The Central Idea

A quantity is periodic when its behavior repeats after a fixed interval.

A basic sinusoidal model can be written as

\[ \boxed{ y=A\sin(B(x-C))+D } \]

or

\[ \boxed{ y=A\cos(B(x-C))+D }. \]

The parameters describe the geometry of the repeating pattern:

When the independent variable is time, these same parameters describe oscillation.

What Makes a Function Periodic?

A function \(f\) is periodic if there is a positive number \(T\) such that

\[ \boxed{ f(x+T)=f(x) } \]

throughout its domain.

The smallest positive value of \(T\), when one exists, is called the fundamental period.

Examples of Periodic Functions

Sine and cosine satisfy

\[ \sin(x+2\pi)=\sin x \]

and

\[ \cos(x+2\pi)=\cos x. \]

Their fundamental period is

\[ \boxed{2\pi}. \]

Tangent satisfies

\[ \tan(x+\pi)=\tan x, \]

so its fundamental period is

\[ \boxed{\pi}. \]

Periodic Does Not Mean Constant

A periodic quantity may change continuously while still returning to the same pattern.

For example,

\[ y=\sin x \]

rises, falls, becomes negative, rises again, and repeats.

Repetition refers to the entire pattern, not to a fixed output.

Cycle

A cycle is one complete repetition of a periodic pattern.

For a sine wave, one convenient cycle is

\[ 0\leq x\leq2\pi. \]

During that interval the function passes through its full sequence of midline, maximum, midline, minimum, and return to the midline.

Amplitude

The amplitude measures the maximum displacement from the midline.

For

\[ y=A\sin(B(x-C))+D, \]

the amplitude is

\[ \boxed{|A|}. \]

Large amplitude means a larger vertical excursion; small amplitude means a smaller excursion.

Midline or Equilibrium Level

The horizontal center of a sinusoidal oscillation is

\[ \boxed{y=D}. \]

In an applied model, this may represent an average level, equilibrium position, baseline temperature, mean depth, resting height, or another central value.

Maximum and Minimum

For a sinusoidal model with amplitude \(|A|\) and midline \(D\),

\[ \boxed{\text{maximum}=D+|A|} \]

and

\[ \boxed{\text{minimum}=D-|A|}. \]

Therefore,

\[ \boxed{ \operatorname{range} = [D-|A|,\ D+|A|] }. \]

Recovering Amplitude from Data

If a periodic data set has observed maximum \(M\) and minimum \(m\), then

\[ \boxed{ \text{amplitude}=\frac{M-m}{2} }. \]

The midpoint of the extremes is

\[ \boxed{ D=\frac{M+m}{2} }. \]

Period

The period \(T\) is the horizontal distance required for one full cycle.

For

\[ y=A\sin(B(x-C))+D \]

or

\[ y=A\cos(B(x-C))+D, \]

the period is

\[ \boxed{ T=\frac{2\pi}{|B|} }. \]

Frequency

Frequency counts cycles per unit of the independent variable.

If \(T\) is the period,

\[ \boxed{ f=\frac1T }. \]

Thus period and frequency are reciprocal descriptions of repetition.

Frequency and Time

When the independent variable is time in seconds, frequency is often measured in hertz:

\[ \boxed{ 1\text{ Hz}=1\text{ cycle per second} }. \]

A frequency of \(5\text{ Hz}\) means five cycles occur each second.

The corresponding period is

\[ T=\frac15\text{ s}=0.2\text{ s}. \]

Angular Frequency

A time-based sinusoid is often written

\[ y=A\sin(\omega t+\phi)+D. \]

The parameter

\[ \omega \]

is the angular frequency.

It measures phase change in radians per unit time.

The relationships are

\[ \boxed{\omega=2\pi f} \]

and

\[ \boxed{ T=\frac{2\pi}{|\omega|} }. \]

Phase

Phase describes position within a cycle.

Two signals can have identical amplitude and frequency but reach their maxima at different times.

That offset is a phase difference.

