the standard \(0\) to \(2\pi\) range.
Worked Example
Solving with a Double Angle
Solve: \(\cos(2x) = 1/2\) for \(0 \leq x < 2\pi\).
- Step 1: Find solutions for the "block" \(2x\).
- \(2x = \pi/3, 5\pi/3\).
- BUT, since \(x\) goes to \(2\pi\), \(2x\) goes to \(4\pi\). We need to add \(2\pi\) to our solutions to get the next cycle.
- \(2x = \pi/3, 5\pi/3, 7\pi/3, 11\pi/3\).
- Step 2: Divide everything by 2.
- \(x = \pi/6, 5\pi/6, 7\pi/6, 11\pi/6\).
- Result: Four solutions!
The Bridge to Quantum Mechanics
In Chapter 11, we will study a particle in a box. The allowed wavefunctions are \(\psi_n = \sqrt{2/L} \sin(n\pi x / L)\). To find where the probability is highest, we have to solve the equation for the "Multiple Angle" \(\frac{n\pi x}{L}\). As \(n\) increases (higher energy), the "frequency" inside the sine function gets higher, creating more "nodes" and "peaks" in the box. This lesson shows you how to handle those multiple peaks, which are the physical representation of high-energy quantum states.