Why are cosine and sine functions used when representing a signal or a wave? [closed]

Question

Actually, in the mathematics sine and cosine functions are defined based on right angled triangles. But how will the representation of a wave or signal say based on these trigonometric functions (we can't draw any right angled triangles in the media, i.e., the air) then how can we say that?

Answer 1

While Sine and Cosine functions were originally defined based on right angle triangles, looking at that point of view in the current scenario isn't really the best thing. You might have been taught to recognize the Sine function as "opposite by hypotenuse", but now it's time to have a slightly different point of view.

Consider the unit circle $x^2+y^2=1$ on a Cartesian plane. Suppose a line passing through the origin makes an angle $\theta$ with the $x$-axis in a counterclockwise direction, the point of intersection of the line and the circle is $(\cos{\theta},\sin{\theta})$.

Think about it. Does this point of view correlate with the earlier one? Both of the definitions are the same.

So you'd wonder, why do we need this point of view? Well, I'd say it's easier to understand how Sine waves are actually important in many common phenomena. Suppose we start to spin the line, by making $\theta$ increase linearly. You'd get something like this:

The Sine and Cosine functions are arguably the most important periodic functions in several cases:

1. The periodic functions of how displacement, velocity, and acceleration change with time in SHM oscillators are sinusoidal functions.

2. Every particle has a wave nature and vice versa. This is de-Broglie's Wave Particle Duality. Waves are always sinusoidal functions of some physical quantity (such as Electric Field for EM Waves, and Pressure for Sound Waves).

Sound itself is a pressure disturbance that propagates through material media capable of compressing and expanding. It's the pressure at a point along the sound wave that varies sinusoidally with time.

Speech signals are not perfectly sine waves. A pure sound, from a tuning fork would be the perfect sine wave. Regular talking is not a pure sine wave as people don't maintain the same loudness or frequency. As a result, this is what noise looks like, compared to pure frequencies.

Notice the irregularities in the amplitude and the frequency of the noise wave.

3. Alternating voltages used in your everyday plug sockets are infact sinusoidally varying voltages as a function of time.

tl;dr:

Considering Sine waves as "opposite by hypotenuse" is far from the best comparison when dealing with everyday applications of physics.

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Further reading:

1. History of Trigonometry
2. Sound
3. Wave model of Electromagnetic radiation
4. Alternating current
5. Simple Harmonic Motion

Answer 2

The trigonometric functions form a basis for the space of "reasonable signals". (For the purposes of this answer, "reasonable signals" are continuous functions having finite energy and bounded power.) The word "basis" here is meant exactly the way it is used in linear algebra. (This is explicitly discussed on the linked page.)

Why would anyone use this basis?

tl;dr: This basis captures our experience of signal decomposing into spectral or frequency components. Also, it has mathematical properties making some physically relevant differential equations easy to solve.

Physically, it corresponds to our experience that sounds are combinations of frequencies at various amplitudes and phases. A musical chord is a simple example of this. The differences between musical instruments playing the same note is largely about the amplitudes of higher frequency components also produced by the instrument when playing the note. It also corresponds to our experience looking at light after it has passed through a prism. (We now know that) This light has been split into its various frequency components. So these phenomena suggest it would be convenient/fruitful to find a way to express a complicated signal as a sum of simple periodic functions. Note that both of these phenomena are transported through media in which there is no physical triangle corresponding to the observed signal. (For the sound, the longitudinal waves seem to lead to degenerate triangles with all three sides parallel to the direction of travel. For the light, the "triangles" point in the direction of the electric or magnetic fields, which are, in some sense, perpendicular to space.)

Mathematically, this basis is convenient because differentiation converts basis elements to vectors in the same basis with the same frequency. For instance, $\frac{\mathrm{d}}{\mathrm{d}t} \sin(9t) = 9 \cos(9t)$. This is unlike what happens to polynomials, where differentiation takes polynomial basis elements (which are $t^n$ for integers $n$) to polynomials of a different degree, e.g., $\frac{\mathrm{d}}{\mathrm{d}t} t^2 = 2 t$. There are of course (infinitely many) other bases, but most of them use functions you don't know or wouldn't recognize. (Examples: Daubechies wavelets, coiflets) The first basis you ever use could be called "the Dirac basis" (although no one does). In this basis, a signal is a sum of time-shifted and amplitude scaled copies of the function $p(t) = \begin{cases} 1 & t=0 \\ 0 &\text{otherwise} \end{cases}$, one for each instant of time, that each specify the amplitude of the signal at that time. Unlike the trigonometric basis elements, the $p$'s derivative isn't even a function, so is a difficult beast to work with. Generally, elements of other bases do not differentiate as nicely as the trigonometric functions. So while there are many periodic bases we could use to represent a signal, the trigonometric function basis has some nice properties that the others don't.

