Intro
Trigonometry marks in the HSC come from knowing a small set of identities cold and choosing the right one to simplify or solve equations. Extension 1 focuses on identities, equations, and graphs; Extension 2 adds inverse trig and deeper manipulation. NSW Year 12 HSC Mathematics. Keywords: HSC trig identities, trig equations, math revision notes.
Summary
Memorise Pythagorean, compound-angle, and double-angle identities. To solve equations, isolate a single trig function, find the reference angle, then use ASTC for all solutions in the domain. Extension 2 requires careful domain restriction for inverse functions.
Trigonometry underpins waves, integration, and complex arguments. A weak identity sheet slows every later topic — invest time here early in Year 12 regardless of which Extension courses you take.
Key Points
- Pythagorean: sin²θ + cos²θ = 1 — the starting point for most simplifications.
- Compound angle: sin(A ± B), cos(A ± B), tan(A ± B) — essential for proof and solve questions.
- Double angle: sin 2θ, cos 2θ, tan 2θ — often follows from compound angles.
- R sin(θ + α) form solves a cos θ + b sin θ = c style equations.
- Check for extraneous solutions after squaring both sides of a trig equation.
- Revise systematically in the HSC Trigonometry booklet.
Worked example
Question. Solve cos 2θ = cos θ for 0 ≤ θ ≤ 2π.
Solution.
- Use double angle: cos 2θ = 2 cos²θ − 1, so 2 cos²θ − 1 = cos θ.
- Rearrange: 2 cos²θ − cos θ − 1 = 0. Let u = cos θ: 2u² − u − 1 = 0.
- Factor: (2u + 1)(u − 1) = 0, so cos θ = −1/2 or cos θ = 1.
- cos θ = 1 → θ = 0, 2π. cos θ = −1/2 → θ = 2π/3, 4π/3.
Answer. θ = 0, 2π/3, 4π/3, 2π.
Takeaway. Substituting a compound or double-angle identity often reduces the equation to a quadratic in sin θ or cos θ.
Exam Preparation
Trig rewards daily micro-practice: five identity simplifications and two equation solves per session. Before the HSC, tabulate identities on one A4 sheet and drill ASTC quadrant rules until they are instant.
Build revision around exam question types: prove identities, solve equations in a given domain, and sketch related graphs. Each type uses different identity subsets — sort past-paper questions into these three buckets and drill the weakest bucket first.
- Identity flashcards. Test compound and double-angle forms both directions.
- Equation drills. Solve with explicit domain stated; box all solutions.
- Graph and inverse trig. Extension 2: sketch inverses with restricted domains labelled.
Extension 1 papers frequently combine identities with calculus — differentiating sin(nx) or integrating cos²θ using double-angle identities. Build a one-page identity sheet and test yourself weekly. For equations with multiple angles (e.g. sin 2θ = sin θ), factor rather than divide to avoid losing solutions. Inverse trig in Extension 2 requires stating principal values explicitly; write the restricted domain beside each inverse function before solving.
Mini-FAQ
How many solutions should I expect?
Unless restricted, expect multiple solutions in [0, 2π] — one per valid quadrant for each reference angle.
When do I use the t-formulae?
When substituting t = tan(θ/2) simplifies a messy equation — common in Extension 2.
Do I lose marks for missing a solution?
Yes — always scan all quadrants and verify each candidate in the original equation.
Common mistakes to avoid
- Using degrees when the question specifies radians (or vice versa).
- Dropping solutions when dividing by cos θ or sin θ without checking.
- Applying inverse sin to a value outside [−1, 1].
- Forgetting that tan θ is undefined at odd multiples of π/2.
Trigonometry rewards daily micro-drills: five identity simplifications each morning and two equation solves each evening build fluency faster than weekly marathons. Extension 2 students should add inverse-function sketches to every session — domain restrictions are a frequent source of lost marks in trial exams across NSW schools.
Past Extension 1 papers combine trig with calculus and sequences — keep identities accessible year-round, not only during the trig assessment block. When practising, state the domain at the start of every equation solve; markers deduct for extra solutions outside the given interval.
Link each identity you memorise to one past-paper question where it appeared — context helps recall under pressure.
Practice on Vu's Maths Hub
Need more practice on this topic? Open the free HSC Trigonometry booklet on Vu's Maths Hub — worked examples and exam-style questions, readable in your browser with no account required.
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