How Do Trigonometric Functions Model Real-Life Situations Like Tides or a Ferris Wheel?

Trigonometric functions model tides and Ferris wheels because both involve approximately repeating, or periodic, change. Sine and cosine curves represent how a quantity rises and falls smoothly around a central value over regular time intervals.

The Model and Its Parameters

In IB Mathematics AA SL 3.7, a standard model is

$y=a\sin\bigl(b(x+c)\bigr)+d.$

Each parameter represents a feature of the situation:

The mechanism comes from uniform circular motion. As a passenger moves around the wheel, their vertical height is the vertical coordinate of a rotating radius. That coordinate varies as sine or cosine, so equal time intervals produce a smooth, repeating pattern rather than a straight-line change.

For example, consider a Ferris wheel with radius $12$ m, centre $15$ m above the ground and rotation time $40$ seconds. If a passenger starts at the lowest point, an appropriate cosine model is

$h(t)=15-12\cos\left(\frac{2\pi t}{40}\right).$

At $t=0$,

$h(0)=15-12=3,$

so the passenger begins $3$ m above the ground. After half a revolution, at $t=20$,

$h(20)=15-12\cos(\pi)=27,$

which is the maximum height.

A tide can be modelled similarly: the midline is the average water level, the amplitude measures the variation from that average, and the period is the time between successive high tides.

A common misconception is that $b$ is the period. It is not: the period is $\frac{2\pi}{\lvert b\rvert}$. Also, real tides may not be perfectly sinusoidal because weather, geography and multiple tidal cycles affect them.

Exam Technique

This AA SL 3.7 content may appear on Paper 1 or Paper 2. State the amplitude, period, midline and phase information, build the function, and interpret calculated values with correct units and within the model's domain.