Sinusoidal Functions as Harmonic Oscillations


December 10, 2022

The sinusoidal functions (like sin and cosine) normally first come up when students are learning about angles of right angled triangles. However there are many other ways to define them; as projections of points on a circle, as the real and imaginary parts of the mapping \(f(x) = e^{ix}\), in terms of addition law, as a power series or as inverse functions of the anti-derivatives of \(f(x) = \frac{1}{\sqrt{1-x^2}}\). A physically motivated way is through the Simple Harmonic Oscillator and differential equations.

Consider an object at rest that stays near its initial state after a small perturbation, like a ball at the bottom of a hill given a small kick, a swing hanging straight down given a small push, or a chair given a small push on its back. Representing the potential energy at \(x\) by \(V(x)\) then by Newton’s equations of motion \(m\frac{{\rm d}^2x}{{\rm d}t^2} = -\frac{{\rm d}V}{{\rm d}{x}}\). Since the object isn’t initially accelerating the first derivative of V must be zero, and since a small perturbation accelerates the object back towards its initial state the second derivative of V must be non-negative. So locally \(V(x) \approx c + k^2 x^2 + O(x^3)\) where \(c\) and \(k\) are arbitrary constants. Under this approximation then the equations of motion are \(m\frac{{\rm d}^2x}{{\rm d}t^2} = -k^2 x\). Without loss of generality assume \(m = 1\), then we are trying to solve the equation \[x''(t) = -k^2 x(t)\] for some arbitrary \(k\).

We can find a power series solution to this equation; setting \[x(t) = \sum_{n=0}^{\infty} a_n \frac{t^n}{n!}\] then a change of variables gives \[x'(t) = \sum_{n=0}^{\infty} a_{n} n \frac{t^{n-1}}{n!} = \sum_{n=0}^{\infty} a_{n+1} \frac{t^n}{n!}\] and similarly \[x''(t) = \sum_{n=0}^{\infty} a_{n+2} \frac{t^n}{n!}\] for all values of t. Then the equation becomes \[ \sum_{n=0}^{\infty} (a_{n+2} + k^2 a_{n}) \frac{t^n}{n!} = 0\] and since this is true for all values of \(t\) then it must hold for each coefficient \(n\), \(a_{n+2} = -k^2 a_n\). Thus \(a_{2n} = \left(-k^2\right)^{2n} a_0\) and \(a_{2n+1} = \left(-k^2\right)^{2n} a_1\). A set of solutions is thus \[x(t) = a_0 \left(\sum_{n=0}^{\infty} (-1)^n \frac{(kt)^{2n}}{(2n)!} \right) + a_1 \left( \sum_{n=0}^{\infty} (-1)^n \frac{(kt)^{2n+1}}{(2n+1)!}\right)\] for some arbitrary constants \(a_0\) and \(a_1\). We then define \(\cos(t) = \sum_{n=0}^{\infty} (-1)^n \frac{t^{2n}}{(2n)!}\) and \(\sin(t) = \sum_{n=0}^{\infty} (-1)^n \frac{t^{2n+1}}{(2n+1)!}\) making the solution \(x(t) = a_0 \cos(kt) + a_1 \sin(kt)\).

Now we have defined sin and cos as solutions of the Simple Harmonic Oscillator, we need to discover some of their properties in order to understand the oscillator. First note that the power series converges for all \(t\) by the ratio test, so the functions are well defined on the whole real line. Note from the power series that \(\sin(0) = 0\) and \(\cos(0) = 1\), and that \(\sin(-t) = \sin(t)\) and \(\cos(-t) = \cos(t)\) (that is \(\sin\) is an odd function and \(\cos\) is an even function). Taking derivatives term-wise in the power series immediately shows \(\cos'(t) = -\sin(t)\) and \(\sin'(t) = \cos(t)\) (which are consistent with our original differential equation).

