Combining Angles
What is \(\sin(A + B)\)? Warning: It is not \(\sin A + \sin B\). Waves don't add that simply. We use the Sum Formula:
\[\sin(A + B) = \sin A \cos B + \cos A \sin B\]
\[\sin(A - B) = \sin A \cos B - \cos A \sin B\]
Worked Examples
Example 1: Non-Standard Angles
Find the exact value of \(\sin(75^\circ)\).
- \(75^\circ = 45^\circ + 30^\circ\).
- \(\sin(45+30) = \sin(45)\cos(30) + \cos(45)\sin(30)\).
- \(= (\frac{\sqrt{2}}{2})(\frac{\sqrt{3}}{2}) + (\frac{\sqrt{2}}{2})(\frac{1}{2})\).
- \(= \frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{4} = \frac{\sqrt{6} + \sqrt{2}}{4}\).
Example 2: Simplifying a Wave
Simplify: \(\sin(x)\cos(\pi) + \cos(x)\sin(\pi)\).
- This is the sum formula for \(\sin(x + \pi)\).
- Since \(\cos(\pi) = -1\) and \(\sin(\pi) = 0\), the expression is just \(-\sin(x)\).
The Bridge to Quantum Mechanics
In Quantum Mechanics, we often study "Superposition"—what happens when two different quantum states exist at once. If State A is \(\sin(kx)\) and we introduce a shift, the resulting interaction follows these sum and difference formulas. This is how we calculate Interference Patterns. When you see dark and light bands in a laser experiment, you are seeing the sum and difference of sine waves physically manifesting in the real world. This formula is the math of "Mixing States."