Waves as Polynomials
Because the derivatives of sine and cosine are cyclic (sin \(\to\) cos \(\to\) -sin \(\to\) -cos \(\dots\)), their Maclaurin series are also cyclic and easy to remember.
- Sine (Odd): \(\sin(x) = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \dots\)
- Cosine (Even): \(\cos(x) = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \dots\)
Worked Examples
Example 1: The Small Angle Approximation
For very small angles (\(x \ll 1\)), \(\sin(x) \approx x\). This is why a pendulum's motion is a perfect sine wave only for small swings. If the swing is too large, the \(x^3/6\) term becomes important, and the math becomes much harder.
Example 2: Visualizing Symmetries
Notice that \(\sin(x)\) only has odd powers of \(x\). This is why \(\sin(-x) = -\sin(x)\) (Odd Symmetry). \(\cos(x)\) only has even powers, so \(\cos(-x) = \cos(x)\) (Even Symmetry).
The Bridge to Quantum Mechanics
In Quantum Mechanics, the Parity of a particle (whether its state is even or odd) determines what kind of light it can absorb. Because sine is odd and cosine is even, these series expansions immediately tell us the parity of any wave state. Furthermore, when we solve the "Radial Equation" for an atom, we often find ourselves with a solution that is "almost" a sine wave. We use these series expansions to find the tiny corrections caused by the nucleus's finite size. This is how we achieve the extreme precision required for atomic clocks.