Trigonometric equation — Target at amplitude

Question

Express as and solve for .

This practice question was inspired by Question 11(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.

Set up the solution: rewrite the equation into a familiar trig form, solve each branch, then filter by the interval., .
Finish the calculation, then check that the result meets the question’s conditions..
State the final answer clearly in the form the question requested.Only in the expanded interval that lands in range: .

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

Verification: Maximum of the left side is , attained once per period in .

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