LEARN · EXPLAIN · REVISE

Two daily temperature observations fit a sinusoid. Explain why more observations are needed before predicting its period.

Read the idea, work independently, then explain what changed.

高一必修第一册(A版).pdf · 5.7 · PDF 249 / printed page 242

TOPIC 01

Applications of trigonometric functions

Build understanding of applications of trigonometric functions through definitions, contrasting cases and justified applications.

Use applications of trigonometric functions with explicit angle units and domains.
Connect representations and justify the steps, including boundary cases.
Explain a solution and apply the idea to a changed situation.

Use one hint at a time. A correction explains what changed, not just the final answer.

01 / Transfer#Worked example

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Define the geometric or periodic quantities with units, then choose a trigonometric relation.

Hint 2

Count the unknown parameters in a general sinusoid.

Worked solution

Define the geometric or periodic quantities with units, then choose a trigonometric relation.
Record enough observations over several cycles and compare a fitted curve with residual errors and context.
A visual fit alone does not establish a unique periodic model.

Several amplitudes, phases and periods can fit two observations.

Checks and common pitfalls: A visual fit alone does not establish a unique periodic model.

Reasoning checklist · self / teacher assessment

Think first. Reveal a hint when the class is ready.

Teacher preparation and assessment

What must be true before using the main rule for applications of trigonometric functions?
Which representation makes this task easier, and why?
Change one assumption. Does the conclusion survive?

Use applications of trigonometric functions with explicit angle units and domains.
Geometry and units: Choose the appropriate side ratio and angular unit.
Periodic models: Interpret amplitude, midline, period and phase in the context.
Close with: conditions → representation → reasoning → check.

Expected reasoning: Choose the appropriate side ratio and angular unit.
Expected reasoning: Interpret amplitude, midline, period and phase in the context.
Expected correction: Fitting a curve does not prove a physical law.