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 noteslist 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 confidencethat 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!