Trigonometric equation — Shift by π/4

Question

(i) Express as , where and .

(ii) Hence solve for .

This practice question was inspired by Question 12(e) 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. and coefficient matching gives .
Simplify the previous line carefully, keeping exact values where possible.Part (ii) becomes .
Simplify the previous line carefully, keeping exact values where possible..
Finish the calculation, then check that the result meets the question’s conditions.Filtering both branches to the interval gives .
State the final answer clearly in the form the question requested.Final answer: (i) ; (ii) or .

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

Verification: Expanding part (i) recovers the original coefficients; substituting both x-values gives 1.

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