Trigonometric Modelling

Many real-world quantities change in patterns that repeat over time. Daylight hours increase and decrease each year, tides rise and fall, a Ferris wheel repeatedly carries passengers up and down, and vibrating objects move back and forth around an equilibrium position.

These are examples of periodic phenomena.

Sine and cosine functions are especially useful for modelling smooth repeating patterns because they naturally describe periodic motion.

A general sinusoidal model can be written as:

y = A sin(B(x − C)) + D

or:

y = A cos(B(x − C)) + D

The parameters A, B, C, and D describe important features of the real system.

Recognising Periodic Behaviour

A phenomenon is periodic when it repeats after a regular interval.

Examples include:

• rotation of a wheel
• motion of a pendulum
• seasonal daylight patterns
• approximately seasonal temperature patterns
• tides
• alternating electrical current
• some biological rhythms
• vibrating springs
• sound waves

If the pattern is smooth and oscillates around a central value, a sine or cosine model may be appropriate.

A repeating pattern does not have to start at zero.

It might begin at:

• a maximum
• a minimum
• the midline while increasing
• the midline while decreasing

The starting position helps us choose the most convenient equation.

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The General Sinusoidal Model

Consider:

y = A sin(B(x − C)) + D

The parameters have different roles.

For sine and cosine:

amplitude = |A|

period = 2π/|B|

phase shift = C

midline = y = D

These mathematical features often have direct physical meanings.

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Amplitude in a Model

The amplitude measures the maximum distance between the midline and the highest or lowest point.

If the maximum and minimum are known:

amplitude = (maximum − minimum)/2

Suppose a periodic quantity varies between:

maximum = 18

minimum = 6

Then:

amplitude = (18 − 6)/2

= 12/2

= 6

The amplitude tells us the size of the oscillation around the central value.

Finding the Midline

The midline is halfway between the maximum and minimum.

Calculate:

D = (maximum + minimum)/2

Using:

maximum = 18

minimum = 6

we get:

D = (18 + 6)/2

= 24/2

= 12

Therefore, the model oscillates around:

y = 12

Its maximum is:

12 + 6 = 18

and its minimum is:

12 − 6 = 6

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Maximum and Minimum

For:

y = A sin(B(x − C)) + D

or the equivalent cosine model:

maximum = D + |A|

minimum = D − |A|

This means that amplitude and midline completely determine the vertical range of a sinusoidal model.

For example:

y = 7cos(Bx) + 20

has:

amplitude = 7

midline = 20

maximum = 27

minimum = 13

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Period

The period is the amount of input required for one complete cycle.

For example:

• Earth's seasonal cycle is approximately 1 year
• a Ferris wheel might complete a revolution every 60 seconds
• a pendulum might repeat its motion every 2 seconds
• a periodic signal might repeat every 0.01 seconds

For:

y = A sin(B(x − C)) + D

the period is:

P = 2π/|B|

Therefore, if the period is known:

|B| = 2π/P

This equation is extremely useful when building models from data.

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Finding B from the Period

Suppose a periodic phenomenon repeats every:

12 hours

Then:

P = 12

Use:

B = 2π/P

B = 2π/12

= π/6

So the model could contain:

sin((π/6)t)

or:

cos((π/6)t)

depending on its starting position.

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Frequency

Frequency describes how often a cycle occurs.

A useful relationship is:

frequency = 1/period

or:

f = 1/P

If a machine completes 5 cycles every second:

f = 5 Hz

Its period is:

P = 1/5

= 0.2 s

Frequency is especially important when modelling waves, vibrations, sound, and electrical signals.

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Phase Shift

The phase shift determines the horizontal position of the periodic pattern.

In:

y = A sin(B(x − C)) + D

C represents the phase shift.

For example:

y = 4sin((π/6)(t − 3)) + 10

has a phase shift of:

3 units to the right

In a real-world model, phase shift often tells us when a particular stage of the cycle occurs.

For example, it might describe:

• when high tide occurs
• when maximum temperature occurs
• when a Ferris wheel rider reaches the midline
• when daylight begins increasing from its average
• when a seasonal population reaches its peak

Choosing Sine or Cosine

Sine and cosine can model the same type of periodic behaviour.

The choice often depends on where the data begin.

Cosine is convenient when the cycle begins at:

• a maximum
• a minimum

Sine is convenient when the cycle begins at:

• the midline while increasing
• the midline while decreasing

For example, if a Ferris wheel rider starts at the top, cosine is usually convenient.

