Modelling with Trigonometric Functions

students, have you ever noticed how the height of the Sun changes during the day, how a Ferris wheel rises and falls, or how ocean tides repeat in a pattern? 🌞🎡🌊 These are all examples of quantities that change in a regular, repeating way. Trigonometric functions are ideal for modelling this kind of behaviour.

In this lesson, you will learn to:

• explain what it means to model real situations with trigonometric functions;
• identify key features such as amplitude, period, midline, and phase shift;
• build and interpret trigonometric models from data or a situation;
• connect these models to graphs, equations, and applications in geometry and trigonometry;
• use trigonometric reasoning to solve real-world problems.

Modelling is an important part of IB Mathematics: Analysis and Approaches HL because it connects pure mathematics to real situations. Trigonometric models often describe motion, seasonal change, sound waves, and circular motion. The goal is not just to draw a curve, but to choose a function that reasonably represents what is happening in the real world.

What It Means to Model with Trigonometric Functions

A model is a mathematical description of a real situation. When we model with trigonometric functions, we use functions like $y=A\sin(Bx+C)+D$ or $y=A\cos(Bx+C)+D$ to represent repeated patterns.

These functions are useful because they are periodic, which means they repeat after a fixed interval. This repeating nature matches many real phenomena. For example, the height of a point on a rotating wheel repeats every full turn, and daylight hours repeat across the seasons each year.

The main features of a trigonometric model are:

• Amplitude: $|A|$, the distance from the midline to a maximum or minimum value.
• Midline: $y=D$, the central horizontal line around which the graph oscillates.
• Period: the length of one complete cycle, given by $\frac{2\pi}{|B|}$ for sine and cosine functions.
• Phase shift: the horizontal shift caused by $C$, which moves the graph left or right.

A good model should match the shape, scale, and timing of the data. If the data repeats but is not perfectly smooth, a trigonometric function can still give a strong approximation. Real-world data often has small irregularities because of measurement error, weather changes, or other influences.

Building a Trigonometric Model from a Situation

To create a model, first identify the pattern in the problem. Ask these questions:

1. Does the quantity repeat?
2. What is the maximum value?
3. What is the minimum value?
4. How long does one full cycle take?
5. What is the average value around which the data varies?

Suppose the temperature in a city varies between $12^\circ\text{C}$ and $28^\circ\text{C}$ over a 24-hour period. The amplitude is

$$A=\frac{28-12}{2}=8$$

and the midline is

$$D=\frac{28+12}{2}=20.$$

Because one cycle takes 24 hours, the period is $24$. For a sine or cosine model in hours, we need

$$\frac{2\pi}{|B|}=24,$$

so

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

A possible model is

$$T(t)=8\sin\left(\frac{\pi}{12}t\right)+20,$$

where $t$ is time in hours.

This model is not unique. A cosine model could also work, depending on when the maximum or minimum occurs. For example, if the temperature is highest at $t=6$, then a shifted cosine model may fit better.

One of the most important skills in this topic is choosing the correct function form. If the graph starts at a maximum, cosine is often convenient. If it starts at the midline and rises, sine may be simpler. Both can represent the same phenomenon with a suitable shift.

Interpreting Key Features in Context

When students reads a trigonometric graph, each feature should have a real meaning.

• Amplitude tells how far the quantity moves above or below the average level. In a tide model, a larger amplitude means a bigger difference between high tide and low tide.
• Period tells how long one complete cycle lasts. In circular motion, it may represent the time for one rotation.
• Midline represents the average or equilibrium value. In daily temperature data, this is often the average temperature.
• Maximum and minimum values show the extremes of the situation.
• Phase shift tells when the cycle begins relative to the chosen time axis.

Consider a Ferris wheel with radius $15\text{ m}$ and center $18\text{ m}$ above the ground. If a rider starts at the lowest point and completes one revolution in $40$ seconds, the height can be modelled by

$$h(t)=15\cos\left(\frac{\pi}{20}t+\pi\right)+18.$$

Here, the amplitude is $15$, the midline is $h=18$, and the period is $40$ seconds because

$$\frac{2\pi}{\pi/20}=40.$$

The model works because the rider’s vertical motion is periodic. The cosine graph is shifted so that at $t=0$, the rider is at the bottom, not the top.

This is a strong example of how geometry and trigonometry connect. The circular motion of the wheel is geometric, while the vertical height over time is trigonometric.

