Trigonometric equation — Sum of four phases invariant

Question

For written in the four phase forms with phases in , the sum of the four phases is

This practice question was inspired by Question 10 in the 2024 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.The invariant is independent of the nonzero coefficients .
Finish the calculation, then check that the result meets the question’s conditions.As in the exam, the sum is .
Match the result to the choices and state the correct option.Therefore option A is correct.

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

Verification: Same structure as Question 10 with .

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