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← 2025 HSC Extension 1 paper Question 11(c)

Variation 3 · answer guide

Trigonometric identity — An auxiliary angle

Question

Express as .

This practice question was inspired by Question 11(c) in the 2025 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..
  2. Simplify the previous line carefully, keeping exact values where possible. and .
  3. Finish the calculation, then check that the result meets the question’s conditions., so the expression is .
  4. 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: Expanding the shifted sine reproduces both coefficients.

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

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