Trigonometric identity — Identity then integral to pi/3

Question

(i) Show that .

(ii) Hence evaluate .

This practice question was inspired by Question 13(b) in the 2024 NSW HSC Mathematics Extension 1 examination, © NESA. It is an original TestMum variation that practises the same mathematical idea, not an official NESA question.

Set up the solution: apply the named identity first and simplify before integrating or solving.(i) . Then .
Finish the calculation, then check that the result meets the question’s conditions.(ii) Hence .
State the final answer clearly in the form the question requested.Answer: (i) as shown; (ii) .

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

Verification: Dependent structure: identity first, then definite integral with new upper limit .

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