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2023 HSC Mathematics Extension 1

Question 7:
Inverse trigonometric functions

1 markinverse trigonometric functions

Which statement is always true for real numbers and where ?

How to recognise this question

Question type: inverse trigonometric functions

  • The command and notation point to inverse trigonometric functions.
  • The requested response is a multiple choice.

How to handle it: use the restricted domain and range that make the inverse function single-valued.

Watch out: Do not select a formula until its domain, interval, sign and units match the question.

Step-by-step answer

  1. The function is strictly increasing on , so forces .
  2. Option A: is even about and not monotone on (e.g. , gives equal secants).
  3. Option C: is strictly decreasing, so when .
  4. Option D: for all , so both sides equal .
  5. Answer: B. (NESA multiple-choice key: B.)

Common mistake: choosing a familiar formula before checking that its conditions, signs, interval and units match this question.

The paper records 1 total mark 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

For , which is always true?

See the step-by-step answer guide →

Question 2

For all , equals

See the step-by-step answer guide →

Question 3

For real , which is always true?

See the step-by-step answer guide →