Using Data to Create and Test a Model

In IB Mathematics: Analysis and Approaches HL, you may be given data in a table, graph, or description and asked to create a model. A common process is:

1. Identify the period from the repeating pattern.
2. Find the maximum and minimum values.
3. Calculate the amplitude and midline.
4. Determine whether sine or cosine is the best starting point.
5. Apply a phase shift if needed.
6. Check whether the model fits the given information.

For example, suppose a sound wave has a maximum displacement of $4$ and a minimum displacement of $-4$, and one cycle lasts $0.02$ seconds. Then the amplitude is $4$, the midline is $y=0$, and the period is $0.02$. Since

$$\frac{2\pi}{|B|}=0.02,$$

we get

$$B=100\pi.$$

A possible model is

$$y=4\sin(100\pi t).$$

This kind of function is useful in physics and engineering because waves, vibrations, and alternating currents often behave periodically.

Sometimes you are asked to solve for unknown values using the model. For instance, if a tide model is

$$h(t)=3\cos\left(\frac{\pi}{6}t\right)+5,$$

then the highest tide is $8$ and the lowest tide is $2$. To find when the tide is at height $6$, solve

$$3\cos\left(\frac{\pi}{6}t\right)+5=6.$$

This gives

$$\cos\left(\frac{\pi}{6}t\right)=\frac{1}{3}.$$

Using inverse trigonometric reasoning, students can find the times in one or more cycles where this occurs.

Trigonometric Reasoning and Graph Interpretation

A major part of this lesson is understanding that trigonometric models are not just formulas; they are graphs with meaning. The graph shows how a quantity changes over time or angle.

In trigonometric reasoning, you may need to use identities, inverse functions, or algebra to interpret a model. For example, if a model is written as

$$y=2\sin\left(3x-\frac{\pi}{2}\right)+1,$$

then the period is

$$\frac{2\pi}{3},$$

and the phase shift is

$$\frac{\pi}{6}\text{ to the right}.$$

This is because

$$3x-\frac{\pi}{2}=3\left(x-\frac{\pi}{6}\right).$$

Understanding this form helps students see how the algebra changes the graph.

It is also important to recognize domain and range. If the model is for time, the domain may be restricted to realistic values such as $0\le t\le 24$. The range depends on amplitude and midline. For a model

$$y=A\sin(Bx+C)+D,$$

the range is

$$[D-|A|,\,D+|A|].$$

This tells you the possible values the quantity can take.

Why This Topic Matters in Geometry and Trigonometry

Modelling with trigonometric functions sits naturally inside Geometry and Trigonometry because it links angles, circles, and periodic motion.

A point moving around a circle creates sine and cosine patterns when its coordinates are observed separately. If a point on a circle of radius $r$ moves with constant angular speed, its horizontal and vertical coordinates can be written using trigonometric functions. This is one of the clearest examples of how circular geometry becomes a trigonometric model.

The topic also supports later work in calculus and statistics. In calculus, trigonometric models may be differentiated or integrated to study changing rates and accumulated quantities. In statistics, models may be fitted to data and compared with observations.

The key idea is that trigonometric models give a compact way to represent repeated change. They are especially powerful when the exact values are hard to predict but the overall pattern is clear.

Conclusion

Modelling with trigonometric functions helps students turn real-world repeating behaviour into mathematics. By identifying amplitude, period, midline, and phase shift, you can build models that describe motion, waves, tides, temperatures, and many other patterns. The topic strengthens understanding of graphs, circular motion, and trigonometric reasoning, while also showing how mathematics can describe the world in a structured way. In IB Mathematics: Analysis and Approaches HL, this skill is essential because it combines algebra, geometry, and interpretation into one powerful method. 📈

Study Notes

• Trigonometric models describe repeating patterns using functions such as $y=A\sin(Bx+C)+D$ and $y=A\cos(Bx+C)+D$.
• The amplitude is $|A|$.
• The midline is $y=D$.
• The period is $\frac{2\pi}{|B|}$.
• A phase shift comes from the term inside the trigonometric function and moves the graph left or right.
• Choose sine or cosine based on the starting point and shape of the situation.
• Real-world examples include tides, temperature changes, sound waves, and Ferris wheel motion.
• Use the maximum and minimum values to find the amplitude and midline.
• Check that the model makes sense in context, including the domain and range.
• Trigonometric modelling connects geometry, circular motion, and periodic change.
• In IB Mathematics: Analysis and Approaches HL, this topic helps students interpret, construct, and solve realistic mathematical models.