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← 2020 HSC Extension 1 paper Question 12(d)

Variation 2 · answer guide

Trigonometric identity — Frequencies 3 and 1

Question

Evaluate .

This practice question was inspired by Question 12(d) in the 2020 NSW HSC Mathematics Extension 1 examination, © NESA. It is an original TestMum variation that practises the same mathematical idea, not an official NESA question.

Step by step

  1. Set up the solution: apply the named identity first and simplify before integrating or solving.Product-to-sum: .
  2. Simplify the previous line carefully, keeping exact values where possible.Integrate: .
  3. Simplify the previous line carefully, keeping exact values where possible.At : .
  4. Simplify the previous line carefully, keeping exact values where possible.At : .
  5. Finish the calculation, then check that the result meets the question’s conditions.Difference: .
  6. State the final answer clearly in the form the question requested.Final answer: .

Common mistake: changing so many features that the new example no longer tests the same mathematical idea.

Verification: Alternatively via gives .

Any mark labels are a TestMum study aid, not an official NESA marking allocation.

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