Phase Shift

In

\[ y=A\sin(B(x-C))+D, \]

the horizontal phase shift is

\[ \boxed{C}. \]

A positive \(C\) moves the pattern right, while a negative \(C\) moves it left.

Phase-Angle Form

A model may instead be written

\[ y=A\sin(Bx+\phi)+D. \]

Factoring \(B\) gives

\[ Bx+\phi = B\left(x+\frac{\phi}{B}\right). \]

Therefore the horizontal phase shift is

\[ \boxed{ -\frac{\phi}{B} }. \]

Temporal Period versus Spatial Period

A repeating pattern can be described in time or in space.

If the horizontal variable is time, the repeat interval is a temporal period.

If the horizontal variable is position, the repeat distance is a spatial period.

In wave language, spatial period is usually called wavelength.

Wavelength

The wavelength, commonly denoted by \(\lambda\), is the spatial distance over which a wave pattern repeats.

It may be measured from one crest to the next, one trough to the next, or any point in the cycle to the corresponding point one cycle later.

\[ \boxed{\lambda=\text{spatial length of one cycle}}. \]

Wavenumber

A spatial sinusoid may be written

\[ y=A\sin(kx+\phi). \]

The coefficient \(k\) is called the wavenumber in many wave contexts.

If the pattern repeats after wavelength \(\lambda\), then

\[ \boxed{ k=\frac{2\pi}{\lambda} }. \]

A Traveling-Wave Form

A simple idealized traveling wave may be represented by

\[ \boxed{ y(x,t)=A\sin(kx-\omega t+\phi) }. \]

Here:

This formula is an introduction to wave modeling rather than a complete theory of physical waves.

Wave Speed

If one spatial cycle of length \(\lambda\) passes a point in one temporal period \(T\), then the propagation speed is

\[ v=\frac{\lambda}{T}. \]

Since

\[ f=\frac1T, \]

we obtain

\[ \boxed{v=f\lambda}. \]

In terms of angular frequency and wavenumber,

\[ \boxed{ v=\frac{\omega}{k} } \]

for this simple traveling-wave form.

Wave Vocabulary

Several geometric terms recur in wave descriptions:

Crest-to-Crest and Trough-to-Trough Distance

Corresponding points one cycle apart are separated by one wavelength.

Therefore:

\[ \boxed{ \text{crest-to-next-crest distance}=\lambda } \]

and

\[ \boxed{ \text{trough-to-next-trough distance}=\lambda }. \]

Crest-to-Trough Distance

A crest and the immediately following trough are half a cycle apart.

Therefore their horizontal separation is

\[ \boxed{\frac{\lambda}{2}} \]

for a spatial wave, or

\[ \boxed{\frac{T}{2}} \]

for a time graph.

Quarter-Cycle Structure

A sinusoid naturally divides into quarter cycles.

If the period is \(T\), the key-point spacing is

\[ \boxed{\frac{T}{4}}. \]

For a spatial wave, the corresponding spacing is

\[ \boxed{\frac{\lambda}{4}}. \]

Sine as a Periodic Model

Sine is convenient when a periodic quantity begins at the midline.

A positive-amplitude sine model

\[ y=A\sin(B(x-C))+D \]

begins at the shifted midline when \(x=C\) and initially increases.

If \(A<0\), it initially decreases instead.

Cosine as a Periodic Model

Cosine is convenient when a periodic quantity begins at an extreme.

For positive \(A\),

\[ y=A\cos(B(x-C))+D \]

begins at a maximum when \(x=C\).

For negative \(A\), it begins at a minimum.

Sine and Cosine Can Model the Same Cycle

Sine and cosine differ only by phase:

\[ \sin x = \cos\left(x-\frac{\pi}{2}\right). \]

Therefore a sinusoidal data set can usually be modeled by either sine or cosine if the phase is chosen appropriately.

Example: Ferris-Wheel-Type Height Model

Suppose a rotating point moves between heights \(2\) meters and \(18\) meters and completes one revolution every \(40\) seconds.

