Trigonometric Models (AI HL)
Anything that repeats on a regular cycle - tides, daylight hours, a Ferris wheel, a pendulum, temperature over a day - is a candidate for a sinusoidal model. This topic covers building a sine or cosine model from real-world information like a maximum and minimum, reading off amplitude, period and the principal axis, and solving for when the model reaches a given value using your GDC.
What the syllabus says
This topic maps onto two points in the official IB Applications & Interpretation syllabus - the SL foundation, extended at HL.
| Code | Syllabus content |
|---|---|
| SL2.5 | Sinusoidal models \(f(x)=a\sin(bx)+d\), \(f(x)=a\cos(bx)+d\). Students find amplitude (\(a\)), period (\(\tfrac{360^\circ}{b}\)), or the equation of the principal axis (\(y=d\)). Contexts include tides, weather patterns, and the motion of Ferris and bicycle wheels. |
| AHL2.9 | Extends to \(f(x)=a\sin(b(x-c))+d\). Radian measure is assumed unless a degree symbol is used. In radians the period is \(\tfrac{2\pi}{b}\). A horizontal translation of \(c\) can be referred to as a phase shift. |
Sine rule / cosine rule and graphical solving of trig equations in a finite interval (AHL3.4) are closely linked and appear in Geometry & Trigonometry.
Key terms
Five words worth knowing cold before you touch the formulas below - each with a worked example showing exactly what it means.
What is a sinusoidal model?
A sinusoidal model is a function of the form \(f(x)=a\sin(b(x-c))+d\) used to describe anything that repeats in a regular, wave-like cycle - tides, daylight hours, temperature, or the height of a point on a rotating wheel.
e.g. For \(T(t) = 5\sin(2t) + 15\), \(T(0) = 5\sin(0)+15 = 15\).
What is amplitude?
Amplitude is how far the curve swings above and below its midline - half the distance between the maximum and minimum values, \(a = \dfrac{\text{max}-\text{min}}{2}\).
e.g. For \(h(t)=7+3\sin(30t^\circ)\), max \(=10\), min \(=4\), so \(a = \dfrac{10-4}{2}=3\).
What is the period?
The period is the time (or distance) for one complete cycle of the model. In degrees it's \(\tfrac{360^\circ}{b}\); in radians it's \(\tfrac{2\pi}{b}\).
e.g. For \(h(t)=7+3\sin(30t^\circ)\), period \(= \dfrac{360}{30}=12\) hours.
What is the principal axis (midline)?
The principal axis, \(y=d\), is the horizontal line the curve oscillates evenly around - the average of the maximum and minimum values, \(d = \dfrac{\text{max}+\text{min}}{2}\).
e.g. For \(h(t)=7+3\sin(30t^\circ)\), midline \(d = \dfrac{10+4}{2}=7\).
What is a phase shift?
A phase shift is the horizontal translation \(c\) in \(f(x)=a\sin(b(x-c))+d\) - it moves the whole cycle earlier or later without changing its shape, amplitude or period.
e.g. For \(h(t)=2.4\cos(30(t-3)^\circ)+3.8\), the phase shift \(c=3\) moves the peak (high tide) to \(t=3\) hours.
Key formulas
One model to build from context, and three properties to read straight off a maximum, minimum and period.
Formula reference
The general sinusoidal model is on the AI formula booklet; the period, amplitude and midline relationships are follow-on properties assumed as prior knowledge from SL.
| Formula | Used for | Booklet? |
|---|---|---|
| \(f(x)=a\sin(b(x-c))+d\) | General sinusoidal model | ✓ Yes |
| \(a = \dfrac{\text{max}-\text{min}}{2}\) | Amplitude from max/min | Not in the formula booklet - prior knowledge |
| \(d = \dfrac{\text{max}+\text{min}}{2}\) | Principal axis from max/min | Not in the formula booklet - prior knowledge |
| Period \(=\dfrac{2\pi}{b}\) (rad) or \(\dfrac{360^\circ}{b}\) (deg) | Period from \(b\) | Not in the formula booklet - prior knowledge |
Sine-start vs cosine-start models
The same wave can be written starting from either function - the choice usually depends on where in the cycle \(t=0\) falls.
