2024 HSC Mathematics Extension 1
Question 6:
Trigonometric equation
How many real value(s) of satisfy the equation , where and is not zero?
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
- Since , divide both sides by : .
- So , i.e. .
- Equivalently when for integer , i.e. .
- For , one has . The odd multiples of in that interval are — eight values.
- Each gives a distinct in . Endpoints: at , ; at , . Neither endpoint solves the equation, but both are already excluded from the eight odd multiples listed.
- 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
How many solutions does have for ?
Question 2
Number of with is
Question 3
How many solutions of lie in ?