Sine wave
From HvWiki
Any signal of the form Asin(ωt + φ). Hence the cosine function is considered a sine wave, since shifting a cosine along results in a sine.
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Asin(ωt + φ)
A plot of just sin(t) is shown above.
In this expression A is the amplitude and represents the "height" of the wave. Increasing A is like increasing the AC voltage or sound signal loudness.
ω is called the angular frequency and is equal to 2πf.
t is time in seconds.
φ is the phase shift. A positive φ simply shifts the whole wave to the left (negative t direction) or, in other words, makes each peak come sooner.
Mathematicians generally use the radian measure for angles and it will be used on this page. In this system, a full revolution is equal to 2π radians. The Units page should have more details one day.
Voltage
The three most important values are:
- Vp - peak voltage
- Vrms - effective voltage (root mean square)
- Vav - average voltage
General Properties
The maximum and minimum values of Asin(ωt + φ) are A and - A respectively.
Asin(ωt + φ) is equivalent to Acos(ωt + φ - π / 2). So cos is just a phase shifted version of sin.
Adding and Multiplying
Any product of sine waves is equivalent to a sum and, conversely, any sum of sine waves is equivalant to a product. See this List of trig identities for more details.
Sum to Product
Product to Sum
Means
Arithmetic
The overall average of any sine wave is zero, however, each hump has an average of 2A / π. This is only sometimes useful in engineering practice.
Root mean square
Commonly abbreviated RMS.
To calculate the RMS of a wave you square it, find the average and then square root. This is not the same thing as just averaging. For example, the RMS of three discrete values is
The RMS value of a sine wave is 
. Note that the RMS value of the wave is often given instead of the amplitude, A. This is important to remember when building a rectifier (See Diode) or considering voltage ratings. The peak of an AC voltage is 
times the quoted RMS value.
Sine Waves and Calculus
Differentiation and Integration
The derivative of sin is cos. The derivative of cos is − sin. Remember, "cos-us to sinus gets you a minus." This has the result that integrating or deriving a sine wave simply causes a phase shift. To be pedantic, it may also change the amplitude depending on what the frequency is.
Fourier
Any physically realisable signal can be decomposed into a sum of sine waves. This allows any theory applying to sinusoidal signals in a linear system to be generalised to all signals. See Fourier Analysis.
