What you have probably learned is that the autocorrelation function (ACF) of a stationary autoregressive (AR) process decays relatively fast towards zero. But this does not mean that the ACF of any stationary process takes this shape. For example, the ACF of a moving-average process of order $q$ is truncated at lag $q$.
You found another stationary process for which the ACF does not follow a fast decaying pattern. The sine wave that represents your process is stationary because the joint probability distribution does not change over time (i.e., if we take blocks of length, say, 10 observations at different points of the sample, the mean, variance and other moments remain constant across these blocks).
Interestingly, the sine wave can be approximated by a stationary AR process of order 2. For example, an AR(2) process with coefficients 1 and -0.9 generates a cycle of period 6 observations with damping factor $\sqrt 0.9 = 0.95$, i.e., relatively persistent. The theoretical ACF of this process is shown below. The ACF exhibits a periodic pattern. Observe that this AR process is stationary since all the roots lie outside the unit circle.
R code that reproduces the figure above:
arcoefs <- c(1, -0.9)
acf <- c(arcoefs/(1-arcoefs), arcoefs^2/(1-arcoefs)+arcoefs)
for (i in seq_len(20))
acf <- c(acf, arcoefs*acf[1+i] + arcoefs*acf[i])
acf <- c(1, acf)
plot(c(seq(0, length(acf)-1)), acf, type="h", xlab = "Lag", ylab = "acf",
main = "Theoretical ACF of the AR(2) process")
abline(h = 0, lty = 3)
Mod(polyroot(c(1, -arcoefs))) # roots outside the unit circle
# 1.054093 1.054093
# ACF for simulated data (instead of the theoretical ACF)