Trigonometric equation — Polynomial in t=sin x

Question

Let . Factor completely over the reals and state the roots.

This practice question was inspired by Question 11(g) in the 2021 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.Possible rational root : .
Finish the calculation, then check that the result meets the question’s conditions.Divide: (as in the exam).
State the final answer clearly in the form the question requested.Roots , .

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

Verification: Three real roots for .

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