If the rider begins at the wheel's centre height while travelling upward, sine is usually convenient.

Different correct equations can sometimes describe the same sinusoidal pattern.

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Building a Model from Maximum and Minimum Data

Suppose a quantity varies between:

maximum = 34

minimum = 10

Amplitude:

A = (34 − 10)/2

= 12

Midline:

D = (34 + 10)/2

= 22

We already know that a possible model has the form:

y = 12sin(B(x − C)) + 22

or:

y = 12cos(B(x − C)) + 22

We still need the period and phase information to determine B and C.

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A Step-by-Step Modelling Strategy

When constructing a sinusoidal model from data, use this process:

Identify maximum and minimum → Find amplitude → Find midline → Find period → Calculate B → Determine phase shift → Choose sine or cosine → Check the model

This systematic method works for many problems.

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Worked Example: Ferris Wheel

A Ferris wheel has:

maximum rider height = 46 m

minimum rider height = 6 m

one revolution = 80 seconds

A rider begins at the highest point.

Find the Amplitude

A = (46 − 6)/2

= 40/2

= 20 m

Find the Midline

D = (46 + 6)/2

= 52/2

= 26 m

This also represents the height of the wheel's centre.

Find B

P = 80

B = 2π/80

= π/40

Since the rider begins at a maximum, cosine is convenient.

Therefore:

h(t) = 20cos((π/40)t) + 26

Interpreting the Ferris Wheel Equation

Consider:

h(t) = 20cos((π/40)t) + 26

Each part has meaning.

20 = radius of the Ferris wheel

π/40 = controls the rotational period

80 s = time for one revolution

26 = height of the centre of the wheel

At:

t = 0

cos(0) = 1

so:

h(0) = 20 + 26

= 46 m

This confirms that the rider begins at the highest point.

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Predicting with the Ferris Wheel Model

What is the rider's height after 20 seconds?

Use:

h(t) = 20cos((π/40)t) + 26

Substitute:

t = 20

h(20) = 20cos((π/40)(20)) + 26

= 20cos(π/2) + 26

= 20(0) + 26

= 26 m

After one-quarter of the 80-second cycle, the rider is at the wheel's midline.

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Understanding Quarter-Periods

One complete sinusoidal cycle can be divided into four equal sections.

If:

period = P

then:

quarter-period = P/4

For the Ferris wheel:

P = 80 s

quarter-period = 80/4

= 20 s

Starting at the maximum:

This pattern makes many modelling problems easier to solve without extensive calculation.

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Worked Example: Daylight Hours

Suppose a simplified model of daylight has:

maximum daylight = 15 hours

minimum daylight = 9 hours

period = 12 months

Amplitude:

A = (15 − 9)/2

= 3

Midline:

D = (15 + 9)/2

= 12

Period:

P = 12

Therefore:

B = 2π/12

= π/6

If maximum daylight occurs at:

t = 6 months

a convenient cosine model is:

D(t) = 3cos((π/6)(t − 6)) + 12

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Interpreting the Daylight Model

In:

D(t) = 3cos((π/6)(t − 6)) + 12

the amplitude 3 means daylight varies approximately:

3 hours above and below the midline

The midline 12 represents:

12 hours of daylight

The period is:

12 months

The phase shift places the maximum at:

t = 6

This demonstrates how the parameters connect mathematical features to real-world meaning.

Worked Example: Tides

Suppose water depth at a harbour varies approximately between:

1.5 m and 5.5 m

and the simplified period between successive high tides is:

12 hours

Suppose high tide occurs at:

t = 2 hours

Amplitude:

A = (5.5 − 1.5)/2

= 2 m

Midline:

D = (5.5 + 1.5)/2

= 3.5 m

Period:

P = 12

B = 2π/12

= π/6

Since high tide represents a maximum, cosine is convenient:

h(t) = 2cos((π/6)(t − 2)) + 3.5

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Using the Tide Model

Find the predicted water depth at:

t = 5 hours

Use:

h(t) = 2cos((π/6)(t − 2)) + 3.5

Substitute:

h(5) = 2cos((π/6)(3)) + 3.5

= 2cos(π/2) + 3.5

= 3.5 m

Three hours after the high tide, the model predicts that the water level has reached its midline.

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Models from Tables

Sometimes we are given data rather than a description.

