# Thread: The equation of a Tide: with cos or sin?

1. ## The equation of a Tide: with cos or sin?

The question says to create an equation to describe the depth of water as a function of time. The description of the tide goes like this:

The wter depth in a harbour is 8 m at low tide and 20 m at high tide. One cycle is completed every 12 hours.

The question asks us to "Find an equation for the water depth, d(t) meters, as a function of time (t) hours, after high tide, which occured at 03:00 .

What I know from this is that the amplitude is 6 m. Since we are finding the tide relative to high tide, we can use cos as our function, correct? Then our period (lets say k) equals 2(pi)/k=12hours, and we work that out to be pi/6 .

So in my attempt, I came up with the equation: d(t)=6cos((pi)/6)t+12

The back of the book gives different. Can anyone explain what I did wrong?

2. ## depth of water during tidal cycles

"It has been shown," said my math instructor, "that most students forget all the trigonometry they learned about a month after the class ends."

One way to answer one of your questions would be to build a table of values for tide highs and lows and times they occurred as would be predicted from your original description, then extrapolate to fill in the water depths between the highs and lows at any increment of time you choose.

Unfortunately a lot of science is grunt work so to speak, going through mounds of data attempting to find a predicatable pattern.

Does the answer in the book predict the the water depth at specifed points?

Your question presupposes that at least one of the answers is correct. They may both be correct for all I know, or they could both be wrong.

I am not see adept at trigonometry to tell at a glance, or even after long deliberation. Apparently you aren't either, but I am guessing that you are much better at it than I if the first tool you used was trig.

3. Originally Posted by mike_302
The question says to create an equation to describe the depth of water as a function of time. The description of the tide goes like this:

The wter depth in a harbour is 8 m at low tide and 20 m at high tide. One cycle is completed every 12 hours.

The question asks us to "Find an equation for the water depth, d(t) meters, as a function of time (t) hours, after high tide, which occured at 03:00 .

What I know from this is that the amplitude is 6 m. Since we are finding the tide relative to high tide, we can use cos as our function, correct? Then our period (lets say k) equals 2(pi)/k=12hours, and we work that out to be pi/6 .

So in my attempt, I came up with the equation: d(t)=6cos((pi)/6)t+12

The back of the book gives different. Can anyone explain what I did wrong?
I believe that it should be $d(t)=6\cos\left( \frac{\pi}{6}t\right)+14$

We need to shift the graph up 14 so that it will bounce between 20 and 8 rather than 18 and 6 which is what the 12 in yours would do.

But the rest of what you did looks great!!!

Good Job!

Hope this helps.

4. wow. that was half poetry half math help, I must thank you. The back of the book gives the answer to be a sin function, while I made my equation to be cosine... because it says to start from 0300 .

5. Originally Posted by mike_302
wow. that was half poetry half math help, I must thank you. The back of the book gives the answer to be a sin function, while I made my equation to be cosine... because it says to start from 0300 .
Oh, right. I didn't notice that part.

I assumed that it was starting at x=0. If it starts at x=3 hours then that would need to be over 1/4 of the period which would shift it to be a sin wave. You are correct.

I am not a very good poet.

6. OHHH, ok, my next reply was going to be why can you not have 3:00 at x=0 and se the cosine wave... But I guess if you have d as a funcion of time, then you have to hve you're x's equal the value that theyre representing right.. thanks for all your help!