Module 7 Synthesis & Exam Technique
This is the final lesson of Module 7 — everything comes together here. You'll build a complete reference map of the module, work through a multi-part exam question from scratch, and practise the time-management tactics that turn thorough preparation into actual HSC marks. By the end you'll know exactly which technique to reach for, and why.
Module 7 covers a substantial set of techniques. Without looking at notes — list as many distinct technique families as you can recall, and for each, write the key formula or the type of problem it solves.
Module 7 builds a single unified skill: transforming trigonometric expressions into forms that are easier to solve or analyse. Every technique is a different transformation strategy.
The six families:
- Auxiliary angle — combine $a\sin x + b\cos x$ into $R\sin(x \pm \alpha)$
- t-formulae — rational substitution $t = \tan\tfrac x2$
- Inverse trig — principal values, identities, evaluation
- Identities & proofs — double-angle, compound angle, sum-to-product
- Multiple-angle equations — reduce to a standard form via identities
- Combined / harder — chain two or more techniques in one question
Key formulae
- All six technique families and when each applies
- The identity chain: compound $\to$ double-angle $\to$ half-angle
- Inverse trig principal value ranges and key identities
Connections
- How auxiliary angle, t-formulae and identities all stem from the same underlying algebra
- Why "hence" parts must use the previous result, not an alternative method
- How to check solutions and identify missing/extraneous roots
Exam skills
- Identify the correct technique within 30 seconds of reading a question
- Manage time effectively — allocate marks-to-minutes correctly
- Complete a multi-part Module 7 question with full working
Use this map to decide which technique to apply. The diagnostic question is always: what structure does the equation have?
| Question type | Key signal | Technique | Core formula |
|---|---|---|---|
| Linear trig equation | $a\sin x + b\cos x = c$ | Auxiliary angle | $R = \sqrt{a^2+b^2}$ |
| Rational trig equation | fraction in $\sin x$, $\cos x$ | t-substitution | $t = \tan\tfrac x2$ |
| Inverse trig value/identity | $\sin^{-1}$, $\cos^{-1}$, $\tan^{-1}$ | Principal values | Range definitions |
| Double/compound expression | $\sin 2x$, $\cos(A+B)$ etc. | Identities | $\sin(A\pm B)$ etc. |
| Equation in $\sin nx$ or $\cos nx$ | multiple angle, $x \in [0, 2\pi]$ | Expand, substitute | Substitute $u = nx$, solve in $u$ |
| Multi-part "hence" question | part (i) gives a form, part (ii) uses it | Chain techniques | Must use prior result |
Keep a one-page "technique selector" in the front of your notes book: the 6-row table above; For every question: (1) identify structure, (2) select technique, (3) execute, (4) check domain
Pause — copy the 6-row technique selection table into your book, mapping each equation structure (factorisation / quadratic-sub / auxiliary angle / $t$-formula / identity / multiple angle) to its required method.
Quick check: A question says "use $t = \tan\tfrac{x}{2}$ to solve $\ldots$". This instruction tells you to use which technique family?
We just saw the 6-technique selection table mapping every trig equation structure (factorisation, quadratic-sub, auxiliary angle, $t$-formula, identity, multiple angle) to its method. That raises a question: in an HSC exam with linked multi-part questions, how do you manage the information flow between parts without re-deriving results? This card answers it → read all parts first; the result proved in part (i) is the scaffold for part (ii) — cite it, don’t reprove it.
HSC Module 7 questions often have 3–4 linked parts worth 8–12 marks total. The structure is predictable:
- Part (i) (1–2 marks): Express or show. Derive a useful form of the expression. This is the easy entry point — always attempt it.
- Part (ii) (2–3 marks): Hence solve/find. Must use part (i). A domain is given. Expect 2–3 solutions.
- Part (iii) (2–3 marks): Harder application — may involve: maximum/minimum, number of solutions, or a contextual problem.
- Part (iv) (1–2 marks, sometimes): Proof, extension, or "show that".
Strategy: Read all parts before starting. Part (iii) often reveals what "nice form" part (i) should produce. Working backwards from the end gives you strategic information.
Before writing: read all parts of the question; Part (i) is usually the scaffolding for part (ii) — do (i) carefully
Pause — copy the multi-part strategy into your book: read all parts before writing; the result from part (i) feeds directly into part (ii) — never re-derive in (ii) what you proved in (i).
Did you get this? True or false: in a "hence solve" question, you may use any valid method to solve the equation, not just the result from the previous part.
Worked examples · one complete multi-part Module 7 question
(i) Express $f(x) = \sqrt{3}\sin x + \cos x$ in the form $R\sin(x + \alpha)$ where $R > 0$ and $0 < \alpha < \frac{\pi}{2}$. (2 marks)
(ii) Hence, or otherwise, solve $f(x) = \sqrt{2}$ for $x \in [0, 2\pi]$. (3 marks)
(iii) Hence find the maximum value of $f(x)$ and state all $x \in [0, 2\pi]$ where it occurs. (2 marks)
(iv) For how many values of $x \in [0, 2\pi]$ does $f(x) = k$ have solutions when $k > R$? (1 mark)
Show that $\cos 2x = 1 - 2\sin^2 x$, and hence solve $\cos 2x + 3\sin x = 2$ for $x \in [0, 2\pi]$. (3 marks)
Given that $\theta = \cos^{-1}\!\left(-\dfrac{\sqrt{3}}{2}\right)$, find the exact value of $\sin\theta$. (3 marks)
Fill the gap: The maximum value of $a\sin x + b\cos x$ is $= \sqrt{a^2 + b^2}$, and it is achieved when $\sin(x+\alpha) = 1$.