The amplitude is

\[ \frac{18-2}{2}=8. \]

The midline is

\[ \frac{18+2}{2}=10. \]

The angular frequency is

\[ \omega=\frac{2\pi}{40}=\frac{\pi}{20}. \]

A Model Beginning at the Maximum

If the rotating point begins at its maximum height at \(t=0\), a convenient cosine model is

\[ \boxed{ h(t)=8\cos\left(\frac{\pi}{20}t\right)+10 }. \]

Check:

\[ h(0)=8+10=18. \]

Half a period later,

\[ t=20, \]

the point reaches the minimum height:

\[ h(20)=2. \]

Example: Daily Temperature Model

Suppose a simplified daily temperature varies from \(12^\circ\) to \(28^\circ\) and repeats every \(24\) hours.

Then

\[ A=\frac{28-12}{2}=8 \]

and

\[ D=\frac{28+12}{2}=20. \]

The angular frequency is

\[ \omega=\frac{2\pi}{24}=\frac{\pi}{12}. \]

Adding a Time Shift

If the temperature reaches its maximum at \(t=15\) hours, a convenient model is

\[ \boxed{ T(t)=8\cos\left( \frac{\pi}{12}(t-15) \right)+20 }. \]

At \(t=15\),

\[ \cos0=1, \]

so the model gives the maximum \(28^\circ\).

Example: Tidal Depth

Suppose water depth ranges from \(1.8\) meters to \(5.4\) meters and the pattern repeats every \(12\) hours.

The amplitude is

\[ \frac{5.4-1.8}{2}=1.8. \]

The midline is

\[ \frac{5.4+1.8}{2}=3.6. \]

The angular frequency is

\[ \frac{2\pi}{12} = \frac{\pi}{6}. \]

Modeling from a Midline Crossing

Suppose the tidal depth passes upward through the midline at \(t=2\).

A convenient sine model is

\[ \boxed{ d(t) = 1.8\sin\left( \frac{\pi}{6}(t-2) \right) +3.6 }. \]

Reading a Model from a Graph

To reconstruct a sinusoidal equation from a graph, identify:

  1. maximum and minimum;
  2. midline;
  3. amplitude;
  4. period;
  5. a convenient starting feature;
  6. the direction of motion at that feature.

These determine a useful sine or cosine form.

Finding Period from Repeated Peaks

If a graph reaches consecutive maxima at

\[ x=3 \]

and

\[ x=11, \]

then the period is

\[ \boxed{T=11-3=8}. \]

Finding Period from Repeated Midline Crossings

Be careful: not every repeated-looking feature is one full cycle apart.

Two consecutive midline crossings usually occur half a cycle apart.

If upward and downward crossings occur at \(x=2\) and \(x=5\), then

\[ \frac{T}{2}=3 \]

and

\[ \boxed{T=6}. \]

Same-Direction Midline Crossings

Two midline crossings in the same direction are one full cycle apart.

For example, if an increasing midline crossing occurs at \(t=1\) and the next increasing crossing occurs at \(t=9\), then

\[ \boxed{T=8}. \]

Estimating a Sinusoid from Data

Real data rarely follow a perfect sine or cosine curve.

A sinusoidal model may still be useful when the data display:

The model is then an approximation rather than an exact identity.

Model versus Data

A mathematical model is not expected to reproduce every measurement exactly.

Instead, it captures the important structure of the phenomenon.

Differences between observed data and model predictions are called residuals or errors.

Checking a Periodic Model

A useful model should be checked against known features:

Units Matter

In an applied sinusoidal model:

Example: Units in a Vibration Model

Suppose

\[ y(t)=0.04\sin(12\pi t), \]

where \(y\) is measured in meters and \(t\) in seconds.

The amplitude is

\[ \boxed{0.04\text{ m}}. \]

The angular frequency is

\[ 12\pi\text{ rad/s}. \]

The ordinary frequency is

\[ \frac{12\pi}{2\pi} = \boxed{6\text{ Hz}}. \]

Period of the Vibration

Since

\[ f=6\text{ Hz}, \]

the period is

\[ T=\frac16\text{ s}. \]

Equivalently,

\[ T=\frac{2\pi}{12\pi} = \boxed{\frac16\text{ s}}. \]

Example: Spatial Wave

Consider

\[ y(x)=0.3\sin(4\pi x), \]

where \(x\) is measured in meters.

