Inverse trigonometric functions — A principal inverse value

Question

Evaluate .

This practice question was inspired by Question 11(d) 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: use the restricted domain and range that make the inverse function single-valued.The range of is .
Simplify the previous line carefully, keeping exact values where possible..
Finish the calculation, then check that the result meets the question’s conditions.The angle in with this cosine is .
State the final answer clearly in the form the question requested.Final answer:

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

Verification: and is in the principal range.

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