Trigonometric identity — Sine with half-scale roots

Question

(i) Show that .

(ii) By letting in the cubic equation , show that .

(iii) Prove that .

This practice question was inspired by Question 14(b) 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: apply the named identity first and simplify before integrating or solving.(i) Expand (via as in the original).
Simplify the previous line carefully, keeping exact values where possible.Rearrange: , then divide by .
Simplify the previous line carefully, keeping exact values where possible.(ii) Substitute : . Divide by : .
Simplify the previous line carefully, keeping exact values where possible.From (i), , so .
Simplify the previous line carefully, keeping exact values where possible.(iii) Angles give three distinct roots of (each with ).
Finish the calculation, then check that the result meets the question’s conditions.Vieta: sum , sum of products two at a time , so .
State the final answer clearly in the form the question requested.Thus .

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

Verification: Same triple-angle rearrangement as the original; scaled roots give power sum instead of .

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