How are these 2 frequencies aliases of each other?

Question

I'm trying to understand this solved example from a book, given an analog signal $x(t) = 3\cos(100\pi t) $. This signal has a frequency of $50hz$. If sampled at $F_S=75Hz$ ,I obtain the discrete time signal $3\cos(\frac{2\pi n}{3}) $.

Now I am supposed to find the frequency $F $ lying in the range $0 < F < F_S/2 $ that yields identical samples. Now I get that this frequency will be $F = 25Hz$ by using $f = \frac{F}{F_S}$.

However, I learnt earlier that this frequency is supposed to satisfy the relation $F_k = F_o +kF_S$ for integer values for k, and as it is obvious, there seem to be no integer solutions for this. Where am I going wrong ?

Any help would be appreciated.

Book Reference : https://engineering.purdue.edu/~ee538/DSP_Text_3rdEdition.pdf

Answer 1

Note that the cosine has a positive and a negative frequency component. Consequently, in the range $[0,f_s/2]$ you get a component at $-50\textrm{ Hz}+75\textrm{ Hz}=25\textrm{ Hz}$.