2024 HSC Mathematics Extension 1
Question 14(c):
Inverse trigonometric functions
(i) Explain why the equation , where , has exactly one solution.
(ii) Solve .
How to recognise this question
Question type: inverse trigonometric functions
- The command and notation point to inverse trigonometric functions.
- The word “hence” means the later part is intended to reuse the earlier result.
How to handle it: use the restricted domain and range that make the inverse function single-valued.
Watch out: Do not restart the second part with unrelated numbers or discard the result proved in the first part.
Step-by-step answer
- (i) Both and are strictly increasing continuous functions (horizontal compressions of ), each with range .
- Their sum is strictly increasing and, considering the limits as , has range .
- Hence for each the horizontal line meets the graph of the sum exactly once.
- NESA 14(c)(i): 1 mark for a correct explanation.
- (ii) Take tan of both sides: .
- So .
- Factor: or .
- Reject : the sum of arctangents is negative there, while . Accept (both arctangents positive).
- NESA 14(c)(ii): 2 marks correct; 1 mark for the correct quadratic in .
Common mistake: choosing a familiar formula before checking that its conditions, signs, interval and units match this question.
The paper records 3 total marks for this group. Where the official marking guide provides useful partial-credit criteria, those criteria are stated in the steps above.
Try your hand at it again by answering these questions
Question 1
(i) Explain why has exactly one real solution for each . Hence,
(ii) Solve .
Question 2
(i) Explain briefly why has at most one solution for . Hence,
(ii) Solve .
Question 3
(i) Explain why , where , has exactly one real solution.
(ii) Hence solve .