Tables of common Discrete-Time Fourier Transform pairs list the transform of a sine wave:
$ \sin(\omega_0\ n) $ and its transform: $ -j\pi\ [d( \omega\ - \omega_0\ ) - d( \omega\ + \omega_0\ )] $
And the cosine:
$ \cos(\omega_0\ n) $ and its transform: $ \pi\ [d( \omega\ - \omega_0\ ) + d( \omega\ + \omega_0\ )] $
How might the results differ if the sin or cosine is causal? That is, I would like to determine the Fourier Transform of the following (sin or cosine multiplied by the unit step):
$ \sin(\omega_0\ n) u[n] $ or $ \cos(\omega_0\ n) u[n] $
So regarding the DTFT of the signals above, is the only method to realize that multiplication in the time domain is convolution in the frequency domain? Or is there a more simple rule-of-thumb, or property, of the Fourier Transform that I am missing?
(This is not homework, but it is for studying, so it would be helpful to include resources along with answers.)