The amplitude is

\[ 0.3. \]

The wavenumber is

\[ k=4\pi. \]

Therefore,

\[ \lambda = \frac{2\pi}{4\pi} = \boxed{0.5\text{ m}}. \]

Traveling-Wave Direction

In a standard expression such as

\[ y(x,t)=A\sin(kx-\omega t), \]

keeping the phase constant gives

\[ kx-\omega t=\text{constant}. \]

Solving for \(x\),

\[ x=\frac{\omega}{k}t+\text{constant}. \]

The phase therefore moves toward increasing \(x\).

Changing the sign before \(\omega t\) reverses the travel direction under this convention.

Snapshot versus Time Trace

A wave can be graphed in different ways.

If time is fixed and position varies, the graph is a spatial snapshot.

If position is fixed and time varies, the graph records how one location oscillates over time.

These two views have wavelength and period as their respective repeat intervals.

Phase Difference between Two Signals

Suppose

\[ y_1=\sin x \]

and

\[ y_2=\sin\left(x-\frac{\pi}{3}\right). \]

The second signal is shifted right by

\[ \boxed{\frac{\pi}{3}}. \]

They have the same amplitude and period but different phase.

In Phase

Two sinusoidal signals of the same frequency are in phase when corresponding parts of their cycles occur together.

For example,

\[ \sin x \]

and

\[ 4\sin x \]

reach zeros, maxima, and minima at the same inputs even though their amplitudes differ.

Half a Cycle Out of Phase

A phase difference of

\[ \pi \]

corresponds to half of a \(2\pi\) cycle.

Therefore,

\[ \sin(x-\pi)=-\sin x. \]

When one signal is at a maximum, the other is at a minimum.

Quarter-Cycle Phase Difference

Sine and cosine differ by one quarter of a cycle:

\[ \boxed{ \cos x = \sin\left(x+\frac{\pi}{2}\right) }. \]

This is why cosine begins at a maximum while sine begins at a midline crossing.

Introductory Superposition

When two wave-like effects occur simultaneously, their displacements may sometimes be modeled by adding the functions:

\[ y=y_1+y_2. \]

This principle is called superposition in many linear models.

The resulting pattern depends on the amplitudes, frequencies, and phases of the component waves.

Same Frequency, Same Phase

Suppose

\[ y_1=2\sin x \]

and

\[ y_2=3\sin x. \]

Adding gives

\[ y=5\sin x. \]

The amplitudes reinforce because the waves have the same frequency and phase.

Same Frequency, Opposite Phase

Suppose

\[ y_1=3\sin x \]

and

\[ y_2=3\sin(x+\pi). \]

Since

\[ \sin(x+\pi)=-\sin x, \]

the sum is

\[ y=0. \]

In this idealized mathematical case, the two waves cancel completely.

Constructive and Destructive Combination

When wave contributions reinforce one another, the combination is often described as constructive.

When they partially or completely cancel, the combination is often described as destructive.

These ideas follow from ordinary addition of signed displacements.

Not Every Periodic Pattern Is Sinusoidal

Periodicity does not imply that a graph must be a sine or cosine wave.

Square waves, sawtooth patterns, repeating pulses, and many other forms are periodic but not single sinusoids.

Sine and cosine are important because they are simple, smooth periodic building blocks.

Not Every Oscillation Is Perfectly Periodic

Real systems may change over time.

Their amplitude, frequency, midline, or phase may drift.

Some oscillations decay, grow, or respond irregularly to outside forces.

A fixed sinusoidal model is most appropriate when the repeating structure remains approximately stable.

Periodic and Cyclic Are Related but Not Identical Ideas

A cycle describes one repetition of a process.

Periodic means that the mathematical pattern repeats at a consistent interval.

Everyday processes may be called cyclic even when their timing or magnitude varies enough that a strict periodic equation is only an approximation.

Interpolation versus Extrapolation

A periodic model may be used to estimate values between observed data points.

