Modelling with Trigonometric Functions

students, imagine watching a Ferris wheel 🎡, tracking the height of a tide 🌊, or predicting the motion of a swinging playground swing. These situations often repeat in a smooth cycle, and trigonometric functions are one of the best tools for describing them. In this lesson, you will learn how to build a trigonometric model from real-world data, interpret the parts of the model, and connect it to the wider Geometry and Trigonometry topic in IB Mathematics Analysis and Approaches SL.

What you will learn

By the end of this lesson, students, you should be able to:

explain the key ideas and vocabulary used in trigonometric modelling
identify when a situation can be modelled with a trigonometric function
find the amplitude, period, midline, and phase shift of a model
use a trigonometric model to make predictions and answer questions
connect trigonometric modelling to periodic behavior in geometry and real life

The main idea is simple: if a quantity repeats in a regular way, a trigonometric function may describe it well. The goal is not just to memorize formulas, but to understand what each part of the function means in context 📈

Why trigonometric functions are useful

Many real-world patterns are periodic, which means they repeat after a fixed interval. Examples include:

the height of a point on a rotating wheel
daylight hours over a year
the distance of a boat from the shore as waves pass
air temperature during a day

A basic sine or cosine graph already has repeating behavior. That makes functions such as $y = a\sin(bx + c) + d$ and $y = a\cos(bx + c) + d$ especially useful.

The important parameters are:

$a$, which controls the amplitude and reflection
$b$, which controls the period
$c$, which controls the horizontal shift
$d$, which controls the vertical shift or midline

For a model like $y = a\sin(bx + c) + d$:

amplitude is $|a|$
period is $\frac{2\pi}{|b|}$
midline is $y = d$
phase shift is $-\frac{c}{b}$

These features help you match the graph to real data.

Reading the meaning of the parts of a model

Suppose the height of a rider on a Ferris wheel is modelled by $h(t) = 8\sin\left(\frac{\pi}{6}t\right) + 10$, where $h(t)$ is in metres and $t$ is time in minutes.

Let’s interpret it:

amplitude $= 8$, so the rider moves $8$ metres above and below the midline
midline $h = 10$, so the average height is $10$ metres
period $= \frac{2\pi}{\pi/6} = 12$, so one complete rotation takes $12$ minutes
there is no horizontal shift because there is no added constant inside the bracket

The model predicts that the rider’s height oscillates between $10 - 8 = 2$ and $10 + 8 = 18$ metres.

This is a good example of how trigonometry links to geometry. A point moving around a circle has a vertical position that follows a sine or cosine pattern. In fact, a sine graph can be seen as a shadow of circular motion 🌞

Building a trigonometric model from data

In IB Mathematics Analysis and Approaches SL, you may be given a table, a graph, or a real situation and asked to create a model. A standard approach is:

identify the maximum and minimum values
calculate the amplitude using $\frac{\text{max} - \text{min}}{2}$
find the midline using $\frac{\text{max} + \text{min}}{2}$
determine the period from the graph or context
choose sine or cosine depending on where the graph starts
adjust for horizontal shift if needed

Example: A tide rises and falls between $2$ m and $6$ m every $12$ hours. At $t = 0$, the tide is at its minimum.

First, find the amplitude:

$\frac{6 - 2}{2} = 2$

The midline is:

$\frac{6 + 2}{2} = 4$

The period is $12$ hours, so for $y = a\cos(bt) + d$ we use:

$\frac{2\pi}{b} = 12$

which gives:

$ b = \frac{\pi}{6}$

Because the tide starts at a minimum when $t = 0$, cosine is useful with a negative sign:

$T(t) = -2\cos\left(\frac{\pi}{6}t\right) + 4$

Check it: when $t = 0$,

T(0) = -2(1) + 4 = 2

which matches the minimum.

This method is powerful because it turns real measurements into a mathematical description that can be used to predict future values.

Choosing sine or cosine

students, one common question is whether to use $\sin$ or $\cos$. Both can model the same situation, but they may start at different points.

Use cosine when the graph starts at a maximum or minimum.
Use sine when the graph starts at the midline and is increasing or decreasing.