Consider:

The maximum is:

12

The minimum is:

4

Amplitude:

A = (12 − 4)/2

= 4

Midline:

D = (12 + 4)/2

= 8

Maximum values occur at:

t = 2 and t = 10

Therefore:

period = 8 hours

So:

B = 2π/8

= π/4

A convenient cosine model with a maximum at t = 2 is:

h(t) = 4cos((π/4)(t − 2)) + 8

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Models from Graphs

When determining a trigonometric model from a graph, identify:

• maximum
• minimum
• midline
• amplitude
• distance between consecutive peaks
• distance between consecutive troughs
• direction of movement at useful points
• location of a maximum, minimum, or midline crossing

The distance between consecutive peaks or consecutive troughs gives the period.

Once the period is known:

B = 2π/P

Modelling Seasonal Temperature

Seasonal temperatures often show an approximately periodic annual pattern.

Suppose a simplified temperature model is:

T(t) = 11cos((π/6)(t − 7)) + 18

where t is measured in months.

Amplitude:

11°C

Period:

2π/(π/6)

= 12 months

Phase shift:

7 months

Midline:

18°C

Maximum:

18 + 11 = 29°C

Minimum:

18 − 11 = 7°C

The model predicts a maximum at approximately:

t = 7

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Interpreting Parameters in Context

A good modelling answer should not merely calculate parameters. It should explain what they mean.

Suppose:

T(t) = 11cos((π/6)(t − 7)) + 18

Then:

Amplitude = 11°C

The temperature varies approximately 11°C above and below its central level.

Period = 12 months

The pattern repeats annually.

Phase shift = 7 months

The model is positioned so that its maximum occurs around month 7.

Vertical displacement = 18°C

The sinusoidal pattern oscillates around 18°C.

Mathematical parameters should always be interpreted using the appropriate units.

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Modelling Oscillations

Consider a mass attached to a spring.

Its displacement from equilibrium might be modelled by:

x(t) = 0.12cos(4πt)

where:

x = displacement in metres

t = time in seconds

Amplitude:

0.12 m

Angular parameter:

B = 4π

Period:

P = 2π/(4π)

= 0.5 s

Frequency:

f = 1/0.5

= 2 Hz

The mass completes two oscillations each second.

Modelling Sound and Waves

A simple sound wave can be modelled using:

y(t) = A sin(2πft + φ)

where:

A = amplitude

f = frequency

φ = phase constant

t = time

For example:

y(t) = 0.02sin(880πt)

Since:

880π = 2πf

divide by 2π:

f = 440 Hz

This corresponds to 440 oscillations per second.

The period is:

P = 1/440

≈ 0.00227 s

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Comparing Models

Suppose we have:

Model A: y = 4sin(x) + 10

and:

Model B: y = 7sin(x) + 10

Both have the same:

• period
• phase
• midline

But Model B has a larger amplitude.

Therefore, Model B represents greater variation around the average level.

Now consider:

Model C: y = 4sin(2x) + 10

Compared with Model A, Model C has the same:

• amplitude
• midline

but a shorter period.

It oscillates more rapidly.

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Determining Parameters from Real Data

Real data rarely fit a perfect sine or cosine curve.

Suppose measurements show:

maximum ≈ 24.3

minimum ≈ 9.7

Rather than expecting exact values, we might estimate:

Amplitude:

A ≈ (24.3 − 9.7)/2

= 7.3

Midline:

D ≈ (24.3 + 9.7)/2

= 17.0

The use of ≈ is important because the model is an approximation.

Mathematical modelling often involves estimates rather than exact values.

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What Makes a Good Trigonometric Model?

A good sinusoidal model usually requires:

• a repeating pattern
• an approximately constant period
• reasonably stable maximum and minimum values
• a central value around which the data oscillate
• smooth changes rather than sudden jumps

If these conditions are approximately satisfied, a sine or cosine model may be useful.

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Evaluating Model Suitability

A mathematical equation can always produce numbers, but that does not mean it is a good model.

Ask:

Does the phenomenon actually repeat?

Is the period approximately constant?

Are the peaks and troughs reasonably consistent?

Does the model follow the observed data?

Are there important factors the model ignores?

Are predictions being made inside or outside the observed data range?

A model should be judged by how well it represents reality, not simply by how convenient its equation is.

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Residuals

One way to evaluate a model is to compare predicted and observed values.

