Sinusoid

Sinus function for A = ω = 1 and φ = 0.

In mathematics, the curve that graphically represents the sine function and also the function itself is called a sinusoid or sinusoid. It is a curve that describes a repetitive oscillation and soft.

Its most basic form as a function of time (t) is:

${displaystyle y(t)=Aoperatorname {sen}(omega (t+varphi))}$

The sinusoid is important in physics due to the fact described by Fourier's theorem which says that every wave, whatever its shape, can be uniquely expressed as a superposition (sum) of sinusoidal waves of defined wavelengths and amplitudes For this reason, this function is used to represent both sound waves and alternating current waves.

Features

Figure 1: Parameters characteristic of a sinusoidal form.

The sinusoid can be described by the following mathematical expressions:

${displaystyle y(x)=A {rm {sen}}left(omega x+varphi right)}$

${displaystyle y(t)=Aoperatorname {sen}(2pi ft+varphi)}$

${displaystyle y(x)=A {rm {sen}}left({frac {2pi }{T}}x+varphi right)}$

where

• ${displaystyle A}$ is the range of oscillation.
• ${displaystyle omega }$ is the angular speed; ${displaystyle omega =2pi f}$.
• ${displaystyle f}$ is the frequency of oscillation.
• ${displaystyle T}$ is the oscillation period; ${displaystyle T={1}/{f}}}$.
• ${displaystyle omega (x}$ + ${displaystyle varphi)}$ is the oscillation phase.
• ${displaystyle varphi }$ It's the initial phase.

Period (T) in a sinusoid

It's the least set of values ${displaystyle x}$ which corresponds to a complete cycle of function values; in this sense all function of a variable that repeats its values in a complete cycle is a periodic, senoidal or sinusoidal function.

In the graphics of the sin-thing functions the period is ${displaystyle 2pi }$.

Amplitude (A) in a sinusoid

It is the maximum absolute value departure from the curve measured from the x axis.

From a more technical point of view, the breadth of the sinusoid is the rule of the supreme sinusoid: ${displaystyle A=abiyinftyinfty }=sup _{xin mathbb {R} }healthy(x)$

Initial phase (φ) in a sinusoid

The phase gives an idea of the horizontal displacement of the sinusoid. If two sinusoids have the same frequency and the same phase, they are said to be in phase.

If two sinusoids have the same frequency and different phase, they are said to be out of phase, and one of the sinusoids is leading or lagging > with respect to the other.

It makes no sense to compare the phase of two sinusoids with different frequencies, since they are in phase and out of phase periodically.

Sinusoid and cosinusoid

The graphic representation of the breast and the cosine are sinusoidal functions with different phases

Note that the cosinusoid (cosine), or any linear combination of sine and cosine with the same frequency, can be transformed into a sinusoid and vice versa, since:

${displaystyle {A {rm {sen}}left(omega x+varphi right)=M{rm {sen}}}}left(omega xright)+Ncos left(omega xright)}$

being

• ${displaystyle A^{2}=M^{2}+N^{2},}$
• ${displaystyle omega varphi =arctan {frac {N}{M}}}}$

Yeah. MConsider it ${displaystyle omega varphi =arctan {frac {N}{M}}}$

For the particular case ${displaystyle omega varphi =pi /2}$:

${displaystyle {{rm {sen}}left(omega x+pi /2right)={rm {cos}}}}left(omega xright)}}$

i.e., the sinus function and the cosine function is the same sinusoid detached (displaced) ${displaystyle pi /2}$.