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2021 HSC Mathematics Extension 1

Question 9:
Inverse trigonometric functions

1 markinverse trigonometric functions

Which graph represents the function ?

(Four sketched graphs A–D on with range involving are supplied in the paper.)

The four supplied graph options for y = sin inverse of sin x.
The four supplied graph options for y = sin inverse of sin x.
  1. Triangular wave oscillating between and , with on and periodic sawtooth/triangle folding thereafter
  2. A single straight line across the whole domain
  3. A standard sine wave between and
  4. A cosine-like wave between and without the triangular folds

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 range of is , so the graph never leaves those horizontal bounds.
  2. On , (identity on the principal interval).
  3. Outside that interval, is not one-to-one, so the composition folds back: the graph is the classic triangular (sawtooth) wave that reflects into .
  4. Option A is that triangular wave. A straight line (B) is only valid on the principal interval. C confuses range with that of . D lacks the piecewise linear folds.
  5. Answer: A. (NESA multiple-choice key: A.)

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

What is ?

See the step-by-step answer guide →

Question 2

What is ?

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Question 3

On which largest closed interval about is for all in the interval?

See the step-by-step answer guide →