All Solutions and Contextual Interpretation

Lesson 2 of 2: Symmetry, Periodicity, and Meaning

In this lesson:

• Find all solutions in an interval using symmetry
• Add full periods to generate additional solutions
• Filter and interpret solutions in context

What You Will Learn Today

By the end of this lesson, you can:

3. Use sine and cosine symmetry to find all solutions
4. Generate solutions in any interval by adding full periods
5. Filter solutions by the problem's stated domain and interpret in context

Recall: The Ferris Wheel Result

Model:

From Lesson 1: minutes is one time when .

Is this the only time? The wheel keeps turning.

What does the unit circle tell us about sine?

Two Angles Share the Same Sine Value

Two angles per revolution have the same sine value.

Sine Symmetry: The Second Solution Formula

If and , then:

For the Ferris wheel:

Applying Sine Symmetry to Find Second Time

From , solve for :

Two solutions in the first period: and

Periodicity: Adding Full Periods for More Solutions

The period of is minutes.

General solutions (in terms of the inner angle ):

where

All Solutions for a Twelve-Minute Ride

Interval min; period min

Cosine Symmetry: A Different Formula

Cosine symmetry uses the x-axis mirror:

Check: Second Solution with Sine

; calculator gives rad.

What is ?

(A)
(B)
(C)

Choose the correct formula for sine.

Check Answer: Sine Uses Pi Minus First

Verify: ✓

Cosine formula () would give ✗

From Math to Real-World Answers

Mathematical solutions exist for any integer .

Context creates a filter:

• Domain restriction: ride is to min; tide is to hr
• Direction constraint: rising only, falling only, or both?
• Physical sense: height must be non-negative; time after event starts

Ferris Wheel: All Four Valid Solutions

, ride duration min

The rider reaches 40 feet at:

0.70 and 8.70: on the way up. 3.30 and 11.30: on the way down.

Graph Verification: Crossings at 40 Feet

Four intersections confirm the four solutions.

Worked Example: Tide Problem Setup

, height in meters, in hours after midnight

Find when during the first 24 hours.

Isolation:

Tide Problem: Second Solution and Periods

; for cosine:

Solve for :

Period hr. Add : keep solutions in .

Check: Filtering Tide Solutions by Daylight

Solutions in : hours

Daylight hours: 6 AM to 6 PM ( to )

Which of the four solutions fall in daylight?

Identify and state with contextual meaning.

Check Answer: Two Daylight Crossings

hr (10:43 AM) and hr (2:17 PM) are in daylight.

The tide reaches 6 m twice during daylight:

• 10:43 AM — tide rising toward high water
• 2:17 PM — tide falling from high water

The Six-Step Trig Equation Solving Process

1. Set up — substitute target
2. Isolate trig expression
3. Inverse trig — arcsin, arccos, or arctan
4. Second angle — (sine) or (cosine)
5. Add periods — list all solutions
6. Filter — keep those in the stated domain

Five Warnings to Avoid Mistakes

1. Calculator returns one angle — find the second via symmetry
2. Sine: second (not )
3. Cosine: second (not )
4. Report , not the inner angle
5. Negatives are not automatically invalid — check the interval

Extending This Skill to Advanced Applications

Extending this skill:

• Equations with coefficient on : adjust period formula
• Phase-shifted models: solve
• Combining with amplitude/midline to build and solve full models
• Precalculus: identities and multiple-angle equations