(Sine or Cosine) of (Sine or Cosine)

DRAFT

December 2005

Not long ago a colleague was doing a pendulum project with his calculus class. The goal was to model the pendulum and solve the equations of motion, and then collect data using a motion detector, fit the data, and compare the results to the model. The details are not important to this story, but in the end the data fitting led to the function

for the vertical displacement of the pendulum bob. According to the physical model the vertical displacement was given by the function

.

Surprisingly, the graphs of these two functions appear identical! In fact, while f and g are surely not identically equal, they differ by at most 0.005 over the entire real line. While this is good news for the data collection exercise, it raises some questions about how two very different functions − one a standard cosine function and the other a composition of cosine functions − can produce nearly identical values.

The result can be resolved and much more can be learned by considering the four functions

.

Three parameters are involved, all of which we assume to be positive: A and a might be called the outer and inner amplitudes, respectively; and we refer to ω as the frequency. Specifically, we ask when these four functions can be well approximated be a standard sine or cosine function of the form Bcos σt + C or Bsin σt + C, which is what occurred in the pendulum project.

The first observation is that we really need to consider only two of the four functions f, g, h, and p. Using the complement property of the sine and cosine, plus the symmetry of the sine and cosine, we have

Thus the graphs of f and h are related by a translation of π/2ω (which will be shown to be a half-period of f and h). Similarly, the graphs of g and p are related by a translation of π/2ω (which is a quarter-period of g and p. So we restrict our attention to f and g. The graphs of these four functions are shown in Figure 1, with A = 2, a = 0.5, ω = 2.

Figure 1

The next observation is that f and g are both periodic functions, and it is easily shown that

.

So the periods of f and g are at most 2π/ω. However, because f involves an even function, we have

Thus the period of f is π/ω, or half of what one might expect.

Let’s now focus on , its properties, and the question of approximating it by a simpler function. Figure 2 shows the graph of f for A = 2, ω = 2, and a = 0.5, 1.0, 1.2. Note the period of all three functions is π/ω = π/2. Each function has minimum values at t = 0, ±π/2, ±3π/2, …., (or more generally, t = 0, ±π/ω, ±3π/ω), ….which are the points at which the inner cosine function has a maximum. The maximum values of f occur at the points t = 0, ±π/4, ±3π/4, …., (or more generally, t = 0, ±π/2ω, ±3π/2ω,…), which are the points at which the inner cosine function is 0.

Figure 2

Given the function , how can we approximate it by a function of the form F(t) = B cos σt + C?  We observe that the periods of f and F must match. Because the period if f is π/ω and the period of F is 2π/ σ; we have σ = 2ω. The next task is to find B and C in terms of A and a. The simplest conditions are to require that f and F agree at the minimum and maximum values of f:

At t = 0:           f(0) =  F(0) implies that                        A cos a = B + C

At t = π/2ω:     f(π/2ω) = F(π/2ω) implies that A = −B + C.

These two equation are easily solved for B and C; we find that

.

As shown in Figure 3,  (A = 2, a = 0.5) is well approximated by , where B = cos (0.5) −1  ≈ −0.1224 and C = 1 + cos (0.5) ≈1.8776. The maximum difference between these functions is approximately 0.0015. The two function in the pendulum project agree so well because a similar matching of coefficients.

Figure 3

Unfortunately we cannot always expect such good agreement between these two classes of functions. Why does it happen in the cases above? We begin by writing f in the form f(t) = A cos θ, where θ = a cos ωt. Notice that for 0 ≤ t ≤ π/ω (as t increases through a period of f), θ decreases from a to 0 to −a. As a consequence f increases from its minimum value of f(0) = Acos a to its maximum value of f(π/2ω) = A, then back to its minimum value f(0) = f(π/ω).

If a is relatively small (say |a| < 1), the variation in θ is relatively small, and the resulting variation in f is small. It is in this case of small inner amplitudes a, that the approximation of f by functions such as  is good. Figure 4 shows graphs of f in the cases that A = 2, ω = 2, and a = 2, 3, 4. We see that the departure from functions of the form of F increases with a. In fact, for a > π, the function develops additional local maxima and minima within the fundamental period.

Figure 4

These properties can be seen clearly if we assume |a| < 1 and expand f in a Taylor series about t = 0. We have

Simplifying cos² ωt by a double angle formula, we see that

We can now compare this representation of f with the function F(t) = C + B cos 2ωt given above, where . Again assuming |a| < 1, we can expand B and C in Taylor series to see that B = B’ and C = C’ up to terms of order a²; that is,

We conclude that the function  is well approximated by functions of the form  provided the inner amplitude a is small. A similar analysis shows that the function  is well approximated simply by F(t) = Aa sin ωt provided a is small.