This is interpolation.

Using the model beyond the observed interval is extrapolation.

Extrapolation assumes the repeating pattern continues and therefore requires more caution.

Common Error: Using Peak-to-Peak Height as Amplitude

If a wave ranges from \(-4\) to \(4\), the total vertical span is

\[ 8. \]

But the amplitude is half that:

\[ \boxed{4}. \]

Common Error: Confusing Period with Wavelength

Period is a repeat interval in time or in the chosen independent variable.

Wavelength is specifically a spatial repeat distance.

They can be related through wave speed,

\[ v=f\lambda, \]

but they are not the same quantity.

Common Error: Confusing Frequency with Angular Frequency

Ordinary frequency \(f\) counts cycles per unit time.

Angular frequency \(\omega\) measures radians of phase per unit time.

They are related by

\[ \boxed{\omega=2\pi f}. \]

Common Error: Reading Half a Cycle as a Full Period

Consecutive maximum and minimum points are separated by half a period, not a full period.

If a maximum occurs at \(t=2\) and the next minimum at \(t=7\), then

\[ \frac{T}{2}=5 \]

so

\[ \boxed{T=10}. \]

Common Error: Ignoring Direction at a Midline Crossing

An increasing midline crossing and a decreasing midline crossing are only half a cycle apart.

To identify a full period from crossings, compare crossings with the same direction.

Common Error: Treating a Model as Exact Reality

A sinusoidal equation may capture the dominant pattern without explaining every observed variation.

Modeling requires attention to approximation, measurement error, changing conditions, and the limits of the assumptions.

Common Error: Ignoring Units

A frequency of

\[ 4 \]

is incomplete in an applied setting.

It might mean

\[ 4\text{ Hz}, \]

\(4\) cycles per hour, or another rate entirely.

Units are part of the mathematical interpretation.

A Reliable Procedure for Reading a Periodic Graph

  1. Identify the maximum and minimum.
  2. Compute the amplitude.
  3. Compute the midline.
  4. Find two corresponding repeated features.
  5. Measure the period between them.
  6. Compute frequency if meaningful.
  7. Identify a convenient phase reference.
  8. Check units and the direction of change.

A Reliable Procedure for Building a Sinusoidal Model

  1. Find the maximum \(M\) and minimum \(m\).
  2. Compute \(A=(M-m)/2\).
  3. Compute \(D=(M+m)/2\).
  4. Determine the period \(T\).
  5. Compute \(B=2\pi/T\).
  6. Choose sine or cosine based on a convenient observed feature.
  7. Determine the phase shift.
  8. Write the equation.
  9. Test the model against several known points.

A Reliable Procedure for a Simple Wave

  1. Identify amplitude.
  2. Identify wavelength \(\lambda\).
  3. Compute \(k=2\pi/\lambda\).
  4. Identify period \(T\) or frequency \(f\).
  5. Compute \(\omega=2\pi f=2\pi/T\).
  6. Determine the phase or starting position.
  7. Choose a sign convention for travel direction if motion is included.
  8. Check units in the completed model.

Practice 1: Amplitude and Midline

A periodic graph has maximum \(17\) and minimum \(5\).

Find its amplitude and midline.

Practice 2: Period and Frequency

A motion repeats every \(8\) seconds.

Find its period and frequency.

Practice 3: Angular Frequency

A sinusoidal motion has frequency

\[ 3\text{ Hz}. \]

Find its angular frequency.

Practice 4: Wavelength

A spatial wave is modeled by

\[ y=\sin(5\pi x). \]

Find its wavelength.

Practice 5: Wave Speed

A wave has frequency

\[ 4\text{ Hz} \]

and wavelength

\[ 2.5\text{ m}. \]

Find its speed.

Practice 6: Build a Cosine Model

A quantity ranges from \(6\) to \(14\), repeats every \(10\) units, and begins at a maximum when \(x=0\).

Write a cosine model.

Practice 7: Build a Sine Model

The same quantity ranges from \(6\) to \(14\), repeats every \(10\) units, and passes upward through its midline at \(x=2\).

