Trigonometric identity — Identity then tan equals $\sqrt{3}$

Question

(i) The numbers , and are related by and , where is a constant. Show that .

(ii) Hence, or otherwise, solve for .

This practice question was inspired by Question 13(d) 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: apply the named identity first and simplify before integrating or solving.(i) As before: (where defined).
Simplify the previous line carefully, keeping exact values where possible.(ii) , , so and . By (i), .
Simplify the previous line carefully, keeping exact values where possible.General solution: .
Simplify the previous line carefully, keeping exact values where possible.For : . For : , exclude.
Finish the calculation, then check that the result meets the question’s conditions.Check : at each listed value and . All four are valid.
State the final answer clearly in the form the question requested.Solutions: .

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

Verification: (i) Sum-to-product identity. (ii) Four solutions as listed.

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