Why do we always use sine wave to represent a wave in amplitude modulation?
Answer:
Well the tangent function is never used because it is not continuous over more than one period (sinx/cosx = inf if x = pi/2, etc). The cosine wave could be used just as easily as the sine wave, but for generality only the sine function is used (you can get a cosine wave from a sine wave by adding the appropriate phase). So since the sine and cosine waves are basically identical (have same amplitude, and same period, just different phase).
A sine wave looks so elegant and in most of the physical processes, the signal doesn't start with a full amplitude of 1 when time is zero. So, a sine wave represents the real life better than a cosine wave. A tan wave goes to infinity and so is difficult to represent.
textbook wise, it's the easiest way to show the relationship between the real and frequency domain. Remember, Fourier analysis is very important in audio engineering/mathematics. If a more complex method was used it would make subsequent calculations a major pain in the **** and you would eventually have to shift back to a harmonic function (solves the laplace equation) anyway.
It also makes power determination easier: eg: Paley -Weiner criterion etc.
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