finding the frequency of signal without using FFT

hi all

I have sine signal with noise where the number of points of noise per oscillation should be the same in my code five points per oscillation) , I want to change each time the number of oscillation :2,3,4….15 ( in my code changing the vairable "random")

At each number of oscillation I shall extract the amplitude as a function of frequency < unfortunately for few oscillation FFT wouldn't work there are too few points in the signal , so I have to fit the signal ( sine with noise ) to sine wave in order to compare the frequency of new signal to the frequency of the sine wave

see my code? how can I do the fitting ,so I can extract the frequency of the signal

clc
random=40;
f=5; % the frequency of the sine wave also the number of points per oscillation

%the number of oscillation is random/f

t = (1:random)';
 X = ones(random,2);


 y_1= sin((2*pi)/f*t);
  X(:,2) = y_1;
 y=y_1+randn(random,1);
 y = y(:);
 beta = X\y;
 yhat = beta(1)+beta(2)*sin((2*pi)/f*t);
 figure
 plot(t,y,'.b','markersize',12);
 hold on
 plot(t,yhat,'r','linewidth',2);