Find Sine function without given Minimum So, recently for a grade 11 school math project, I collected data, which I knew took a form of a sine graph, but the values which experimented to find were chosen randomly. Faith the data collected, I got the maximum point, but did not include the minimum point. So is there a way to find an equation of sine regression if no minimum is given/shown? 
 A: Given $n$ data points $(x_i,y_i)$ for a sine wave means fitting the model
$$y=A+B\sin(Cx+D)$$ and it is a quite difficelt task (google for fitting a sine wave).
From a practical point of view, it is easier to expand the sine term and consider instead
$$y=a+b\sin(cx)+d\cos(cx)$$ Taking into account the fact that your $x_i$'s are given in degrees, I prefer to rewrite the model as
$$y=a+b \sin \left(c\frac{\pi   x}{180}\right)+d \cos \left(c\frac{\pi  x}{180}\right)$$This problem is nonlinear (because of the $c$ parameter). If $c$ was known, it would just be a multilinear regression easy to do.
So, for the time being, give $c$ a value; for this value, compute $a,b,d$ and the sum of squares $SSQ$ (all of that can easily be done using Excel). Now, plot $SSQ$ a a funtion of $c$ and  locate its minimum. When done, you have all the estimates required for the nonlinear regression for a fine tuning of the parameters.
Using the data you posted, $c=4.5$ seems to be a good candidate. Using this value and the corresponding $a,b,d$ obtained by the preliminary  multilinear regression, a nonlinear regression will give
$$a=34.442 \qquad b=6.987 \qquad c=4.526 \qquad d=8.002$$ For these values, the table below reproduces your data and the predicted values from the regression
$$\left(
\begin{array}{ccc}
 x & y & \text{predicted} \\
 90 & 48.88 & 45.003 \\
 68 & 33.98 & 33.820 \\
 41.2 & 26.58 & 25.704 \\
 33.2 & 30.72 & 30.960 \\
 20.4 & 38.59 & 41.098 \\
 0 & 40.28 & 42.445
\end{array}
\right)$$ which is not fantastic even if $R^2=0.9968$ seems to be be quite good.
Now, taking the derivative of $y$ with respect to $x$
$$y'=\frac{\pi  c}{180}  \left(b \cos \left(c\frac{\pi  x}{180}\right)-d \sin
   \left(c\frac{\pi   x}{180}\right)\right) $$and solving the trigonometric equation
$$b \cos \left(c\frac{\pi  x}{180}\right)-d \sin
   \left(c\frac{\pi   x}{180}\right)=0 \implies x=\frac{180 }{\pi  c}\tan ^{-1}\left(\frac{b}{d}\right)$$ gives $y_{max}=45.066$ at $x=88.63 ^{\circ}$ and  $y_{min}=23.819$ at $x=48.86 ^{\circ}$.
