When studying physics, I often come across sines and cosines, and while I understand they're an elegant and useful way to describe systems with periodic characteristics, I can't help but wonder whether there are other ways to describe these systems mathematically without using these functions. All functions in this list of periodic functions from Wikipedia seem to contain references to trigonometry.
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$\begingroup$ Suppose you could just use complex exponentials all the time but it's really just trig in disguise. $\endgroup$– TriatticusCommented Jan 28, 2021 at 3:49
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$\begingroup$ @Triatticus So there's no escaping trigonometry? $\endgroup$– polythenemanCommented Jan 28, 2021 at 4:13
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$\begingroup$ @Triatticus Or maybe trig is complex exponentials in disguise, but we insist that plane geometry is more fundamental and ignore the reality of $i^2=-1$ :) $\endgroup$– Bill NCommented Jan 28, 2021 at 4:49
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$\begingroup$ The conflict might go away if you consider sin and cos to be circular functions and get rid of "trigonometry" and it's implied connection to triangles. $\endgroup$– garypCommented Jan 28, 2021 at 12:43
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1$\begingroup$ @BillN certainly from our point of view that's true, but for most people a complex exponential certainly would make their work more complicated. I don't disagree with that point as the commplex exponential was one of the coolest things I remember first learning about. $\endgroup$– TriatticusCommented Jan 28, 2021 at 16:14
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2 Answers
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The reason why trigonometry comes up al the time is that every periodic function has an associated Fourier Series; which essentially a frequency spectrum, composed of harmonic waves. And harmonic waves are inherently connected to triangles through the unit circle. We can have alternate definitions of what sin(x) and cos(x) mean, but the connections between periodicity, harmonic waves and trigonometry is inherent.
To elaborate on how harmonic waves can be They can be represented in general using differential equations; for continuous periodic functions, the simplest case would be:
$$\frac{d^2f}{dx^2} = -\omega^2f$$
Has solutions:
$$f=A\text{cos}(\omega x - \phi)$$
Where $ A$ and $\phi$ depend on initials conditions.This is the general form of 'sine wave'; and from this definition we can obtain everything else as well without refference to nay triangles (although the connection is still there)
Personally; I find the inherent connection to triangles one of the most beautiful aspects of physics and mathematics; and that results of people drawing shapes in the sand 2000+ years ago permeates physics, math and engineering at almost every level.
answered Jan 28, 2021 at 4:22
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$\begingroup$ Or you can do it with $e^{\pm i \omega x}$ without reference to trig functions. $\endgroup$– Bill NCommented Jan 28, 2021 at 4:51
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$\begingroup$ @BillN: I'd argue that $\exp(\mathrm ix)$ is the original trigonometric function. You can construct all other trig functions from it using only field operations and field automorphisms. You're not escaping trig functions by using the complex exponential, you're just unmasking them. $\endgroup$ Commented Jan 28, 2021 at 9:08
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$\begingroup$ @Vercassivelaunos Maybe so. :) They (sin, cos, $e^{ix}$) are all just infinite power series in a wrapper, convenient for manipulation. I like the triangle based definitions, then I watch them connect to other domains. It is indeed beautiful. $\endgroup$– Bill NCommented Jan 28, 2021 at 14:08
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Yes, it is possible. Fourier series is just a way to represent a periodic function as an infinite sum of sinus and cosinus functions, but it is not unique. From the mathematical point of view, trigonometric functions (or complex exponentials) correspond to a specific choice of a basis in an infinite-dimensional vector space (Hilbert space) of periodic functions. However, such a basis is not unique. Indeed, other representations are possible. A trivially simple alternative to represent any periodic function of period $L$ is $$ f(x)=\sum_{n=-\infty}^{\infty} \phi(x-n L), $$ where $\phi(x)$ is a function, defined in the interval $[0,L]$, corresponding to the behavior of the periodic function over a single period. Therefore, any choice of a basis in the interval $[0,L]$ would result in a corresponding basis for periodic function.
For example, any basis $P_n(x)$, made by orthogonal polynomials over $[0,L]$, would originate a basis for periodic functions: according to the previous formula.
The Fourier series made by trigonometric function is dominant in studying periodic phenomena due to different reasons. Two important reasons are the following. In some cases, the periodic phenomenon is due to linear differential equations whose solutions are combinations of trigonometric functions. Moreover, there are very efficient ways to manipulate Fourier series (Fast Fourier transform algorithms) numerically.
answered Jan 28, 2021 at 8:49
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