Trigonometric equation — A shifted sine equation

Question

Express as a single sine and solve for .

This practice question was inspired by Question 11(b) 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.

Set up the solution: rewrite the equation into a familiar trig form, solve each branch, then filter by the interval.
Simplify the previous line carefully, keeping exact values where possible.
Simplify the previous line carefully, keeping exact values where possible.Solve both sine branches and filter 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.Final answer: or

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

Verification: Direct substitution gives at both values.

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