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It is a series of impulses (Dirac Delta functions), each positioned at the point where the square wave changes amplitude. For a negative change in amplitude, the impulse is -inf; likewise, for a positive change in amplitude, the impulse is +inf.
The fourier expansion of the square wave is:
4/pi sum_{k=1}^\infty [ sin((2k-1)*w*t) / (2k-1) ] Likewise, its derivative is:
4/pi*w sum_{k=1}^\infty [ cos((2k-1)*w*t) ]
If you plot, the first 50 terms, you will get a pretty good idea of what it looks like. impulse
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How is trig used in life?
The obvious answer is the relationships between the sides and angles of triangles. Waves in the sea are an example of a sine wave. Tidal Experts and Meterologists alike use sine waves to help predict tides. Music will also emit waves that may often look like a sine wave and pure notes will look like sine or cosine waves. The speed of a swinging pendulum can be plotted as a sine wave as well as the sound of a tuning fork.
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