What is the sinusoidal equation for this wave?

Question

What is the sinusoidal equation for this wave? I don't need the exact equation, as there isn't one. I know how to solve it, but whenever I do I mess up somewhere because I keep getting 1.875 cos(π/7)+2.95 but when I graph it, it doesn't match the table.

Answer 1

As said in comments

• we do not know if the $x$ values are degrees or radians
• you need more than likely a phase shift

Anyway, in order to introduce the phase shift, I would write the model as $$y=a+b \sin(cx)+d\cos(cx)$$ This model is nonlinear with respect to its parameters only because of $c$. If you fix $c$, the problem reduces to a simple multilinear regression. So, consider now the sum of the squares $$SSQ(c)=\sum_{i=1}^n \left(a+b \sin(cx_i)+d\cos(cx_i)-y_i \right)^2$$

• run the linear regression for different values of $c$
• for each value of $c$, recover the value of the corresponding sum of squares
• plot $SSQ(c)$ as a function of $c$ and try to locate (more or less) a minimum
• when this will be done, you have all elements (reasonable starting guesses for $(a,b,c,d)$) to start a nonlinear regression. If you cannot use nonlinear regression, continue with the linear regression zooming more and more.
• when finished, recombine $b \sin(cx)+d\cos(cx)$ to get the phase shift.

Edit

I did follow the steps described above and used for $c$ a step size equal to $0.01$. Below are reproduced the results around the minimum value of $SSQ$. $$\left( \begin{array}{cc} c & SSQ & a & b & d \\ 0.40 & 11.4585& 1.71838& 1.93529& -1.40612 \\ 0.41 & 8.41521& 1.79334& 1.78015& -1.71673 \\ 0.42 & 6.43999& 1.85654& 1.49990& -1.96209 \\ 0.43 & 5.50826& 1.90446& 1.13659& -2.11907 \\ 0.44 &\color{red} {5.43097}& 1.93687& 0.73446& -2.18405 \\ 0.45 & 5.97479& 1.95542& 0.32929& -2.16538 \\ 0.46 & 6.93773& 1.96229& -0.05319& -2.07640 \\ 0.47 & 8.17545& 1.95950& -0.39469& -1.93206 \\ 0.48 & 9.59722& 1.94889& -0.68194& -1.74886 \\ 0.49 & 11.1494& 1.93240& -0.90729& -1.54526 \\ 0.50 & 12.7983& 1.91225& -1.07229& -1.33913 \end{array} \right)$$

Now, the nonlinear regression leads to $R^2=0.916515$, $SSQ=5.3744$ and the following values $$\begin{array}{clclclclc} \text{} & \text{Estimate} & \text{Standard Error} & \text{Confidence Interval} \\ a & +1.92546 & 0.26644 & \{+1.29543,+2.55549\} \\ b & +0.90042 & 0.72962 & \{-0.82487,+2.62570\} \\ c & +0.43593 & 0.01529 & \{+0.39977,+0.47208\} \\ d & -2.16832 & 0.36072 & \{-3.02130,-1.31534\} \\ \end{array}$$

and we can notice that parameter $b$ is non significant.

Repeating without $b$ in the model, what is obtained is $R^2=0.900114$, $SSQ=6.4302$ and the following values $$\begin{array}{clclclclc} \text{} & \text{Estimate} & \text{Standard Error} & \text{Confidence Interval} \\ a & +1.96425 & 0.26957 & \{+1.34262,+2.58589\} \\ c & +0.45242 & 0.00933 & \{+0.43090,+0.47393\} \\ d & -2.14451 & 0.35010 & \{-2.95185,-1.33717\} \\ \end{array}$$