| Feature | Sine model \(a\sin(bt)+d\) | Cosine model \(a\cos(bt)+d\) |
|---|---|---|
| Value at \(t=0\) | Midline, \(d\) | Maximum, \(a+d\) (if \(a>0\)) |
| Best used when... | The cycle starts at the average value | The cycle starts at a peak or trough |
| Example context | Temperature rising through the midline at midnight | A Ferris wheel starting at its lowest point |
Key features of a sinusoidal model
Four numbers, \(a\), \(b\), \(c\) and \(d\), fully describe any sinusoidal model - each one has a distinct visual and physical meaning.
Amplitude
\[a = \dfrac{\text{max}-\text{min}}{2}\]
The height of the wave above (and depth below) the midline.
Not in the formula booklet - prior knowledgePeriod
\[\text{period} = \dfrac{2\pi}{b} \text{ or } \dfrac{360^\circ}{b}\]
Time for one full cycle - rearrange to find \(b\) from a stated period.
Not in the formula booklet - prior knowledgePrincipal axis
\[y = d, \quad d = \dfrac{\text{max}+\text{min}}{2}\]
The horizontal line the model oscillates around - also the mean value over a full cycle.
Not in the formula booklet - prior knowledgePhase shift
\[c \text{ in } a\sin(b(x-c))+d\]
Shifts the whole cycle left or right in time, without changing its shape.
✓ Part of the booklet's general modelSetting up a model from a context
Most exam questions give you real-world facts rather than a ready-made equation - you have to build the model first.
From a maximum and minimum
Use \(a = \tfrac{\text{max}-\text{min}}{2}\) and \(d = \tfrac{\text{max}+\text{min}}{2}\) to find the amplitude and midline directly.
Method, not a formulaFrom a stated period
Rearrange \(\text{period} = \tfrac{2\pi}{b}\) (or \(\tfrac{360^\circ}{b}\)) to find \(b\) once you know how long a full cycle takes.
Method, not a formulaSolving equations from a model
Once a model is set up, most remaining questions ask you to find a specific time - this is almost always faster done graphically on a GDC than by hand.
Finding when a model reaches a value
Graph the model and \(y=\text{target}\) as two functions and find their intersection(s), or use an equation solver.
GDC techniqueDegree vs radian mode
Check whether the model uses a degree symbol (e.g. \(30t^\circ\)); if not, radian measure is assumed - set your GDC's mode to match before evaluating or solving.
Essential checkWorked examples
Two full exam-style questions, marked exactly like the real thing. Try each one yourself before checking the worked solution.
A pendulum's angular displacement is \(\theta(t) = 0.2\cos(3t)\) (radians, \(t\) seconds).
(a) State the maximum displacement.
(b) Find the period.
(c) Find the first time \(\theta = 0\).
Worked solution
(a) Maximum \(= 0.2\) rad. A1
(b) Period \(= \dfrac{2\pi}{3}\) M1
\(\approx 2.09\) s. A1
(c) \(\cos(3t) = 0 \Rightarrow 3t = \tfrac{\pi}{2} \Rightarrow t = \tfrac{\pi}{6}\) M1
\(\approx 0.524\) s. A1
A Ferris wheel has diameter 40 m, lowest point 2 m above ground, and one revolution every 4 minutes. A rider starts at the lowest point at \(t = 0\).
(a) Find the centre height.
(b) Write a model \(h(t) = a\cos(bt) + d\) (with \(b\) in deg/min), explaining the sign of \(a\).
(c) Find the height after 1 minute.
