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the average value of sin(t) over a 2Π period equals zero and not 2Π. This is because ∫sin(t) dt from 0 to 2Π = cos(0)-cos(2Π)=1-1=0.
The rule is that the average value of a sinusoid over one complete period is zero. Note that the period of sin(t) is 2Π.
Also the period of sin(2t) is Π. If you integrate this signal over 2Π period i.e. over two complete cycles you also get a zero.
try in this way. For the sine wave upper and lower half are equal so find the area of upper or lower part of the signal multiply it with 2 to get the total area and then divide by the base which is 2Π. You will get the average value as 2/Π...
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