A residual can be calculated as:

residual = observed value − predicted value

Suppose:

observed temperature = 25.4°C

predicted temperature = 24.8°C

Then:

residual = 25.4 − 24.8

= 0.6°C

A small residual means the prediction is close to the observation.

A large residual suggests a poorer fit at that point.

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Residual Patterns

Residuals are especially useful when viewed across many data points.

If residuals are:

• relatively small
• scattered above and below zero
• without an obvious pattern

then the model may be reasonable.

If residuals show a strong pattern, the model may be missing something important.

For example, if residuals repeatedly become positive near each peak and negative near each trough, the model may have the wrong amplitude.

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Interpolation

Interpolation means using a model to estimate values within the range of the data used to construct it.

Suppose temperature measurements were collected from January through December.

Using the model to estimate the temperature in June is interpolation.

Interpolation is often relatively reliable if the model fits the data well.

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Extrapolation

Extrapolation means using a model to predict beyond the range of observed data.

Suppose we use one year of measurements to predict conditions 50 years later.

That is extrapolation.

Extrapolation is generally less reliable because the underlying system may change.

For example:

• climate conditions may change
• machinery may wear
• biological populations may shift
• human behaviour may change
• environmental conditions may change

Models should therefore be used cautiously outside the data range.

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Real Phenomena Are Often More Complicated

A simple sine curve may approximate tides, but real tides are influenced by:

• the Moon
• the Sun
• coastline shape
• water depth
• weather
• atmospheric pressure
• local geography

Similarly, temperature depends on more than seasonal position.

A simple sinusoidal model captures the main periodic pattern, not every detail.

This is an important principle:

A model is a useful simplification of reality, not reality itself.

Sinusoidal Regression

When many real data points are available, technology can be used to find a sinusoidal regression model.

Instead of estimating every parameter manually, software can find values for A, B, C, and D that provide a good fit.

A regression equation might look like:

y = 7.42sin(0.53(x − 2.18)) + 16.37

The decimals reflect the fact that real data do not usually fit perfectly.

Even when technology creates the equation, we still need to interpret:

• amplitude
• period
• phase shift
• midline
• predictions
• limitations

Technology does not replace mathematical interpretation.

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Choosing the Correct Domain

Suppose a Ferris wheel ride lasts only:

0 ≤ t ≤ 240 seconds

The trigonometric equation itself continues forever mathematically.

But the real-world model should only be used during the relevant time interval.

This is called a domain restriction.

Likewise, a seasonal model may be mathematically defined for any value of t but only be supported by the years for which data were collected.

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Solving a Modelling Problem

Suppose:

h(t) = 6cos((π/10)t) + 8

models the height of an object in metres.

When does:

h = 11 m?

Set:

11 = 6cos((π/10)t) + 8

Subtract 8:

3 = 6cos((π/10)t)

Divide by 6:

1/2 = cos((π/10)t)

We know:

cos(π/3) = 1/2

So one solution is:

(π/10)t = π/3

Multiply by 10/π:

t = 10/3 s

≈ 3.33 s

However, because cosine is periodic, there may be another time during the same cycle when the object has the same height.

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Finding the Second Solution

The cosine function has the same value at two points in most cycles.

For:

cos(θ) = 1/2

within:

0 ≤ θ < 2π

the solutions are:

θ = π/3

and:

θ = 5π/3

Therefore:

(π/10)t = 5π/3

t = 50/3 s

≈ 16.67 s

So during one 20-second cycle, the object reaches 11 m at approximately:

3.33 s and 16.67 s

This demonstrates why real-world trigonometric problems can have multiple valid answers.

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Checking Whether an Answer Makes Sense

After solving a modelling problem, ask:

• Is the answer inside the allowed domain?
• Are the units correct?
• Is the value between the model's maximum and minimum?
• Does the timing make sense within the cycle?
• Should there be more than one answer?
• Does the graph support the calculation?

For example, if a Ferris wheel has maximum height 40 m, a calculated rider height of 65 m indicates an error.

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Model Parameters Tell a Story

Suppose:

P(t) = 250sin((π/6)(t − 2)) + 1200

models a seasonal population.

We can interpret:

250 — the population varies by about 250 above and below its central level.

π/6 — gives a period of 12 time units.

2 — shifts the timing of the cycle.

1200 — the population oscillates around approximately 1200.

