Sinusoidal Functions

Sinusoidal functions model smooth repeating behaviour using amplitude, midline, period, and phase shift.

Amplitude, period, and phase shift describe a repeating graph. A wave repeats across the axis. A vertical measure marks amplitude, a horizontal bracket marks one period, and a second faint wave shows how a phase shift moves the cycle.

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.

Finding amplitude and midline from extrema

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

|a|=\frac{M-m}{2}

and

c=\frac{M+m}{2}.

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

D(t)=3\cos\left(\frac\pi6(t-6)\right)+12.

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

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.