For example, if a wheel starts with a rider at the highest point, a cosine model is natural. If it starts at the middle height and moves upward, a sine model is often simpler.

You can always rewrite one as the other using a phase shift. For example:

$\sin\left(x\right) = \cos\left(x - \frac{\pi}{2}\right)$

So the choice is often about convenience and how well the model matches the starting condition.

Solving equations in a modelling context

After building a model, you may be asked when a quantity reaches a certain value. This means solving a trigonometric equation.

Example: If

$H(t) = 3\sin\left(\frac{\pi}{4}t\right) + 5$

find when $H(t) = 8$.

Set up the equation:

$3\sin\left(\frac{\pi}{4}t\right) + 5 = 8$

Subtract $5$:

$3\sin\left(\frac{\pi}{4}t\right) = 3$

Divide by $3$:

$\sin\left(\frac{\pi}{4}t\right) = 1$

Sine is equal to $1$ when its angle is:

$\frac{\pi}{2} + 2k\pi$

for integer $k$. So:

$\frac{\pi}{4}t = \frac{\pi}{2} + 2k\pi$

Multiply by $\frac{4}{\pi}$:

$ t = 2 + 8k$

So the height is $8$ metres at $t = 2, 10, 18, \dots$.

In real-world interpretation, the answer may need to fit the time interval given in the question. Always check whether the solution is meaningful in context.

Common modelling assumptions and limitations

A model is not the same as reality. It is a simplified description. students, you should be able to explain what assumptions are being made.

Common assumptions include:

the motion is perfectly periodic
the maximum and minimum values stay constant
the cycle length does not change
external effects are ignored

For example, real tides are influenced by weather, coast shape, and other forces. A trigonometric model may still be useful, but it is only an approximation.

This is important in IB work: you should comment on the suitability of the model and the reasonableness of results. If a model predicts a negative height for something that cannot be below zero, then the model may only be useful over part of the time interval.

Connecting to geometry and trigonometry

Trigonometric modelling is closely linked to the geometry of circles. If a point moves around a circle of radius $r$, then its vertical coordinate can be written using sine, and its horizontal coordinate can be written using cosine.

For a circle centered at the origin:

x = r$\cos$$\theta$, \quad y = r$\sin$$\theta$

As $\theta$ changes uniformly, the coordinates vary periodically. This is the geometric reason behind sine and cosine waves.

Circular measure is also essential. Since trigonometric models often use angles in radians, you must be comfortable with relationships like:

$\text{arc length} = r\theta$

and

$\text{period} = \frac{2\pi}{|b|}$

when working with $y = a\sin(bx + c) + d$ or $y = a\cos(bx + c) + d$.

This connection shows that trigonometric modelling is not isolated. It is part of the bigger picture of Geometry and Trigonometry, linking angles, circles, graphs, and real-world change.

Conclusion

Modelling with trigonometric functions helps you describe repeating patterns in a precise and useful way. students, when you recognize periodic behavior, you can use sine or cosine to build a model, interpret amplitude, period, and shifts, and then solve practical questions using the model. The key is to connect the graph to the situation and always check that your answers make sense in context.

In IB Mathematics Analysis and Approaches SL, this topic shows how algebra, graphs, and geometry work together. A strong understanding of trigonometric models will help you handle real data, interpret periodic change, and explain your reasoning clearly ✅

Study Notes

Periodic behavior means a pattern repeats after a fixed interval.
Common trigonometric models are $y = a\sin(bx + c) + d$ and $y = a\cos(bx + c) + d$.
Amplitude is $|a|$.
Period is $\frac{2\pi}{|b|}$.
Midline is $y = d$.
Phase shift is $-\frac{c}{b}$.
Sine is often used when the graph starts at the midline.
Cosine is often used when the graph starts at a maximum or minimum.
A trigonometric model is an approximation, not a perfect copy of reality.
Real-world examples include tides, Ferris wheels, seasonal temperature changes, and rotating objects.
Trigonometric modelling connects to circles, radians, and coordinate geometry.