How can i calculate the Fourier transform of a delayed cosine? I haven't found anywhere how to do that.
This is my attempt in hoping for a way to find it without using the definition:
$$ x(t) = cos(2\pi f_ct -θ) = cos\bigg(2\pi f_c\bigg(t -\frac{θ}{2πf_c}\bigg)\bigg) $$ ($f_c$ stands for the fundamental frequency of the signal and $θ$ is the phase shift) Now using the Fourier time-shift property $:$ $$ x(t-θ) \longrightarrow X(f)e^{-j2πfθ} $$ and knowing the fourier transform of $$cos(2πf_ct) = \frac{1}{2}δ(f-f_c)+\frac{1}{2}δ(f+f_c)$$ i get: $$ cos\bigg(2\pi f_c\bigg(t -\frac{θ}{2πf_c}\bigg)\bigg) = \bigg[ \frac{1}{2}δ(f-f_c)+\frac{1}{2}δ(f+f_c) \bigg]e^{-j2πf\frac{θ}{2πf_c}} = $$ $$ = \frac{1}{2}δ(f-f_c)e^{-\frac{jfθ}{f_c}} +\frac{1}{2}δ(f+f_c)e^{-\frac{jfθ}{f_c}} $$
Is this a way to find it? If yes can i simplify it further? And what happens with the simplifications when you're given the $f_c$? Thanks in advance.