Final answer:
The statement that the number of resolvable frequencies/harmonics equals the number of data points is false. In Fourier analysis, more data points are needed to precisely resolve multiple frequencies due to the need to account for both sine and cosine terms for each frequency component.
Step-by-step explanation:
The question concerns the representation of a function using a series of cosines and sines, a principle that falls under the study of waves and harmonics in Physics. A common misconception might be that the number of frequencies/harmonics that can be resolved is exactly equal to the number of data points provided. However, this is not necessarily true. In reality, to resolve N frequencies accurately, you would typically need at least 2N data points; this is because each frequency component is described by both a sine and a cosine term, and thus requires two data points for accurate representation.
According to Fourier analysis, both sine and cosine functions can be used to represent a function or wave since they are phase-shifted versions of each other. The wave function oscillates and repeats its pattern regularly, which can be described with these trigonometric functions. Moreover, harmonics are integral multiples of the fundamental frequency, not necessarily mapped one-to-one with the overtones, which is a common point of confusion.
The most important point here is that the mathematical relationships used to represent functions as sines and cosines are valid for all simple harmonic motion, allowing for a wide range of waveforms, including complex ones, to be expressed through these basic functions.