According to the book "Linear Partial Differential Equations for Scientists and Engineers" by Tyn Myint-U. and Lokenath Debnath, on page 446
Many simple functions, such as a constant function, $sin ωx$, and $x^n H (x)$, do not have Fourier transforms even though they occur frequently in applications
But from some tables of Fourier trasnfrom
$$\mathscr{F}_x(1)=\delta(k)$$
and
$$\mathscr{F}_x[\sin(x)](\omega)=i\sqrt{\frac{\pi}{2}}(\delta(\omega-1)-\delta(\omega+1))$$
Can someone please explain?