2021 HSC Mathematics Extension 1
Question 13(d):
Trigonometric identity
(i) The numbers , and are related by and , where is a constant.
Show that .
(ii) Hence, or otherwise, solve for .
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) LHS (provided and ).
- NESA (i): 2 marks; 1 for substituting , and attempting a sum-to-product identity.
- (ii) Here , , so midpoint .
- By (i), LHS .
- General solution: . For : . For : .
- For : (since ). For : negative.
- Also need (i.e. ). Both listed solutions are valid on .
- Solutions: .
- NESA (ii): 2 marks; 1 for identifying to use part (i).
Common mistake: choosing a familiar formula before checking that its conditions, signs, interval and units match this question.
The paper records 4 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) The numbers , and are related by and , where is a constant. Show that .
(ii) Hence, or otherwise, solve for .
Question 2
(i) The numbers , and are related by and , where is a constant. Show that .
(ii) Hence, or otherwise, solve for .
Question 3
(i) The numbers , and are related by and , where is a constant. Show that .
(ii) Hence, or otherwise, solve for .