How can I derive the magnitude of the fourier transform of sine wave?

Question

building on this answer https://math.stackexchange.com/a/2152573/845257 . I was deriving the Fourier transform of sine wave y(t) = sin(2.pi.f.t) and I got it right. Now, I want to derive the magnitude I can't find how he got the magnitude of the sine Fourier transform like that.

Answer 1

It is well known that if $\,z\,$ is a complex number with real part $\,a\,$ and imaginary part $\,b\,$ then the magnitude of $\,z\,$ is $\,\sqrt{a^2+b^2}.\,$ so far so good. Note that if you have a complex function $\,f(\omega)\,$ then to get its magnitude you have to find the real and imaginary part for each value of $\,\omega.\,$ In the case of interest here we have $$ f(\omega) := -\frac{i}{2}\delta(\omega -\omega_0) + \frac{i}{2}\delta(\omega + \omega_0). $$ For each value of $\,\omega\,$ we have to find the imaginary and real parts of $\,f(\omega).\,$ Notice the Dirac delta function $\,\delta()\,$ which means that the function is zero everywhere except at the two values $\,\omega=\omega_0\,$ and $\,\omega=-\omega_0\,$, which implies the magnitude is zero also except at the two values. Now evaluate the functional expression for those two exceptional values of $\,\omega\,$ and you will get the expression given in the answer to the question you refer to.