Simplification after sampling an analog signal
Just for explanation: What you experience here, is aliasing. Look at this code and its output:

Fs = 80 # the sampling frequency
f = 50 # the frequency of the cos wave
t = np.arange(0, .1, 1./Fs)  # the sample times
t2 = np.arange(0, .1, 1./(10*Fs))  # higher sampling frequency illustration purposes

plt.stem(t, np.cos(2*np.pi*f*t), label='sampled points')  # plot sampled values
plt.plot(t2, np.cos(2*np.pi*f*t2), label='50Hz cos wave')
plt.plot(t2, np.cos(2*np.pi*(Fs-f)*t2), label='30Hz cos wave')
plt.legend()




What I show is a 50Hz cos wave (blue) and the sampling points when sampling with a frequency of 80Hz (blue dots). I also draw a cos wave with frequency 30Hz on top of the diagram (green). 

As you can see, at the sampling points the green and blue line intersect. I.e. when your sampled values do not differ, if you sample a 50Hz or 30Hz wave. This is the aliasing effect: You cannot distinguish between waves that have a different frequency, since the sampling rate $F_s$ is too low. In general, given a sampling frequency $F_s$, waves with frequencie $f$ and $F-f$ create the same sampled values.