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Trigonometric modelling

Build and interpret trigonometric models using amplitude, midline, period and phase. Explore circular motion manually, fit simple observations and check units, domains and model assumptions.

Before you startRadians, trig graph transformations and the R formula

01 / Connect a repeating situation to a graph

Define the variables before selecting a formula.

A trigonometric model approximates a repeating change: a height, displacement, reading or spatial profile. Begin by naming what the input and output measure, their units, and the part of the situation the model is intended to describe.

h(t) = H + A sin(ωt + φ)

Here H is the midline, |A| the amplitude, ω the angular frequency and φ the phase. The manual model connects the height of a point moving on a circle to a periodic graph.

Build a height modelExplore

Radius A (m)Centre height H (m)Period T (s)Start positionTime through one turn

h(t) = 6 − 2cos(2πt/8), with t in seconds and angles in radians.

t = 0/12 × T = 0.0000 s; h = 4.0000 m.

Range: 4 ≤ h ≤ 8 metres. One revolution takes 8 seconds.

Move time yourself. The circle and graph show the same height. This model assumes a fixed centre and constant angular speed; the notes remain readable independently.

02 / Read amplitude, midline and period separately

The maximum is the midline plus the amplitude.

Midline H = (maximum + minimum)/2
Amplitude |A| = (maximum − minimum)/2
Period T = 2π/|ω| in radian form

A repeating signal varies between 3 and 15 units and completes one cycle in 10 seconds.Worked example

H = (15 + 3)/2 = 9

The middle value is not the amplitude.

|A| = (15 − 3)/2 = 6

Amplitude is half the total vertical range.

ω = 2π/10 = π/5 radians per second

One full angular turn corresponds to one time period.

A phase or starting condition is still needed

The extrema and period alone do not locate the curve horizontally.

03 / Keep angle units and time units consistent

The coefficient converts an input into an angle.

If the trig function is interpreted in radians, use ω = 2π/T. In a degree-based formula the angular coefficient is 360°/T. These are two representations of the same cycle, but their numerical coefficients must not be mixed.

One 12-second cycle:
sin(πt/6) in radians
sin(30°t) in degree notation, with t measured in seconds

The first coefficient has units radians per second; the second has degrees per second. If you change t from seconds to minutes, change the coefficient accordingly. Always state which convention you use.

04 / The initial value may not determine the direction

Use whether the quantity is rising or falling.

Choose a model with midline 6, amplitude 4 and period 12 seconds.Worked example

Starting at the lowest point: h(t) = 6 − 4cos(πt/6)

At t = 0 the cosine is 1, so the height is 2.

Starting at the highest point: h(t) = 6 + 4cos(πt/6)

The starting height is 10.

Starting at the midline and rising: h(t) = 6 + 4sin(πt/6)

A small positive input makes sine positive.

Starting at the midline and falling: h(t) = 6 − 4sin(πt/6)

The same initial height now has the opposite direction.

A single starting height can leave two possible phases. An observed direction, a peak time or another suitable observation can resolve that ambiguity.

05 / Model a point moving uniformly on a vertical circle

Radius determines amplitude; centre height sets the midline.

A point travels on a circle of radius 4 m, centred 7 m above level ground. One revolution takes 20 s, starting at the lowest point.Worked example

h(t) = 7 − 4cos(πt/10)

The minus cosine puts the start at the bottom.

h(0) = 3 m and h(10) = 11 m

These match the lowest and highest points.

h(5) = h(15) = 7 m

The point crosses the centre height twice per turn.

The range is [3, 11] m and the period is 20 s

These features match the geometry.

This idealisation assumes a rigid circle, a fixed centre and constant angular speed. Stops, acceleration or movement of a hanging seat would need additional modelling.

Watch circular motion trace a height graph

Pause, replay or seek freely. The notes explain the same idea and stay in view.

06 / Use the time of a known maximum or minimum

Write the shift inside the whole angle.

