Tau is the clear winner in periods of trig functions. The two most important ones, sine and cosine, have a period of tau, and overall 2/3 of them have periods of tau, and the ones that have periods of pi only have them because the values repeat halfway around the circle. Additionally f(x)=f(x+τ) applies for all trig functions. f(x)=f(x+π) only works for tangent and cotangent. If we go outside the six basic trigonometric functions, we see even more wins for tau. The complex exponential has a period of tau, and the phase angle of it has both a period and a range of tau. Pi gets second place here because the periods of the other 1/3 of the trig functions is pi, and eta gets last.
Tau also wins when looking at special values. Every special value of any trig function can be expressed as τ(n/k+c) when k is the number of times that special value shows up on the unit circle, c is the offset (meaning cτ is the first time that special value shows up), and n is any integer. For example the maximum of the cosine function, one, shows up once every time you go around the unit circle, so k=1, and the first value of cosine that equals one is cos0=1, so c=0. This means that to find every value for which cosθ=1, you use nτ (for every integer n). Eta gets second place here because most special values occur at multiples of eta, and pi gets last place.