Trig graphs are similar to the Unit Circle in the sense that they both possess the values of the quadrants. The period for sine and cosine is 2π because the sign of the function changes with every two quadrants. The period for tangent and cotangent is π because the sign of the functions changes with every other quadrant.
The fact that sine and cosine have amplitudes of 1 as seen in the Unit Circle means that they start at zero, or the origin of a coordinate plane. Since the other trig functions have amplitudes outside the values of the Unit Circle means that the entirety of their graphs shifts along the y-axis accordingly.