Bandpass filter on noisy sinusoid that predicts turning points
Assuming your change of amplitude is very slow when compared to the frequency of the sinusoid, this problem can be posed as finding frequency of a noisy sinusoid. You can take small windows of the signal and then take it's fourier transform. Since we know that the amplitude is almost constant and that there is only one sinusoid, we can look for two peaks in the DFT of the signal with some side lobes(Due to windowing). Your goal would be to find those two peaks, which would reduce the error energy in your estimate. Once you have the frequency and the phase information, finding turning points is very easy.

Hope this helps.