A hypothetical periodic reading has minimum 2, maximum 14 and period 18 s. A maximum occurs at t = 3 s.Worked example

H = 8 and A = 6

Use the midpoint and half-range.

Choose cosine because its maximum occurs at input 0

Shift that maximum to t = 3.

h(t) = 8 + 6cos(π(t − 3)/9)

The radian coefficient is 2π/18.

Check h(3) = 14 and h(12) = 2

The minimum comes half a period later.

Writing cos(πt/9 − 3) would mean a phase of 3 radians, not a time shift of 3 seconds. Factor the input coefficient when translating between these descriptions.

07 / Fit a simple model to a stated repeating pattern

Check more than the points used to find its parameters.

Suppose a smooth signal is specified to complete one cycle in 12 s. It starts at 6 and rises, reaches 10 at t = 3, returns to 6 at t = 6, reaches 2 at t = 9 and returns to 6 at t = 12.

h(t) = 6 + 4sin(πt/6)

This model matches all five stated observations. It predicts h(1.5) = 6 + 2√2. That prediction is a consequence of assuming a sinusoidal shape; the observations alone would not prove that the true signal has exactly this form.

A useful validation checkWorked example

Compare the prediction with an additional observation

Use an input that was not used to choose the parameters.

Look for systematic differences, not only rounding noise

A changing period or asymmetric rise/fall may make one sine wave unsuitable.

State the intended time interval

Repeated extrapolation assumes the same behaviour continues.

08 / Interpret a model written with both sine and cosine

The R formula reveals the combined amplitude and phase.

Interpret s(t) = 10 + 3sin(πt/6) + 4cos(πt/6), where s is in cm and t in seconds.Worked example

Let α satisfy cos α = 3/5 and sin α = 4/5

Choose the acute phase α = arctan(4/3).

s(t) = 10 + 5sin(πt/6 + α)

The two components have the same input frequency.

Midline 10 cm; amplitude 5 cm; period 12 s

The amplitude is not 3 + 4.

The full-model range is [5, 15] cm

A shorter observation window may not attain both bounds.

s(0) = 14 cm

This checks the phase against the original expression.

09 / A repeating input can be distance rather than time

Read the input unit before calling a cycle length a period.

Interpret y(x) = 1.2 + 0.3sin(2πx/5), with x and y measured in metres.Worked example

Midline 1.2 m and amplitude 0.3 m

These describe the vertical profile.

The pattern repeats every 5 m along x

This is a spatial wavelength, not a five-second period.

The range is [0.9, 1.5] m

This follows from the sine bounds.

A spatial profile by itself does not specify a travelling speed. A time-dependent model would need information linking position and time.

10 / Check that the model makes sense in its context

A mathematically valid formula may make impossible predictions.

If a model describes a nonnegative depth, a negative predicted depth is a sign that the formula or its permitted interval needs reconsideration. Constant amplitude and period are assumptions; many real repeating quantities have noise, trends or changing cycles.

Inspect d(t) = 2 + 3cos t as a full-cycle depth model.Worked example

Its mathematical range is [−1, 5]

The formula is valid as a real-valued function.

A negative physical depth is not compatible with the stated meaning

State two assumptions behind the simple circular height model.

Hint

Consider the circle and how the angle changes with time.

Worked solution

Examples: the centre remains fixed at a constant height; the radius is fixed; angular speed is constant; the point follows the circle without additional seat motion. Any two clearly stated assumptions are sufficient.

12 / Recap

Parameters describe a situation only after units and assumptions are fixed.

Define input/output quantities, units and the intended domain.
Get the midline from the midpoint and amplitude from the half-range.
Use the correct angular coefficient for the period and angle units.
Choose a phase from a starting value, direction or peak time.
Validate extra predictions and state limitations instead of assuming exact real-world repetition.

Section 1 of 12 · Connect a repeating situation to a graph