A sinusoidal wave is a finest waveform that oscillates means moves above and below zero periodically which is shown in figure. This kind of wave pattern occurs in wind, sound and light etc. The alternating changing of voltage and current are also kind of sinusoidal wave (sine wave). The sine wave shows the how the amplitude changes with the time. The graphically representation of alternating power is also a sine wave. The ideal natural wave is distortion less or harmonics consist of only the fundamental frequency. But in practically there could be distortions.
What is Sinusoidal Wave or Sine Wave Signal?
A sinusoidal wave or sine wave signal is a mathematical curve that explains a smooth oscillation over a time period. A sine wave is continuous and its graph based on sine or cosine function. Sine Wave or Sinusoidal Wave Signal is a special kind of signal. It is given by the function. Its mathematical expression and figure of sine function is given below:
Y(t)= A sin(2πft+ φ)=A sin(ωt+ φ)
Where
A is the amplitude,
F is the frequency,
ω = 2πf, angular frequency,
φ is phase
What is a Signal?
In surrounding world, there are different measurable quantities some quantities are constant like acceleration due to gravity velocity of sound, velocity of light and many more. Some are time varying quantity like AC voltage, temperature, pressure. These quantities are changes with time or it is time dependent quantity.
“Signal is a value of quantity that has taken over a period of time”
Signal is time dependent quantity in nature. It is plotted between values at different time instants. This is known as graphical representation of signal.
The AC wave consists of frequencies that are multiplier of fundamental frequency. For example the fundamental frequency is 5 kHz the multiplier 10 kHz, 15 kHz, 20 kHz, and so on. The harmonics also is present in the waveform. The harmonics produce a heating problem in power line, transformer, load and connecting wire. The heating problem reduce by trip the circuit quickly and also reduce the power factor.
The harmonic are reduced by using the filter circuit and filter circuit consists of resistor, capacitor and inductor. The use of filter circuit is block a frequency that are not usable or use less for the system.
In figure shows the filtering of harmonics frequency to get pure sinusoidal wave and the filtering circuit is consist of RC filter circuit. The function of inverter circuit is generating different kinds of sine wave. The types of inverter are pure sine wave, square wave and modified wave. These three waves are described below:

Pure Sine Wave
The pure sine wave is produced by Power Plant Company and some electronics sine wave generator/inverters. The costs of inverters are high. The inverter is most advantages in many ways because it works without any problem or any distortion in wave. The uses of pure sine wave are to run complex devices, medical equipment, microwave and many more.

Square wave
It is cheapest among all inverters are square wave. The use of square wave inverter in many electrical equipment. It is not reliable or less reliable from the others and it is also unsafe for some application especially for the electronics equipment.

Modified sine wave
It is also called quasisine wave. It is similar to the square wave but not exactly the square wave. It is much distorted sine wave.
Generation of Wave
The sine wave is generated by various methods. These method are listed below
 Phase Shift Oscillator
 Quartz Crystal Oscillator
 Wein Bridge Oscillator.
 Negative Resistance Oscillator
 Basic Single Coil AC Generator
The basic method for generating sine wave is “Basic single coil generator” which is already explained below.
Basic Single Coil AC Generator
If a conducting rod or wire is moved in a stationary magnetic field “EMF” (ElectroMotive Force) is induced within the conductor due to the movement of the conductor through the magnetic flux.
From these phenomena we can establish relation between electricity and magnetism. The English scientists Michael Faraday discover the effect of “Electromagnetic Induction” and establish relation between electricity and magnetism. The electromagnetic induction is a basic principle of electrical machine and generation of a sine wave or sinusoidal wave in our AC supply.
According to the electromagnetic induction theory, a conducting rod is placed in a magnetic flux. When conducting rod is rotate a magnetic flux cut by conducting rod, due to this an EMF (electromotive force) induces in conducting rod. When the conducting rod is parallel with magnetic flux lines no flux cut by conducting rod hence no EMF induce in conductor, but if the conducting rod moves at right angle to the magnetic flux means conductor become at point C and D, the maximum amount of flux cut and produce maximum EMF voltage in a conductor. The value of EMF induce in a conductor is depends upon the angle between the conductor and magnetic flux line and well as strength of magnetic flux.
An AC generator uses the principle of Faraday’s electromagnetic induction to convert mechanical energy into electrical energy. A simple generator consists of pair of permanent magnetic material that produce magnetic flux between North Pole to South Pole.
What is R.P.M.?
The R.P.M. is a rotation per minute. It means the number of revolution in one minute is called “RPM”. Suppose the shaft of motor is completing 100 rotations in one minute, than it is known as the speed of motor is 100 RPM.
In a motor the number of poles is always even.
We know ω = 2πf, but in case of rotation takes place because of magnetic poles,
ω = (2 / n) [2πf] where n is the no of poles
we know n = 60 f
Number of revolutions,
N _{p} =(2×60)f/p
ω_{rotor}=(2/Poles) x 2 πf(rad/sec)
N_{p}=120f/Poles (RPM)
Where
 ω is the angular velocity
 N is the number of poles.
 F is the frequency of sine wave.
 π is a constant value which is 3.1416.
The number of poles vs Revolution per mints when frequency is 60 Hz.
P (Number of poles)  N (Revolution per mints) 
2  3600 
4  1800 
6  1200 
8  900 
10  720 
The number of poles vs Revolution per mints when frequency is 50 Hz.
P (Number of poles)  N (Revolution per) 
2  3000 
4  1500 
6  1000 
8  750 
10  600 
Instantaneous Voltage
Instantaneous voltage is found out between two points at a particular moment in time. The voltage of a sine wave at a instant given time is called “Instantaneous voltage”.
In the above diagram the instantaneous voltage of sine wave is v1, v2, v3, v4……
The mathematical formula of instantaneous voltage is given below:
Instantaneous voltage = Max voltage x sin θ
V_{inst }= V_{Max }x sin θ
Angular Velocity
The angular velocity is the rate of change of angular position with respect to time. It is denoted by ω and its unit is rad/sec.
ω = 2π f (rad/s)
The frequency slandered of AC current in India is 50 Hz, and then angular velocity is 314.16 rad / sec.
Frequency is inversely proportional to time i.e. f = 1 / T.
ω=2 π / T (rad/s)
Sinusoidal wave formula
RMS value of sin wave
We know the instantaneous value of voltage/current of sinusoidal wave is:
I = I _{m} sin θ
The mean of square of instantaneous value of current over half or full cycle is
I_{RMS} = 0.707 x Max Value of Current
RMS Value for Different types of Wave form
In below given table shows formula of different types of wave forms:
Waveform Type  Formula for RMS Value (V_{RMS})  Formula for Average Value (V_{av}) 
Sine Wave  V_{Max} / √2  2V_{Max} / √2 
Half wave rectified sine wave  V_{PK} / 2  V_{Max} / π 
Full wave rectified sine wave  V_{Max} / √2  2V_{Max} / π 
Square wave  V_{Max}  V_{Max} 
Triangle waveform  V_{Max} / √3  V_{Max} / 2 
Sawtooth waveform  V_{Max} / √3  V_{Max} / 2 
Half square wave  V_{Max} / √2  V_{Max} / 2 
Half sawtooth wave  V_{Max} / √6  V_{Max} / 4 
Also read: Amplitude modulation, Frequency modulation.