Tide and Trigonometric functions I have a tide guide that gives me four readings for the day - 2 high tides and two low tides. This means it completes two full revolutions within a day. What I'm having trouble with is taking the four measurments and making a graph and equation of the entire function for that specific day. 
One example is...
$$\begin{array}{c|c}
6{:}00\text{am} & 17.1\text{ft}\\\hline
11{:}17\text{am} & 11.8\text{ft}\\\hline 
16{:}01\text{pm} & 15.7\text{ft}\\\hline 
23{:}22\text{pm} & 1.0\text{ft}
\end{array}$$
How do I use these to complete the sine or cosine function?
 These would be random points on the graph which follows the $f(x) = a \sin(bx + c)+ d$ general form. 
 A: It is not possible to solve this system of equations with a function of the form given. When plotting the points, observe that any sine or cosine function that has a minimum of 11.8 cannot possibly reach a value of 1. 
It is possible however, to write a linear combination of such functions up to a constant that give values very close to those desired. In fact, Fourier analysis allows one to construct such a linear combination for any set of points. Consider the function given by $$ f(x)=9 \cos \left[\frac{2 \pi}{44}(x-10)\right]+3.5\cos \left[\frac{2 \pi}{11}(x-5.5)\right]+6.3$$

Then $f(6)=17.08$, $f(11.283)=11.73$, $f(16.017)=15.7$, and $f(23.367)=1.07$. These fit the values desired within a margin of error less than an inch. 
A: Your table of data gives you four values for $(x,f(x))$, once you decide where $0$ is. Suppose we let $x$ represent hours after midnight. Then you've got the data points $(6,17.1), (11.283,11.8), (16.017,15.7), (23.367,1)$
You want to determine the constants $a,b,c,d$, and you have four data points to work with. Plug each data point into your general form equation, and you'll get four equations in four unknowns. The equations might not be easy to solve without mechanical assistance, however. Typing them into something like Wolfram Alpha would probably get you a numerical solution, anyway.
Does this help?
ETA:
Let's write down the details. The resulting equations are:
$a\sin(6b + c) + d = 17.1\\
a\sin(11.283b + c) + d = 11.8\\
a\sin(16.017b + c) + d = 15.7\\
a\sin(23.367b + c) + d = 1$
I don't know a way to solve for $a,b,c$ and $d$ from those equations. :/  I tried feeling them into Wolfram Alpha, but it gave up after thinking for a few seconds. You might try searching for information on "fitting a sine curve to data", and there might be a built-in function in Excel that will do this.
Sorry that I can't be more help.