The equation is therefore not merely a collection of numbers. Each parameter describes part of the system.

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Comparing Two Periodic Models

Suppose:

A(t) = 5sin((π/6)t) + 20

and:

B(t) = 5sin((π/6)(t − 2)) + 20

Both have:

amplitude = 5

period = 12

midline = 20

The only difference is the phase shift.

Model B occurs:

2 time units later

than Model A.

This could represent two locations experiencing similar seasonal patterns at different times.

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Multiple Periodic Effects

Some real systems contain more than one periodic pattern.

For example, a temperature model might contain:

• a yearly cycle
• a daily cycle

A simple model could involve multiple trigonometric terms:

T(t) = 20 + 8sin(2πt/365) + 2sin(2πt)

where the time units are chosen appropriately.

The first sine term represents a longer cycle.

The second represents a shorter cycle.

More advanced modelling often combines several periodic functions.

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Trigonometric Models in Science and Engineering

Sinusoidal models appear throughout science.

Physics

They describe:

• waves
• oscillations
• circular motion
• alternating current
• simple harmonic motion

Biology

They can approximate:

• circadian rhythms
• seasonal populations
• some heart and respiratory patterns

Earth Science

They can model:

• tides
• seasonal temperatures
• daylight cycles

Engineering

They are used for:

• vibration analysis
• rotating machinery
• electrical signals
• communications
• structural oscillations

Astronomy

Periodic functions can help describe:

• rotating objects
• repeating brightness patterns
• orbital phenomena

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Common Mistakes

Using maximum − minimum as the amplitude

Correct:

amplitude = (maximum − minimum)/2

Using the maximum as the midline

Correct:

midline = (maximum + minimum)/2

Assuming B is the period

B controls the period, but:

P = 2π/|B|

Forgetting units

A period might be measured in:

seconds, hours, days, months, or years.

Ignoring phase shift

A model may have the correct amplitude and period but still be badly positioned horizontally.

Automatically choosing sine

Cosine may produce a much simpler model if the data begin at a maximum or minimum.

Assuming sine and cosine models must look different

A sine model and appropriately shifted cosine model can describe the same curve.

Treating a model as exact

Real-world data contain variation, uncertainty, and measurement error.

Ignoring residuals

Differences between predictions and observations help reveal how well a model fits.

Extrapolating too far

A model fitted to current data may not remain accurate indefinitely.

Ignoring domain restrictions

A model should only be used where it makes physical and contextual sense.

Forgetting multiple solutions

A periodic system may reach the same value more than once during a cycle.

Rounding parameters too early

Early rounding can noticeably change later predictions.

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Key Terms

Mathematical model — A mathematical representation of a real system or phenomenon.

Trigonometric model — A model using trigonometric functions such as sine or cosine.

Periodic phenomenon — A phenomenon that repeats after a regular interval.

Sinusoidal model — A model having the wave-like form of sine or cosine.

Parameter — A value in a model controlling one of its features.

Amplitude — The maximum distance from the midline.

Period — The input interval required for one complete cycle.

Frequency — The number of cycles occurring per unit of input.

Phase shift — A horizontal displacement of a periodic model.

Vertical displacement — A vertical shift of a graph.

Midline — The central value around which a sinusoidal model oscillates.

Maximum — The greatest predicted value during a cycle.

Minimum — The smallest predicted value during a cycle.

Cycle — One complete repetition of a periodic pattern.

Quarter-period — One-fourth of a complete period.

Oscillation — Repeated movement or variation around an equilibrium or central value.

Regression — A statistical method for fitting a mathematical model to data.

Sinusoidal regression — A method for fitting a sine or cosine model to periodic data.

Observed value — A value actually measured or recorded.

Predicted value — A value produced by a mathematical model.

Residual — The difference between an observed and predicted value.

Interpolation — Prediction within the range of observed data.

Extrapolation — Prediction outside the range of observed data.

Domain restriction — A limitation placed on input values so that a model matches its real-world context.

Model suitability — How appropriately and accurately a model represents the phenomenon being studied.

Approximation — A value or model that is close to, but not exactly equal to, reality.

Equilibrium — A central or resting position around which an oscillating system moves.

Angular frequency — A measure of the rate of oscillation commonly represented in radians per unit time.

