Kenneth ChengMathematical modeling
← Senior Mathematics Studio

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Evaluate tan(−π/4).

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

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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.

Try it. Leave your reasoning visible.

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

01 / 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.

Focus on one question

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.

No sign-in. Work stays in this browser. Export before clearing browser data. Written reasoning is assessed with a checklist.

Curriculum and source notes ↗