2020 HSC Mathematics Extension 1
Question 14(b):
Trigonometric identity
(i) Show that .
(ii) By letting in the cubic equation , show that .
(iii) Prove that .
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) Expand .
- Use and to obtain .
- Rearrange: , then divide by to get the required form.
- NESA (i): 2 marks; 1 for expanding .
- (ii) Substitute : . Divide by : .
- From (i), , so .
- NESA (ii): 2 marks.
- (iii) From , candidate angles include giving three distinct roots of the cubic .
- For that cubic, sum of roots and sum of products two at a time .
- Hence .
- But , so the sine sum is .
- NESA (iii): 3 marks; partial for identifying roots / using root–coefficient relations.
Common mistake: choosing a familiar formula before checking that its conditions, signs, interval and units match this question.
The paper records 7 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) By letting in the cubic equation , show that .
(iii) Prove that .
Question 2
(i) Show that .
(ii) By letting in the cubic equation , show that .
(iii) Prove that .
Question 3
(i) Show that .
(ii) By letting in the cubic equation , show that .
(iii) Prove that .