One consequence of this nice interaction with differentiation is that one can reduce some differential equations to algebraic equations. In particular, the operation $\frac{\mathrm{d}}{\mathrm{d}t} f(t)$ in the Dirac basis becomes $2 \pi \mathrm{i} \omega \hat{f}(\omega)$ (where the "hat" means the version of $f$ after changing basis to the trigonometric functions and $\omega$ is a frequency). Say we want to solve $$ \frac{\mathrm{d}^2}{\mathrm{d}t^2} f(t) = -k^2 f(t) \text{,} $$ the sort of equation that would come up in studying resonance (for instance in musical instruments or in radio transmitters and receivers). If we change basis to the trigonometric functions, this becomes $$ (2 \pi \mathrm{i} \omega)^2 \hat{f}(\omega) = -k^2 \hat{f}(\omega) \text{.} $$ Notice there isn't any calculus left; this is just algebra. $$ \left( (2 \pi \mathrm{i} \omega)^2 + k^2 \right) \hat{f}(\omega) = 0 \text{,} $$ so either $\hat{f}(\omega) = 0$ for all $\omega$, meaning the signal is zero, or $(2 \pi \mathrm{i} \omega)^2 + k^2 = 0$. Simplifying: \begin{align*} 0 &= (2 \pi \mathrm{i} \omega)^2 + k^2 \\ &= k^2 - (2 \pi \omega)^2 \\ &= (k+2 \pi \omega)(k - 2 \pi \omega) \text{.} \end{align*}

So if $\omega$ is either $\pm \frac{k}{2 \pi}$ (in units of Hertz), we find a solution to our system. That is, there is only one frequency that is a solution to our resonance equation. (It is present in either increasing phase or decreasing phase forms, but both have the same frequency.)

But the point is, switching from the Dirac basis to the trigonometric basis made it much easier to solve this problem. Instead of having to solve a differential equation, we just had to do a little algebra.

Technical comment:

There are several conventions for Fourier transforms (used to go to the trigonometric basis above). You may be used to a different one. In my work, I'm usually using $$ \mathcal{F}_{t \rightarrow \omega}(f)(t) = \int_{-\infty}^{\infty} \; f(t) \mathrm{e}^{-2 \pi \mathrm{i} \omega t} \, \mathrm{d}t \text{.} $$ In Mathematica, this corresponds to FourierParameters -> {0, -2 Pi} in FourierTransform and InverseFourierTransform. The same form is used in the second column of the table here. There are other conventions for how the "$2 \pi$" is split between the forward and inverse transforms and whether the $\omega$ is Hertz or radians per second.

Answer 3

Mathematics has progressed from geometry to calculus and differential equations. It is established that differential equations whose solutions describe waves have sinusoidal functions in those solutions. This should not be surprising, as waves are periodic in time or space, and sines and cosines are periodic functions. As a consequence differential equations that have sinusoidal solutions are called wave equations. See also this link .

Answer 4

Periodicity

Some excellent answers on the $\sin x$ and $\cos x$ functions and how they're solutions to the relevant differential equations were already given, but an important point can still be mentioned: Sine and cosine are used because they are periodic and signals/waves are usually considered to be or are approximated by periodic functions.

In many cases polynomials are the go-to functions for approximating other functions $f(x)$ (see, e.g., Taylor series), but they typically have the disadvantage of diverging for values of $x$ away from a small chosen range. Sine and cosine, being periodic, do not have this problem.

Answer 5

Sines, as functions of time, are not just features of geometric interest. The behavior of some time-dependent electrical components (capacitors, inductors) are simple (linear, in a sense) if and only if one excites those components with a known signal frequency, such as the function sine(omega * t + phi). When 'phi' = pi/2, that's cosine(omega * t).

Much signaling is done with modulated narrow-band signals, because the narrow band allows for efficient rejection of noise. Noise, unlike a signal, is present at ALL frequencies simultaneously, thus is mainly outside the sensitive frequency range of a narrow-band receiver. Radio, television, hard disk magnetization patterns, Ethernet, Wi-Fi... all the best modern communication channels rely on frequency-selection to some extent. Understanding those methods, and the inductors and capacitors that implement them, requires a selected-frequency test signal for each calculation. A pure frequency being a sine(omega *t + phi).

This implies a lot of discussion of sinewaves, generators of sinewaves, and graphs against frequency (meaningful only after decomposition of a complex function into... a multiplicity of sinewaves).

Answer 6

First I like to point out the ambiguity of your question. Smoke signals do not use sine or cosine functions. Neither does an ocean wave! So, I am going to assume you mean constant amplitude, periodic, sound or electromagnetic waves used as signals.
Lets start by not knowing what shape a wave, as described above, has. We then use an apparatus that measures the amplitude of the wave as a function of time (or distance) from the source. We start measuring when the wave has o amplitude and measure subsequent points. We then graph the points and then try to fit a known trigonometric function and find that sine(t) fits as accurately as we can make the measurements!
We then conclude that this wave can be accurately represented by a trigonometric function.
We also use a simple permanent magnet generator and measure the output voltage as the rotor is rotated. We then graph the voltage amplitude as a function of rotating angle $\phi$ and as before, we find we can fit a trigonometric function sine($\phi$), accurately. This reinforces the notion that waves are accurately represented by the (sine/cosine) trigonometric functions.