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Key Takeaways

• Trigonometric functions are useful for modelling repeating phenomena.
• Sine and cosine are particularly useful for smooth periodic patterns.
• A general model is:

y = A sin(B(x − C)) + D

or:

y = A cos(B(x − C)) + D

• The amplitude is:

|A|

• From maximum and minimum:

amplitude = (maximum − minimum)/2

• The midline is:

D = (maximum + minimum)/2

• Maximum:

D + |A|

• Minimum:

D − |A|

• Period is the length of one complete cycle.
• The relationship between B and period is:

P = 2π/|B|

• Therefore:

|B| = 2π/P

• Frequency and period are related by:

f = 1/P

• Phase shift determines the timing of the cycle.
• Cosine is often convenient when the model begins at a maximum or minimum.
• Sine is often convenient when the model begins at its midline.
• Different sine and cosine equations can represent the same periodic pattern.
• Parameters should always be interpreted in context and with units.
• Models can be constructed from descriptions, tables, graphs, or measured data.
• Real-world data rarely follow perfect sinusoidal curves.
• Residuals measure the differences between observed and predicted values.
• Small, randomly distributed residuals generally support the suitability of a model.
• Interpolation uses a model within the observed data range.
• Extrapolation predicts beyond the observed data and is generally less reliable.
• Real systems often contain influences that a simple sinusoidal model ignores.
• A model's mathematical domain may be larger than its meaningful real-world domain.
• Periodic equations can produce multiple solutions to real-world problems.
• Trigonometric modelling is widely used in physics, engineering, biology, Earth science, astronomy, and signal analysis.
• A mathematical model is a simplification of reality, not a perfect copy of it.

A useful modelling process is:

Data → Pattern → Parameters → Equation → Prediction → Evaluation

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Check Your Understanding

1. What makes a phenomenon periodic?

2. Give three real-world examples of periodic phenomena.

3. For:

y = 6sin(2x) + 10

identify the amplitude and midline.

4. Find the maximum and minimum of the function in Question 3.

5. Find the period of:

y = 4cos(3x)

6. A periodic phenomenon has a period of 10 seconds. Find B.

7. A quantity varies between 8 and 24. Find its amplitude.

8. Find its midline.

9. A periodic pattern repeats every 12 months. Find B.

10. Explain what phase shift means in a real-world model.

11. When might cosine be more convenient than sine?

12. A Ferris wheel has maximum height 54 m and minimum height 6 m. Find its amplitude.

13. Find the height of the Ferris wheel's centre.

14. The wheel completes one revolution every 72 seconds. Find B.

15. If a rider begins at the maximum height, write a cosine model for the rider's height.

16. Where will the rider be after one-quarter of the period?

17. A temperature model is:

T(t) = 8cos((π/6)(t − 7)) + 21

Determine:

a. amplitude
b. period
c. phase shift
d. midline
e. maximum temperature
f. minimum temperature

18. Explain the meaning of the period in Question 17.

19. Explain the meaning of the amplitude in Question 17.

20. A tide model is:

h(t) = 1.8cos((π/6)(t − 3)) + 3.2

Determine its maximum and minimum predicted water levels.

21. At what time does the model first predict maximum water level?

22. Explain why a simple sinusoidal tide model may not perfectly predict real tides.

23. Define residual.

24. A model predicts 18.7°C but the measured value is 20.1°C. Calculate the residual using:

observed − predicted

25. Explain the difference between interpolation and extrapolation.

26. Why is long-range extrapolation often unreliable?

27. Explain why a mathematically valid input might not make sense in a real-world model.

28. Describe two characteristics of data that would suggest a sinusoidal model might be appropriate.

29. Explain why a periodic equation may give more than one time when a particular value occurs.

30. Challenge: The depth of water at a harbour varies approximately between 1.4 m and 6.2 m. Successive high tides occur approximately 12.4 hours apart, and one high tide occurs at t = 2.5 hours.

a. Calculate the amplitude.
b. Calculate the midline.
c. Determine the period.
d. Calculate B.
e. Choose an appropriate sine or cosine function and explain your choice.
f. Write a possible model for the water depth h(t).
g. Use the model to predict the water depth 3.1 hours after the high tide.
h. Determine approximately when the first low tide occurs after t = 2.5 hours.
i. Explain what each parameter in your model represents physically.
j. Give at least three reasons why actual harbour measurements might differ from the model's predictions.
k. Explain how residuals could be used to evaluate the model.
l. Explain why predictions several years into the future should be treated cautiously.