2025 HSC Mathematics Extension 1
Question 12(e):
Trigonometric equation
(i) Express in the form , where .
(ii) Hence, or otherwise, solve for .
How to recognise this question
Question type: advanced trig equations
- The command and notation point to advanced trig equations.
- The word “hence” means the later part is intended to reuse the earlier result.
How to handle it: rewrite the equation into a familiar trig form, solve each branch, then filter by the interval.
Watch out: Do not restart the second part with unrelated numbers or discard the result proved in the first part.
Step-by-step answer
- For (i), match .
- The coefficients require and , so .
- For (ii), use part (i): .
- Solving on gives and .
Common mistake: choosing a familiar formula before checking that its conditions, signs, interval and units match this question.
The paper records 3 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) Express as , where and .
(ii) Hence solve for .
Question 2
(i) Express as .
(ii) Hence solve for .
Question 3
(i) Express as .
(ii) Hence solve for .