Sinusoidal Functions

A sinusoidal model compresses an entire repeating story into four meaningful features: centre, size, timing, and starting position.

Recognizing periodic behaviour

A relationship is periodic when its outputs repeat after a fixed input interval. Ferris-wheel height, seasonal daylight, tides, alternating current, and idealized sound waves can all show periodic behaviour.

Not every repeating pattern is sinusoidal. A sine or cosine model is appropriate when changes are smooth and the pattern has a consistent centre, amplitude, and period.

General models

Common forms are

and

The amplitude is $|a|$, the midline is $y=c$, and the phase shift is $d$. In degrees the period is $360^\circ/|k|$; in radians it is $2\pi/|k|$.

For time-based data, it is often clearer to write $k=2\pi/T$, where $T$ is the period.

Finding amplitude and midline from extrema

If a cycle has maximum $M$ and minimum $m$, then

and

The amplitude is half the total vertical range, not the maximum value. The midline is the average of the extremes.

Finding the period and phase

The period is the horizontal distance between matching consecutive points, such as maximum to next maximum. A half-cycle from maximum to minimum covers $T/2$; a quarter-cycle from midline to maximum covers $T/4$.

Phase shift locates a convenient starting feature. A positive cosine model naturally begins at a maximum; a positive sine model naturally begins at an upward midline crossing.

Worked example: Ferris wheel model

Worked example: seasonal data

Suppose daylight varies from $9$ h to $15$ h with a $12$-month period, reaching a maximum at month $6$. Then amplitude is $3$, midline is $12$, and $k=2\pi/12=\pi/6$.

A cosine model is

The phase shift $6$ aligns cosine's maximum with the observed maximum. The model is idealized; actual daylight depends on location and calendar details.

Interpreting parameters in context

Every parameter should be translated into units:

• amplitude: maximum departure from the average output;
• midline: average or equilibrium level;
• period: input time for one full cycle;
• phase shift: timing of a chosen cycle landmark.

The coefficient $k$ is a frequency parameter, not the period itself. Larger $|k|$ means faster cycles and therefore a shorter period.

Solving a model

A question may ask when a model reaches a threshold. Graph the sinusoid and the horizontal target line, or solve algebraically using inverse trig plus periodic solutions.

Because values repeat, there may be two times per cycle or many times over a long interval. State all solutions within the contextual domain and attach units.

Assessing model quality

Compare predictions with observed data and examine residuals. A sinusoidal model may capture a broad seasonal cycle while missing weather, mechanical imperfections, trends, or changing amplitude.

Avoid extrapolating indefinitely when the system's average or period can change. A useful model is a purposeful approximation, not a claim that every data point lies exactly on the curve.

Choosing sine or cosine

Sine and cosine can represent the same sinusoid with different phase shifts. Choose the form that makes the starting condition easiest to express:

• maximum or minimum at a known time: cosine is often convenient;
• midline crossing at a known time: sine is often convenient.

Equivalent forms are not competing answers if they produce the same outputs.

Common mistakes

Using maximum as amplitude. Amplitude is $(M-m)/2$.

Using range as midline. Midline is the average $(M+m)/2$.

Treating $k$ as period. Convert with $T=2\pi/|k|$ or $360^\circ/|k|$.

Forcing a sine model onto non-smooth data. Check whether sinusoidal assumptions are plausible.

Reporting only one repeated solution. Restrict and inspect the full requested interval.

Quick self-check

• Does the phenomenon repeat smoothly?
• What are its maximum, minimum, midline, and amplitude?
• How long is one complete cycle?
• Which starting landmark makes sine or cosine simpler?
• Does the formula reproduce the starting value and a second key point?
• Are all interpretations and solutions reported with units and a realistic domain?

Related topics

• Graphs of Sine and Cosine
• Function Transformations
• Angles in Standard Position
• Unit Circle