Sinusoidal Signals

Consider a continous-time signal

x(t) = cos(w 0 t)

for some real constant w 0. Use Eulers relation to write this as a sum of two complex exponentials, and then use linearity of the CTFT to find

X(w) = p d (w - w 0 ) + p d (w + w 0 )

where d is the Dirac delta function. What this says is that a cosine in the time domain is concentrated at two frequencies in the frequency domain, one the negative of the other (which should not be surprising).