Common misconceptions · the 3 traps that cost most marks
Did you get this? True or false: for $x \in [0, 2\pi]$, the equation $\sin x = \dfrac{1}{2}$ has exactly two solutions.
Activities · consolidation practice
Express $f(x) = \sin x - \cos x$ in the form $R\sin(x - \alpha)$. Find $R$ and $\alpha$ exactly.
Hence solve $f(x) \geq 1$ for $x \in [0, 2\pi]$, expressing your answer as an interval.
Use $\cos 2x = 2\cos^2 x - 1$ to solve $2\cos^2 x + \cos x - 1 = 0$ for $x \in [0, 2\pi]$.
Find the exact value of $\tan\!\left(\cos^{-1}\!\left(\dfrac{3}{5}\right)\right)$, showing all reasoning.
A Module 7 exam question is worth 10 marks and you have 3 hours (180 min) for the paper (100 marks total). How many minutes should you budget for this question?
Odd one out: Three of these statements about Module 7 are correct. Which one is NOT?
Earlier you listed as many Module 7 technique families as you could from memory. The full list is: auxiliary angle, t-formulae, inverse trig, identities/proofs, multiple-angle equations, combined/harder questions. How many did you get?
More importantly: can you now describe, in one sentence each, what triggers you to reach for each technique? Jot your descriptions below — these are the heuristics that translate revision into examination performance.
Pick your answer, then rate your confidence — that tells the system what to drill next.
Q1. Express $\sin x - \cos x$ in the form $R\sin(x - \alpha)$ where $R > 0$ and $0 < \alpha < \dfrac\pi2$. Give exact values for $R$ and $\alpha$. (2 marks)
Q2. Hence solve $\sin x - \cos x = 1$ for $x \in [0, 2\pi]$. (3 marks)
Q3. Find all solutions of $\cos 2x + 3\sin x - 2 = 0$ for $x \in [0, 2\pi]$. (3 marks)
Comprehensive answers (click to reveal)
Activity answers:
1. $R = \sqrt{1+1} = \sqrt{2}$; $\alpha = \pi/4$; $f(x) = \sqrt{2}\sin(x - \pi/4)$.
2. $\sqrt{2}\sin(x-\pi/4) \geq 1 \Rightarrow \sin(x-\pi/4) \geq \tfrac{1}{\sqrt{2}}$. Let $u = x - \pi/4 \in [-\pi/4, 7\pi/4]$. $\sin u \geq \tfrac{1}{\sqrt{2}}$ when $u \in [\pi/4, 3\pi/4]$. So $x \in [\pi/4+\pi/4, 3\pi/4+\pi/4] = [\pi/2, \pi]$.
3. $(2\cos x - 1)(\cos x + 1) = 0$; $\cos x = \tfrac12 \Rightarrow x = \pi/3, 5\pi/3$; $\cos x = -1 \Rightarrow x = \pi$. Solutions: $x = \pi/3, \pi, 5\pi/3$.
4. $\cos\theta = 3/5$, $\theta \in [0,\pi]$. By Pythagoras: $\sin\theta = 4/5$ (positive). $\tan\theta = \tfrac{4/5}{3/5} = \tfrac{4}{3}$.
5. Time per mark $= 180/100 = 1.8$ min. Budget $= 1.8 \times 10 = 18$ minutes.
Q1 (2 marks): $R = \sqrt{1+1} = \sqrt{2}$ [1]; $\tan\alpha = 1 \Rightarrow \alpha = \pi/4$ [1].
Q2 (3 marks): $\sqrt{2}\sin(x-\pi/4) = 1 \Rightarrow \sin(x-\pi/4) = \tfrac{1}{\sqrt{2}}$ [1]. $x-\pi/4 = \pi/4$ or $3\pi/4$ [1]. $x = \pi/2$ or $x = \pi$ [1].
Q3 (3 marks): $\cos 2x = 1-2\sin^2 x$: $1-2\sin^2 x + 3\sin x - 2 = 0 \Rightarrow 2\sin^2 x - 3\sin x + 1 = 0 \Rightarrow (2\sin x - 1)(\sin x - 1) = 0$ [1]. $\sin x = \tfrac12 \Rightarrow x = \pi/6, 5\pi/6$ [1]. $\sin x = 1 \Rightarrow x = \pi/2$ [1]. Three solutions.
Five timed questions spanning all Module 7 techniques. Beat the boss to bank a tier — gold (90% + speed), silver (75%), or bronze (50%). This is your module sign-off.
⚔ Enter the arenaClimb platforms by answering Module 7 synthesis questions. Lighter alternative to the boss.
Mark lesson as complete
Tick when you've finished the practice and review. Congratulations on completing Module 7!