Kenneth ChengMathematical modeling
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Reduction formulas

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

高一必修 第一册(A版).pdf · 5.3 · PDF 195 / printed page 188

Revisit first: Concept of trigonometric functions

TOPIC 01

Reduction formulas

Build understanding of reduction formulas through definitions, contrasting cases and justified applications.

What you will be able to explain

  • Use reduction formulas 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.

Symmetry

Reflections determine which coordinate changes sign.

Periodicity

A full turn preserves sine and cosine; a half turn preserves tangent when defined.

PREDICT → EXPLORE → EXPLAIN → TRANSFER

Make a prediction, then explore the relationship.

Lesson question: Before calculating, predict how the conclusion changes when one defining condition in reduction formulas changes. Record a reason.

Model exploration: predict which displayed result changes with the parameters, check the values, and compare the observation with the lesson question.

θ+φ=45° = 0.7854 rad; sin=0.7071, cos=0.7071. The circle has radius 1.

θ+φ=45° = 0.7854 rad; sin=0.7071, cos=0.7071. The circle has radius 1.

Explain: Compare two admissible cases and one boundary or invalid case. Explain the observed difference using the stated definition.

Transfer: Construct a new example and a tempting incorrect solution. Repair the solution by naming the missing condition.

Try it. Leave your reasoning visible.

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

01 / Foundation#Worked example

Find the transformed sine value.

sin⁡α=3/5;sin⁡(2π+α)\sin\alpha=3/5;\quad\sin(2\pi+\alpha)
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Use unit-circle symmetry and periodicity; determine the sign before simplifying.
Hint 2
A full turn returns to the same unit-circle point.
Worked solution

  1. Use unit-circle symmetry and periodicity; determine the sign before simplifying.

  2. Calculate or simplify this relation.

    sin⁡(2π+α)=sin⁡α=3/5\sin(2\pi+\alpha)=\sin\alpha=3/5

  3. Sine has period 2π.

The requested value is 0.6.

Checks and common pitfalls: Sine has period 2π.

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

02 / Standard#Worked example

Find the reflected cosine value.

cos⁡α=4/5;cos⁡(π−α)\cos\alpha=4/5;\quad\cos(\pi-\alpha)
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Use unit-circle symmetry and periodicity; determine the sign before simplifying.
Hint 2
Reflection across the vertical axis changes x but not y.
Worked solution

  1. Use unit-circle symmetry and periodicity; determine the sign before simplifying.

  2. Calculate or simplify this relation.

    cos⁡(π−α)=−cos⁡α=−4/5\cos(\pi-\alpha)=-\cos\alpha=-4/5

  3. Cosine tracks the horizontal coordinate.

The requested value is -0.8.

Checks and common pitfalls: Cosine tracks the horizontal coordinate.

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

03 / Transfer#Worked example

Disprove the claim that cos(π−α)=cos α for every α.

cos⁡(π−α)=cos⁡α\cos(\pi-\alpha)=\cos\alpha
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Use unit-circle symmetry and periodicity; determine the sign before simplifying.
Hint 2
Choose an angle with nonzero cosine.
Worked solution

  1. Use unit-circle symmetry and periodicity; determine the sign before simplifying.

  2. Calculate or simplify this relation.

    cos⁡π=−1,cos⁡0=1\cos\pi=-1,\quad\cos0=1

  3. The correct identity has a minus sign.

Take α=0: −1≠1.

Checks and common pitfalls: The correct identity has a minus sign.

Reasoning checklist · self / teacher assessment
  • State the relevant definition, condition or model.
  • Show a valid calculation, proof or counterexample.
  • Interpret the conclusion with its restrictions.

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

04 / Foundation#Your turn

Evaluate cos 420° without a calculator.

cos⁡420∘\cos420^{\circ}
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Remove one full turn.
Hint 2
420° and 60° have the same terminal ray.
Worked solution

  1. Remove one full turn.

  2. Calculate or simplify this relation.

    cos⁡420∘=cos⁡60∘=1/2\cos420^{\circ}=\cos60^{\circ}=1/2

  3. Cosine is unchanged by a full turn.

The requested value is 0.5.

