Sinusoidal Modelling

Give every parameter a meaning

A useful form is

The midline is $D$. The amplitude is $|A|$, the vertical distance from the midline to a maximum. For a non-constant model with $\omega\ne0$, the period is $T=2\pi/|\omega|$. The parameter $h$ shifts the timing. In the cosine form with $A>0$, a maximum occurs at $t=h$.

Name the units. If $t$ is measured in seconds and $y$ in metres, then $A$ and $D$ are in metres and $\omega$ is in radians per second. The argument of the cosine is an angle, not a distance.

Choose a starting point that matches the situation

Diagram explained

On this reference graph, sine is the solid curve and cosine is dashed. Both have amplitude one, midline zero and period two pi. The graph is a reference waveform, not the wheel-height model developed below.

Chart values and boundaries

Chart values and boundaries

Angle	Sine	Cosine
$0$	$0$	$1$
$\pi/2$	$1$	$0$
$\pi$	$0$	$-1$
$3\pi/2$	$-1$	$0$
$2\pi$	$0$	$1$

Read amplitude and midline from the extremes

If the observed maximum and minimum are $M$ and $m$, a candidate sinusoid has

The amplitude is half the full vertical range, not the maximum output itself. The midline is the halfway level. It equals the time average over a complete cycle of an ideal sinusoid, but it need not equal the average of an unevenly sampled real data set.

The time between corresponding points in successive cycles gives the period. Use maximum to next maximum, minimum to next minimum, or a midline crossing to the next crossing in the same direction. Opposite-direction midline crossings are separated by half a period.

Worked model: height on a rotating wheel

Consider an idealized observation wheel of radius 10 m, with its centre 12 m above level ground. It completes one revolution in 40 seconds at a constant angular speed. A passenger starts at the lowest point at $t=0$.

The lowest height is 2 m and the highest is 22 m. Therefore the amplitude is 10 m and the midline is 12 m. The angular speed is $2\pi/40=\pi/20$ radians per second. A negative cosine fits the starting minimum:

The minus sign sets the starting position; it does not make the amplitude negative. The amplitude remains $|-10|=10$. Every 40 seconds the model returns to its initial height and direction of motion.

These five quarter-cycle values provide checkpoints for a graph of the model. The formula predicts a smooth curve through them, not straight segments connecting the points.

When is the passenger at least 17 metres high?

For one revolution, solve $H(t)\ge17$ on $0\le t\le40$:

The inequality reverses when dividing by $-10$. Let $u=\pi t/20$, so $0\le u\le2\pi$. Cosine is at most $-1/2$ from $2\pi/3$ to $4\pi/3$. Therefore

The passenger is at least 17 m high from about 13.33 seconds to 26.67 seconds. The two boundary times solve $H(t)=17$; the entire interval between them satisfies the inequality.

An inverse cosine calculation alone would return only the first angle $2\pi/3$. The second time and the interval between them come from the graph’s symmetry and the meaning of the inequality.

Worked model from a small data set

Now consider a hypothetical quantity with these measurements:

The extremes suggest amplitude $(18-6)/2=6$ and midline $(18+6)/2=12$. The repeated maximum from hour 2 to hour 14 suggests a 12-hour period. A cosine model starting at that first maximum is

Substitution reproduces all five listed outputs. In particular, at $t=5$, the angle is $\pi/2$ and the output is 12. At $t=8$, the angle is $\pi$ and the output is 6.

The five points do not prove that the underlying phenomenon is sinusoidal or that 12 hours is its true fundamental period. Many other functions can pass through the same points. More observations, particularly between these times and across several cycles, are needed to test the assumption.

Phase shift and equivalent equations

In the form $A\sin(Bt-C)+D$, factor the argument before reading the shift: $Bt-C=B(t-C/B)$. The shift is $C/B$, not $C$. The expression assumes $B\ne0$.

The same sinusoid can have different-looking equations. A sine model can be rewritten as a shifted cosine; a negative amplitude can be traded for a half-period shift; adding a full period to a phase position does not change the function.

Compare models through their amplitude, midline, period and matching starting behaviour. Do not reject an answer simply because its sine/cosine choice differs from the expected form. Conversely, matching only the maximum and minimum is not enough: a wrong phase can predict the correct heights at the wrong times.

Check the model against the context

Check units, the starting value, the direction immediately after the start, the maximum and minimum, and the time required to repeat. For the wheel, the output must stay between 2 m and 22 m. A prediction below ground would contradict the stated geometry.

For measured data, a residual is observed output minus model output. A pattern in the residuals can reveal an omitted trend, changing amplitude or a poor choice of period. Random-looking small differences do not establish a perfect model, but they are different from a persistent pattern of missed peaks.

The wheel model assumes uniform rotation and treats the passenger as a point fixed to the rim. Starting, stopping and changes in seating position would require additional information. A seasonal or tidal model may need several frequencies rather than one sine wave. Use this model only while its assumptions remain reasonable.

Frequency and angular frequency

Frequency $f=1/T$ counts cycles per unit time, while angular frequency $\omega=2\pi f$ measures radians per unit time. They describe the same repetition using different units. Do not insert a cycles-per-second value directly where the formula expects radians per second.

Related topics

Use Trigonometric Functions for the underlying graphs and Trigonometric Equations for full solution sets. Radians, Arc Length and Circular Motion develops the geometry behind the rotating-wheel model.

Practice questions

1. A periodic quantity has maximum 18 and minimum 6. Find its amplitude and midline.
2. For $H(t)=12-10\cos(\pi t/20)$, find the period and the height at $t=10$.
3. Why does finding one time with inverse cosine not settle when the passenger is at least 17 m high?

Sources and further reading

Further explanations of the definitions and conventions used here:

• OpenStax, Precalculus 2e: Sinusoidal parameters.
• OpenStax, Precalculus 2e: Trigonometric modelling.
• Wolfram Language: Trigonometric Functions. Used for checking exact values, expressions and solutions.