2024 HSC Mathematics Extension 1
Question 10:
Trigonometric equation
For real numbers and , where and , we can find numbers , , , and such that can be written in the following forms:
,\quad ,\quad ,\quad ,
where and .
What is the value of ?
How to recognise this question
Question type: advanced trig equations
- The command and notation point to advanced trig equations.
- The requested response is a multiple choice.
How to handle it: rewrite the equation into a familiar trig form, solve each branch, then filter by the interval.
Watch out: Do not select a formula until its domain, interval, sign and units match the question.
Step-by-step answer
- Write with , so and (matching expansion ).
- Similarly gives and .
- Comparing and signs shows is not forced in every quadrant configuration; a systematic approach uses co-function shifts.
- Note the identities: and , relating the four phase forms.
- For any nonzero , each of the four phases can be chosen in so that is replaced by the consistent total: the four complementary phase choices sum to .
- Concrete check with : , , (since needs , ), , (up to equivalent coterminal choices in ), and .
- Answer: D. (NESA multiple-choice key: D.)
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
If as functions for all with the same , a possible relation is
Question 2
The common amplitude in all four forms of is
Question 3
For written in the four phase forms with phases in , the sum of the four phases is