The most important property is the addition formula, which I can’t find a good motivation for. However if you consider the start of the expansion of \[ \begin{align} \sin(s)\cos(t) &\approx \left(s - \frac{s^3}{3!} + \frac{s^5}{5!} - O(s^7)\ldots \right) \left(1 - \frac{t^2}{2!} + \frac{t^4}{4!} + O(t^6)\right) \\ &\approx s - \frac{st^2}{2!} - \frac{s^3}{3!} + \frac{st^4}{4!} + \frac{s^5}{5!} + \frac{s^3t^2}{2!3!} + O(s^k t^{7-k}) \end{align} \] then it follows that \[ \begin{align} \sin(s)\cos(t) + \cos(s)\sin(t) \approx& s + t - \frac{s^3 + 3 s t^2 + 3 s^2 t + t^3}{3!} + \\ &\frac{s^5 + 5st^4 + 5 s^4 t + 10 s^3 t^2 + 10 s^2 t^3}{5!} + O(s^k t^{7-k}) \\ \approx& (s + t) - \frac{(s+t)^3}{3!} + \frac{(s+t)^5}{5!} + O(s^k t^{7-k}) \\ \approx& \sin(s+t) \end{align} \] for small \(s\) and \(t\). A more rigorous computation of the power series (rearranging the sums into groups of equal sum of power) can show that \[\sin(s+t) = \sin(s) \cos(t) + \cos(s) \sin(t)\] for all \(s\) and \(t\).

Taking the derivative of the sin addition formula with respect to \(s\) gives the cosine addition formula \[\cos(s+t) = \cos(s) \cos(t) - \sin(s) \sin(t)\] for all \(s\) and \(t\). Setting \(s=-t\) gives \[1 = \cos^2(t) + \sin^2(t)\] for all \(t\).

Now there is almost enough to show the periodicity of cos and sin; that is the Simple Harmonic Oscillator will actually oscillate back and forth from its original position (ignoring frictional forces). Assume that \(\cos(t)\) has a zero, and define \(\pi\) as twice the smallest positive zero; \(\cos(\pi/2) = 0\). Since \(\sin^2(t) = 1 - \cos^2(t)\), \(\sin^2(\pi/2) = 1\), and as \(\sin(t)\) is 0 and increasing at the origin, and this is the first zero of its derivative \(\cos\), then the value must be positive; \(\sin(\pi/2) = 1\). The \(\sin(t+\pi/2) = \sin(\pi/2) \cos(t) + \cos(\pi/2) \sin(t) = \cos(t)\) and \(\cos(t+\pi/2) = \cos(\pi/2) \cos(t) - \sin(\pi/2) \sin(t) = - \sin(t)\). Then it follows \(\sin(t + \pi) = \cos(t+\pi/2) = -\sin(t)\) and so \(\sin(t+2\pi) = \sin(t)\). Similarly $(t+) = -(t+/2) = -(t) and so \(\cos(t+2\pi) = \sin(t)\), and so both functions are periodic with period \(2 \pi\) (the period can’t be less than \(2 \pi\) by the translation formula and the definition of \(\pi/2\) as the first positive zero of \(\cos\)).

However we need to actually prove that \(\cos\) has a zero somewhere. Consider \[\cos(2) \approx 1 - \frac{2^2}{2!} + \ldots = -1 + \ldots\] where \(\ldots\) are the higher order terms \((-1)^{n}\frac{2^{2n}}{2n!}\) for \(n=2,3,4,\ldots\). Note that these are less than 1, alternate in sign, and get smaller since \((2n)! > 2^{2n}\) for \(n>=2\), and consequently none of them will change the sign of the result. Thus \(\cos(2) < 0\), and since \(\cos\) is continuous it must have a zero between 0 and 2. Hence by our previous arguments the functions are periodic.

From here it is straightforward to prove other properties of trigonometric and inverse trigonometric functions. There is nothing novel here but this gives a different axiomitisation of the trigonometric functions. It would be interesting to understand how the properties of the trigonometric functions are related to the symmetry of the classical Simple Harmonic Oscillator, analogous to the symmetries of the Quantum Simple Harmonic Oscillator.