Skill: Model periodic phenomena with trigonometric functions

Periodic phenomena repeat in regular cycles, so they can be modeled with a sine or cosine function:

$$y=A\sin(B(x-h))+D$$

or

$$y=A\cos(B(x-h))+D.$$

The important features are:

• Amplitude: $\lvert A\rvert$, half the distance between the maximum and minimum
• Midline: $y=D$, the average of the maximum and minimum
• Period: $\dfrac{2\pi}{\lvert B\rvert}$
• Horizontal shift: $h$
• Range: from $D-\lvert A\rvert$ to $D+\lvert A\rvert$

Worked example

The depth of water at a harbour varies periodically. At low tide, the depth is $2$ m at 3:00 a.m. At high tide, the depth is $8$ m at 9:00 a.m. The pattern repeats every $12$ hours. Create a model for the water depth.

Let $x$ be the number of hours after midnight, and let $y$ be the water depth in metres.

1. Find the amplitude

The maximum is $8$ and the minimum is $2$, so

$$A=\frac{8-2}{2}=3.$$

The water depth varies $3$ m above and below its midline.

2. Find the midline

The midline is the average of the maximum and minimum:

$$D=\frac{8+2}{2}=5.$$

Thus, the average water depth is $5$ m.

3. Find $B$ using the period

The period is $12$ hours. Since

$$\text{Period}=\frac{2\pi}{\lvert B\rvert},$$

we have

$$12=\frac{2\pi}{B}.$$

Solving gives

$$B=\frac{\pi}{6}.$$

4. Choose sine or cosine and determine the shift

At $x=9$, the water is at its maximum of $8$ m. Cosine starts at a maximum, so use a cosine model shifted to $x=9$:

$$y=3\cos\left(\frac{\pi}{6}(x-9)\right)+5.$$

This is a model for the water depth.

5. Check the model

At high tide, $x=9$:

$$y=3\cos(0)+5 =3(1)+5 =8.$$

At low tide, $x=3$:

$$y=3\cos\left(\frac{\pi}{6}(3-9)\right)+5$$ $$=3\cos(-\pi)+5 =3(-1)+5 =2.$$

The model gives the correct high and low tides.

Therefore,

$$\boxed{y=3\cos\left(\frac{\pi}{6}(x-9)\right)+5}$$

where $y$ is the water depth in metres and $x$ is the number of hours after midnight.

Its range is

$$2\le y\le 8,$$

and it repeats every $12$ hours, matching the situation.