Modeling Periodic Phenomena with Trig Functions

In this lesson:

• Identify amplitude, period, and midline in
• Extract parameters from real-world context
• Write and use sinusoidal models for periodic phenomena

Learning Objectives for This Lesson

By the end, you will be able to:

1. Identify amplitude, period, and midline in a sinusoidal equation
2. Extract parameters from a real-world context
3. Write a sinusoidal model from context
4. Choose sine or cosine based on starting position
5. Use a model to predict values

A Repeating Pattern in the Real World

A Ferris wheel seat rises and falls as the wheel turns:

• Maximum height: 52 feet
• Minimum height: 2 feet
• Period: 8 minutes

How could one equation capture the height at any moment?

This is a sinusoidal function — sine or cosine with adjustable parameters.

Anatomy of

Each parameter controls one physical feature:

• = amplitude — half the range (max to min)
• = period — length of one complete cycle
• = midline — vertical center; average of max and min

Same structure applies to .

Parameter Effects on the Graph

Changing one parameter at a time shows each parameter's isolated effect.

Identifying Parameters from an Equation

From :

• → amplitude
• → period
• → midline:
• Maximum ; Minimum

Quick Check: Period = B or 2π/B?

What is the period of ?

• Is it ?
• Or ?

Think about what happens to the graph when B is larger — does it cycle faster or slower?

Extracting Parameters from Max and Min Values

When a problem describes a physical situation, extract parameters using:

Don't estimate — compute from the formulas.

Ferris Wheel Problem: Extracting All Parameters

A Ferris wheel: max height 52 ft, min height 2 ft, period 8 min.

• Midline ft
• Amplitude ft

Second Example: Daily Temperature Cycle

Daily temperature: high 85°F, low 65°F, one cycle per 24 hours.

• Midline °F
• Amplitude °F
• Period hours, so

The model will predict temperature at any time of day.

Quick Check: Extracting Tide Model Parameters

A tide goes from 1.2 ft to 4.8 ft with a period of 12.5 hours.

Find the midline and amplitude.

Use: midline = (max + min)/2 and amplitude = (max − min)/2

Then find B.

How to Choose Sine vs. Cosine

The starting condition at determines the function:

• Midline going up →
• At maximum →
• At minimum →

Choose the form that avoids a phase shift.

Decision Flowchart: Sin or Cos?

The simplest model matches the starting condition without a phase shift.

Building the Model: Board at the Bottom

Ferris wheel: amplitude , , midline . Board at the bottom (start at minimum).

Choice: because we start at the minimum.

Verify: ✓ (minimum height)

Same Wheel, Board at the Midline

Same Ferris wheel; board at midline going up.

Choice: — we start at midline increasing.

Verify: ✓ (midline height)

Same physical wheel — different starting conditions, different equation.

Guided Practice: Write a Temperature Model

High 85°F, low 65°F; period 24 h; max at noon.

• Midline 75, amplitude 10,
• (midnight): midline going down → use

Verify: ✓

Quick Check: Sine, Cosine, or Negative Cosine?

A pendulum starts at its maximum displacement and swings back and forth.

At , displacement is at its maximum.

Which form should you use?

The choice that avoids a phase shift is the simplest model.

Using the Model: Evaluating at a Specific Time

With , find :

Setting Up "When Does ?"

Set the model equal to 40 and solve:

Solving for requires an inverse cosine — this connects to HSF.TF.B.6 and B.7.

Practice Problems: Ferris Wheel Predictions

Using , evaluate:

1. — height at the start
2. — height after half a revolution
3. — height after one full revolution
4. — height at 2 minutes

Show the substitution and evaluation steps for each.

Answers to Ferris Wheel Practice Problems

1. ft ← minimum (boarding)
2. ft ← maximum (top)
3. ft ← back to start
4. ft ← midline

Key Takeaways from This Lesson

• Amplitude , period , midline
• Midline up → sine; max → cosine; min → cosine
• Verify: must match the starting condition

Watch out:

• Period is , NOT
• Amplitude is NOT the maximum value
• Always use radians

What's Next: Inverse Trig Functions

HSF.TF.B.6 — Domain Restrictions for Inverse Trig

• Today you set up but couldn't solve for
• Solving requires inverse cosine:
• But has infinitely many solutions — how do we define a single output?

Domain restrictions make inverse trig functions well-defined.