The period for sine and cosine is 2pi because that's how much the distance is for the pattern of sine and cosine to repeat they have to go a full revolution in the unit circle whereas the period of cotangent and tangent is pi because it only takes a half a revolution for these two to repeat their patterns.
The unit circle only reaches a radius of one, we all know that, and sine and cosine have one as their amplitude because of the unit circles radius. Other trig functions don't have restrictions which is why they don't have amplitudes
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