2024 HSC Mathematics Extension 1
Question 13(b):
Trigonometric identity
(i) Show that .
(ii) Hence, or otherwise, evaluate .
How to recognise this question
Question type: advanced trig identities
- The command and notation point to advanced trig identities.
- The word “hence” means the later part is intended to reuse the earlier result.
How to handle it: apply the named identity first and simplify before integrating or solving.
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) because so .
- Then wait — use alternatively as in NESA: .
- NESA 13(b)(i): 2 marks; 1 mark for expanding or equivalent.
- (ii) . Write : integrand ? NESA path: .
- Evaluate: .
- NESA 13(b)(ii): 3 marks; 2 marks correct primitive; 1 mark for an integral involving .
Common mistake: choosing a familiar formula before checking that its conditions, signs, interval and units match this question.
The paper records 5 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) Show that .
(ii) Hence evaluate .
Question 2
(i) Show .
(ii) Hence evaluate .
Question 3
(i) Show that .
(ii) Hence evaluate .