Worked solution
(a) Radius 20, lowest 2, so centre \(= 22\) m. M1 A1
(b) \(b = 360/4 = 90^\circ\)/min; starting at the lowest point gives \(a = -20.\) \(h(t)\) M1
\(= -20\cos(90t^\circ) + 22.\) A1
(c) \(h(1) = -20\cos(90^\circ) + 22\) M1
\(= 22\) m. A1
Common mistakes
The four slip-ups that account for most of the marks lost on this topic - worth reading before you start practising, not just after you get one wrong.
- Mixing up degree and radian mode. If the argument has a degree symbol like \(30t^\circ\), your GDC must be in degree mode; if not, radians are assumed. Getting this wrong gives a completely different (and wrong) answer.
- Using the wrong sign for \(a\). When a cycle starts at its lowest point, \(a\) is negative for a cosine model; starting at the highest point makes \(a\) positive - always check what \(t=0\) represents in the context.
- Confusing amplitude with the maximum value. Amplitude is the distance from the midline to the peak, \(a = \tfrac{\text{max}-\text{min}}{2}\), not the maximum value itself, which is \(a+d\).
- Only finding one solution to a trig equation. Sine and cosine equations generally have multiple solutions within a given interval - always check the context (e.g. "first two times") for how many values are needed.
Using your GDC
Every step below is a real button sequence, not a vague "use your calculator" hint - covering the TI-84 Plus, TI-Nspire, and Casio fx-9860/fx-CG50. Pick your model to filter down to just the steps that apply to you.
If a question gives you a table of periodic data rather than a stated maximum and minimum, sine regression fits the model directly.
- Enter the data in two lists (x and y).
- STAT → EDIT to enter L1/L2, then STAT → CALC → SinReg.TI-84
- In a Lists & Spreadsheet page enter the data, then menu → Statistics → Stat Calculations → choose Sinusoidal Regression.Nspire
- Main menu → Statistics, enter the data in lists, then CALC → REG and pick Sin.Casio
Tip: Turn DiagnosticOn (TI-84: 2nd → 0 → DiagnosticOn) to see R². Choose the model with the best R² that also makes sense for the context.
After fitting a sinusoidal model, R² tells you how well it actually explains the data - useful if you're deciding between a sinusoidal and another type of model.
- Fit each candidate model in turn and note the R² value each time.
- Turn DiagnosticOn first (2nd → 0, scroll to DiagnosticOn, ENTER) - then R² appears after every regression.TI-84
- R² is shown automatically after each regression calculation in the Statistics menu.Nspire
- R² (displayed as r²) appears in the regression output; run CALC → REG for each model type and compare.Casio
- The model with R² closest to 1 explains the most variation in y - but also consider whether the model makes sense for the context.
Tip: R² alone doesn't tell you whether the model is appropriate - always look at the scatter plot too. A high R² on a model that shouldn't apply (e.g. sinusoidal for steadily growing data) is meaningless.
See the full GDC guide for more calculator models and topics.
Ready to practise properly?
Trigonometric models questions, marked instantly like the real exam.
Quick answers
The questions students on this topic ask most often.
What does each letter in \(f(x)=a\sin(b(x-c))+d\) control?
\(a\) is the amplitude (how far above and below the midline the curve swings), \(b\) controls the period, \(c\) is the horizontal shift (phase shift), and \(d\) is the principal axis - the midline the whole curve oscillates around.
Should I use degrees or radians?
Radian measure is assumed at AHL unless the question uses a degree symbol, like \(30t^\circ\). Always check for that little circle and set your GDC's angle mode to match before solving.
How do I find amplitude and midline from a maximum and minimum?
Amplitude \(a = \dfrac{\text{max}-\text{min}}{2}\) and midline \(d = \dfrac{\text{max}+\text{min}}{2}\). These two values, together with the period, are usually enough to build the whole model.
How do I solve a trig equation from a model on my GDC?
Graph the model and the target value as a horizontal line, then use the intersection or solver tool to find where they meet - much faster and safer than solving trigonometric equations by hand.
Sub-topics
Trigonometric Models broken down into its individual skills, each with its own focused page.
Related topics
More Functions & Modelling topics from the same AI HL syllabus unit, in case you want to keep going.