Tides and Water Depth Trig Problems

Ocean tides rise and fall in a smooth, repeating pattern caused by the pull of the moon and sun. Over the course of a day, the water depth at a dock or harbor climbs to a high tide, drops to a low tide, and climbs again, and it does this on a very regular schedule. That kind of repeating, wave-like behavior is exactly what a sine or cosine curve describes, which is why tide height is one of the classic real-world applications of the cosine graph and its close relative, the sine graph.

The goal in a tides and water depth problem is almost always the same: build a sinusoidal function \(h(t)\) that gives water depth as a function of time \(t\), then use that function to answer a question, such as the depth at a specific hour, or the times when the water is deep enough for a boat to pass safely.

A tide height function is usually written in one of these two equivalent forms:

\( h(t) = A\sin(B(t - C)) + D \)

\( h(t) = A\cos(B(t - C)) + D \)

Each letter carries information straight from the problem:

Amplitude, \(A\): half the distance between the highest tide and the lowest tide. It tells you how far the water level swings above and below the average.

Midline (vertical shift), \(D\): the average depth, found by adding the maximum and minimum depths and dividing by 2. The whole graph is shifted up by \(D\) units.

Period, related to \(B\): the time for one full high-to-low-to-high cycle. If the period is \(P\), then \(B = \frac{2\pi}{P}\). Most locations have a period close to 12 hours between successive high tides.

Phase shift, \(C\): shifts the curve left or right so that it lines up with the actual time of high or low tide given in the problem.

These four values are exactly the same transformations you meet when studying the transformation of trigonometric functions, just applied to a real measurement instead of an abstract graph.

Suppose the water depth at a harbor reaches a maximum of 15 meters and a minimum of 5 meters. High tide occurs at midnight (\(t = 0\)), and the next high tide occurs 12 hours later.

Step 1: Find the amplitude. \( A = \frac{15 - 5}{2} = 5 \)

Step 2: Find the midline. \( D = \frac{15 + 5}{2} = 10 \)

Step 3: Find \(B\) from the period. The period is \(P = 12\) hours, so \( B = \frac{2\pi}{12} = \frac{\pi}{6} \)

Step 4: Choose cosine and set the phase shift. Since high tide happens right at \(t = 0\), a cosine function needs no horizontal shift, because \(\cos(0) = 1\) already gives the maximum value. So \(C = 0\).

Putting it together:

\( h(t) = 5\cos\left(\frac{\pi}{6}t\right) + 10 \)

The graph below shows this tide cycle over 24 hours, with high tide at \(t = 0\) and \(t = 12\), and low tide at \(t = 6\).

Now use the function to answer a typical follow-up question: what is the water depth at \(t = 3\) hours (3 a.m.)?

\( h(3) = 5\cos\left(\frac{\pi}{6}\cdot 3\right) + 10 = 5\cos\left(\frac{\pi}{2}\right) + 10 = 5(0) + 10 = 10 \) meters

At 3 a.m., the water is at its average depth of 10 meters, halfway between the day's high and low tide, since that time falls a quarter of the way through the cycle.

Some questions flip the problem around: instead of "what is the depth at this time," they ask "at what time does the depth equal a certain value." To solve these, set \(h(t)\) equal to the given depth and isolate the trig function, then use an inverse trig function to solve for the angle before dividing by \(B\) to get back to \(t\). Because cosine and sine are periodic, there is usually more than one valid time within a 24-hour window, so it helps to sketch or picture the graph to make sure no solution is missed.

Watch out for these frequent mistakes:

Mixing up amplitude and total range. The amplitude is half the difference between max and min, not the full difference.

Choosing sine when the starting point is a maximum or minimum. Cosine naturally starts at a maximum, so if high tide happens at \(t = 0\), cosine avoids a messy phase shift.

Forgetting units. If time is given in hours, keep \(t\) in hours throughout, and if the period is given in a different unit, convert before finding \(B\).

Tide problems share the exact same structure as other periodic motion questions, such as Ferris wheel trig problems, where height also cycles between a maximum and minimum on a fixed schedule. Practicing one type builds the skills needed for the other.

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