Signal sin(x), traveling at frequency of 0.2 cycles per second, will travel 72 degrees per second.

When sampled each one second, we will read sin(0),sin(72), sin(144), sin(216), sin(288),sin(360),sin(432),sin(504), etc…

i.e. we will read 0, 0.95, 0.5877, -0.5877, -0.95, 0, 0.95

Take the same signal, but let it run at frequency of 0.8 cycles per second. So it will now travel 288 degrees per second. When sampled once per second, we will now read: sin(0), sin(288),sin(576),sin(864),sin(1152),sin(1440). Etc…

i.e. we will read 0, -0.95, -0.5877, 0.5877, 0.95, 0, -0.95

compare the above two, and you will see that there are the same values read, but different signs.

Ignoring the sign, and taking the absolute value, this means that the sin signal, running at a frequency 0.8 cycles per second when sampled once per second, will give the same values are if it was running at frequency 0.2 cycles per second.

Another example.

Signal sin(x), traveling at frequency of 0.2 cycles per second, will travel 72 degrees per second.

When sampled each two second, we will read sin(0),sin(144), sin(288),sin(432), etc…

i.e. we will read 0, 0.5877, -0.95, 0.95

Take the same signal, but let it run at frequency of 0.3 cycles per second. So it will now travel 108 degrees per second. I.e. will travel 0,108,216,324,432,540,648, etc.. degrees per second, When sampled once per two second, we will now read: sin(0), sin(216),sin(432),sin(648),etc..

i.e. we will read 0, -0.5877, 0.95, -0.95

compare the above two, and you will see that there are the same values read, but different signs.

Ignoring the sign, and taking the absolute value, this means that the sin signal, running at a frequency 0.2 cycles per second when sampled twice per second, will give the same values are if it was running at frequency 0.3 cycles per second.

How to find how many such other frequencies exists given a signal, its frequency and the sampling rate, will give the same values?

Use the fourier transform. The fourier transform will give all frequencies that will give the same values for the same sample rate.