Checks and common pitfalls: Cosine is unchanged by a full turn.

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

05 / Foundation#Your turn

Evaluate sin 225° exactly.

sin⁡225∘\sin225^{\circ}
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Identify the reference angle and quadrant.
Hint 2
The angle is in quadrant III.
Worked solution

  1. Identify the reference angle and quadrant.

  2. Calculate or simplify this relation.

    sin⁡(180∘+45∘)=−sin⁡45∘=−2/2\sin(180^{\circ}+45^{\circ})=-\sin45^{\circ}=-\sqrt2/2

  3. Sine is negative in the lower half-plane.

−√2/2.

Checks and common pitfalls: Sine is negative in the lower half-plane.

Reasoning checklist · self / teacher assessment
  • State the relevant definition, condition or model.
  • Show a valid calculation, proof or counterexample.
  • Interpret the conclusion with its restrictions.

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

06 / Foundation#Your turn

Find tangent after a half turn.

tan⁡α=3;tan⁡(π+α)\tan\alpha=3;\quad\tan(\pi+\alpha)
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Use unit-circle symmetry and periodicity; determine the sign before simplifying.
Hint 2
Both sine and cosine change sign.
Worked solution

  1. Use unit-circle symmetry and periodicity; determine the sign before simplifying.

  2. Calculate or simplify this relation.

    tan⁡(π+α)=tan⁡α=3\tan(\pi+\alpha)=\tan\alpha=3

  3. Their quotient stays unchanged when defined.

The requested value is 3.

Checks and common pitfalls: Their quotient stays unchanged when defined.

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

07 / Foundation#Your turn

Evaluate sine at the opposite angle.

sin⁡α=5/13;sin⁡(−α)\sin\alpha=5/13;\quad\sin(-\alpha)
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Use unit-circle symmetry and periodicity; determine the sign before simplifying.
Hint 2
Sine is an odd function.
Worked solution

  1. Use unit-circle symmetry and periodicity; determine the sign before simplifying.

  2. Calculate or simplify this relation.

    sin⁡(−α)=−sin⁡α=−5/13\sin(-\alpha)=-\sin\alpha=-5/13

  3. Negating the angle reflects in the horizontal axis.

The requested value is -0.38461538.

Checks and common pitfalls: Negating the angle reflects in the horizontal axis.

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

08 / Standard#Your turn

Simplify the complementary-angle expression.

sin⁡(π/2−α)\sin(\pi/2-\alpha)
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Use unit-circle symmetry and periodicity; determine the sign before simplifying.
Hint 2
A complementary angle exchanges sine and cosine.
Worked solution

  1. Use unit-circle symmetry and periodicity; determine the sign before simplifying.

  2. Calculate or simplify this relation.

    sin⁡(π/2−α)=cos⁡α\sin(\pi/2-\alpha)=\cos\alpha

  3. The identity holds for every real α.

cos α.

Checks and common pitfalls: The identity holds for every real α.

Reasoning checklist · self / teacher assessment
  • State the relevant definition, condition or model.
  • Show a valid calculation, proof or counterexample.
  • Interpret the conclusion with its restrictions.

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

09 / Standard#Your turn

Evaluate without a calculator.

sin⁡(π)+sin⁡(0)\sin(\pi)+\sin(0)
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Use unit-circle symmetry and periodicity; determine the sign before simplifying.
Hint 2
Both terminal rays lie on the horizontal axis.
Worked solution

  1. Use unit-circle symmetry and periodicity; determine the sign before simplifying.

  2. Calculate or simplify this relation.

    sin⁡(π)+sin⁡(0)=0\sin(\pi)+\sin(0)=0

  3. The sine coordinate is zero at integer multiples of π.

The requested value is 0.

Checks and common pitfalls: The sine coordinate is zero at integer multiples of π.