Write a sine model.

Practice 8: Read Period from Peaks

Consecutive maxima occur at

\[ t=4 \]

and

\[ t=13. \]

Find the period.

Practice 9: Read Period from Maximum to Minimum

A periodic graph reaches a maximum at \(t=3\) and the next minimum at \(t=8\).

Find the period.

Practice 10: Spatial Wave

For

\[ y(x)=0.2\cos(8\pi x), \]

find the amplitude and wavelength.

Practice 11: Traveling-Wave Parameters

Consider

\[ y(x,t)=3\sin(2\pi x-10\pi t). \]

Find the amplitude, wavelength, frequency, and wave speed.

Practice 12: Identify the Error

A graph has a maximum of \(9\) and a minimum of \(1\). A student says its amplitude is

\[ 9-1=8. \]

Explain the error.

Practice Answers

Practice 1

Amplitude:

\[ \frac{17-5}{2} = \boxed{6}. \]

Midline:

\[ \frac{17+5}{2} = \boxed{11}. \]

Practice 2

\[ \boxed{T=8\text{ s}}. \]

Frequency:

\[ f=\frac18 = \boxed{0.125\text{ Hz}}. \]

Practice 3

\[ \omega=2\pi f = 2\pi(3) = \boxed{6\pi\text{ rad/s}}. \]

Practice 4

Here

\[ k=5\pi. \]

Therefore,

\[ \lambda = \frac{2\pi}{5\pi} = \boxed{\frac25}. \]

Practice 5

\[ v=f\lambda = (4)(2.5) = \boxed{10\text{ m/s}}. \]

Practice 6

Amplitude:

\[ A=\frac{14-6}{2}=4. \]

Midline:

\[ D=\frac{14+6}{2}=10. \]

Since

\[ T=10, \]

we have

\[ B=\frac{2\pi}{10} = \frac{\pi}{5}. \]

Beginning at a maximum gives

\[ \boxed{ y=4\cos\left(\frac{\pi}{5}x\right)+10 }. \]

Practice 7

The amplitude, midline, and \(B\)-value remain

\[ 4,\qquad10,\qquad\frac{\pi}{5}. \]

Passing upward through the midline at \(x=2\) gives

\[ \boxed{ y = 4\sin\left( \frac{\pi}{5}(x-2) \right)+10 }. \]

Practice 8

\[ T=13-4 = \boxed{9}. \]

Practice 9

Maximum to the next minimum is half a cycle:

\[ \frac{T}{2}=8-3=5. \]

Therefore,

\[ \boxed{T=10}. \]

Practice 10

Amplitude:

\[ \boxed{0.2}. \]

Wavenumber:

\[ k=8\pi. \]

Therefore,

\[ \lambda = \frac{2\pi}{8\pi} = \boxed{\frac14}. \]

Practice 11

Amplitude:

\[ \boxed{3}. \]

Since

\[ k=2\pi, \]

the wavelength is

\[ \lambda = \frac{2\pi}{2\pi} = \boxed{1}. \]

Since

\[ \omega=10\pi, \]

the frequency is

\[ f = \frac{10\pi}{2\pi} = \boxed{5}. \]

Therefore,

\[ v=f\lambda = 5(1) = \boxed{5}. \]

Physical units depend on the units assigned to \(x\) and \(t\).

Practice 12

The difference

\[ 9-1=8 \]

is the full peak-to-peak vertical range.

Amplitude is half that distance:

\[ \boxed{ \frac{9-1}{2}=4 }. \]

What to Remember

A basic sinusoidal model is

\[ \boxed{ y=A\sin(B(x-C))+D } \]

or

\[ \boxed{ y=A\cos(B(x-C))+D }. \]

Section Perspective

Trigonometry began with ratios in right triangles and expanded through the unit circle into a general language of angle, rotation, coordinates, and periodic behavior.

The same sine and cosine relationships now connect:

This unified viewpoint is the main purpose of Trigonometry and Periodicity: a geometric relationship first seen in a triangle becomes a reusable mathematical language for rotation and repetition.