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

10 / Standard#Your turn

Derive the cotangent reduction identity and state a domain restriction.

cot⁡(π−α)\cot(\pi-\alpha)
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Write cotangent as cosine divided by sine.
Hint 2
Apply the two reflection identities separately.
Worked solution

  1. Write cotangent as cosine divided by sine.

  2. Calculate or simplify this relation.

    cot⁡(π−α)=−cos⁡αsin⁡α=−cot⁡α\cot(\pi-\alpha)=\frac{-\cos\alpha}{\sin\alpha}=-\cot\alpha

  3. Both sides require a nonzero sine denominator.

−cot α, where sin α≠0.

Checks and common pitfalls: Both sides require a nonzero sine denominator.

Reasoning checklist · self / teacher assessment
  • State the relevant definition, condition or model.
  • Show a valid calculation, proof or counterexample.
  • Interpret the conclusion with its restrictions.

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

11 / Standard#Your turn

Evaluate tan(−π/4).

tan⁡(−π/4)\tan(-\pi/4)
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Use the oddness of tangent.
Hint 2
The cosine denominator is nonzero at this angle.
Worked solution

  1. Use the oddness of tangent.

  2. Calculate or simplify this relation.

    tan⁡(−π/4)=−tan⁡(π/4)=−1\tan(-\pi/4)=-\tan(\pi/4)=-1

  3. The result has the sign expected in quadrant IV.

The requested value is -1.

Checks and common pitfalls: The result has the sign expected in quadrant IV.

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

12 / Transfer#Your turn

Simplify sine after a three-quarter-turn shift.

sin⁡(3π/2+α)\sin(3\pi/2+\alpha)
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Use the addition formula or a unit-circle rotation.
Hint 2
sin(3π/2)=−1 and cos(3π/2)=0.
Worked solution

  1. Use the addition formula or a unit-circle rotation.

  2. Calculate or simplify this relation.

    sin⁡(3π/2+α)=(−1)cos⁡α+0sin⁡α\sin(3\pi/2+\alpha)=(-1)\cos\alpha+0\sin\alpha

  3. Quarter-turn shifts exchange the coordinate roles, with a sign determined by orientation.

−cos α.

Checks and common pitfalls: Quarter-turn shifts exchange the coordinate roles, with a sign determined by orientation.

Reasoning checklist · self / teacher assessment
  • State the relevant definition, condition or model.
  • Show a valid calculation, proof or counterexample.
  • Interpret the conclusion with its restrictions.

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

13 / Transfer#Your turn

Find tangent after a half turn.

tan⁡α=4;tan⁡(π+α)\tan\alpha=4;\quad\tan(\pi+\alpha)
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Use unit-circle symmetry and periodicity; determine the sign before simplifying.
Hint 2
Both sine and cosine change sign.
Worked solution

  1. Use unit-circle symmetry and periodicity; determine the sign before simplifying.

  2. Calculate or simplify this relation.

    tan⁡(π+α)=tan⁡α=4\tan(\pi+\alpha)=\tan\alpha=4

  3. Their quotient stays unchanged when defined.

The requested value is 4.

Checks and common pitfalls: Their quotient stays unchanged when defined.

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

Focus on one question

End-of-lesson check and correction

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    Teacher preparation and assessment

    Question sequence

    • What must be true before using the main rule for reduction formulas?
    • Which representation makes this task easier, and why?
    • Change one assumption. Does the conclusion survive?

    Board plan

    • Use reduction formulas with explicit angle units and domains.
    • Symmetry: Reflections determine which coordinate changes sign.
    • Periodicity: A full turn preserves sine and cosine; a half turn preserves tangent when defined.
    • Close with: conditions → representation → reasoning → check.

    Anticipated thinking

    • Expected reasoning: Reflections determine which coordinate changes sign.
    • Expected reasoning: A full turn preserves sine and cosine; a half turn preserves tangent when defined.
    • Expected correction: A reflected cosine often changes sign.

    Assessment checklist

    • 1: identify the givens and required quantity.
    • 1: choose a valid definition, representation or method.
    • 1: present connected, correct reasoning.
    • 1: check conditions and explain the result.

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    Curriculum and source notes ↗