Pure Mathematics 3

Topic Brief: Compound and Double Angle Formulae

Provenance: compiled 2026-09-29, from research/veridian/WMA13-verified-facts.md (this course's primary, provenance-checked WMA13 source — 13 real series checked in total, 8 read start-to-finish with full mark schemes and examiner reports: Oct 2020, Oct 2021, Jan 2022, Jun 2022, Oct 2022, Jun 2023, Jan 2024, Jan 2025; 5 more checked for specific topics only: Jan 2023, Oct 2023, Jun 2024, Oct 2024, Jun 2026), cross-checked against this folder's signal-frequency-and-boundaries.md, forge-exam-technique-guide.md, and codex-deep-dive-compound-and-double-angle-formulae.md so this brief does not restate their content. No new primary-source material was pulled for this pass — this is a synthesis of what those three documents already establish, reorganised as a single fast "what to expect" reference for this one spec point.

#SIGNAL Topic Master Brief: Compound and Double Angle Formulae (WMA13, spec 2.3)

#What this document is

The lesson (compound-and-double-angle-formulae.ts) teaches the derivations and worked method. The CODEX deep-dive builds the full real mark ladder for every anchor. FORGE gives the exam-technique framing. This document does neither — it is the short version: which real series tested this topic, at what question and tariff, in what recurring shape, and what to expect if it appears again. Full workings, mark-by-mark, live in the CODEX deep-dive; this brief only cites and classifies.

#Frequency

Seen in 5 of the 13 WMA13 series checked so far: Oct 2020, Jan 2022, Jun 2022, Jun 2023, Jan 2024. That count is inherited from signal-frequency-and-boundaries.md's own topic-frequency table, not recomputed here. Of those five, only two have their actual question text and mark scheme independently transcribed anywhere in this course's sources — Oct 2020 (three separate real anchors within the one series) and Jan 2022 (one anchor). Jun 2022, Jun 2023 and Jan 2024 are attested as containing a 2.3 question by the facts bank's cross-topic error commentary, but their specific question text was not located and transcribed by this pass or any prior one. This gap is stated plainly rather than filled with an invented question — see "Honest gaps" below.

#The anchors, cited by series and question

Series	Q	Marks	Real content
Oct 2020	Q1	5	Solve $2\cos2x=7\cos x$ for $0°\leqslant x<360°$ — double-angle substitution collapses the equation to a 3-term quadratic in $\cos x$ .
Oct 2020	Q5(a)–(b)	4 + 4	"Show that $\sin3x\equiv3\sin x-4\sin^3x$ ", then hence-integrate $\sin^3x$ over $[0,\pi/3]$ .
Oct 2020	Q7	9 (3+2+4)	Express $\cos x+4\sin x$ as $R\cos(x-\alpha)$ , then apply it to a seabird-height model with a half-angle argument ( $\theta=\tfrac12t$ ) to find a minimum and solve for $t$ .
Jan 2022	Q2	5 total (2 in 2.3)	Reciprocal identity ( $8\cos\theta=3\csc\theta$ ) reduces to $\sin2\theta=\tfrac34$ ; part (b) asks for the smallest positive solution in degrees.
Jun 2022	Q7	not detailed	Attested only via the cross-topic "missing second solution in range" finding; question text not transcribed in any source reviewed.
Jun 2023	—	not detailed	Attested the same way; not transcribed.
Jan 2024	—	not detailed	Attested the same way; not transcribed.

#Mark-tariff pattern

Tariff depends on which of the shapes below is being tested, not on the topic alone:

A bare equation-solving question (double-angle substitution to a quadratic, solve in range): 5 marks, as a standalone question (Oct 2020 Q1) or as the back half of a mixed question (Jan 2022 Q2, 2 of its 5 marks).
A multi-angle "show that" identity, when it appears, carries its own 4 marks, and this course's one located instance is immediately followed by a further 4-mark "hence integrate" part on the same identity — 8 marks total across the pair (Oct 2020 Q5).
Harmonic form constructed and then applied to a modelling context is the heaviest confirmed shape: 9 marks in the one head-on instance located (Oct 2020 Q7), split 3 (construct $R,\alpha$ ) + 2 (apply to a minimum) + 4 (solve for the input variable).

No series in this set tests 2.3 for fewer than 4 marks when it anchors real content — this is a topic that, when it appears, tends to carry real weight rather than a token 1–2 marks.

#Recurring context and angle type

Across the anchors with real, transcribed content, the same handful of shapes recur:

Double-angle substitution collapsing a mixed equation to a 3-term quadratic in one ratio, solved over an explicit range (Oct 2020 Q1). Which $\cos2A$ form to substitute is dictated by what the rest of the equation is already written in — here everything else is in $\cos x$ , so $2\cos^2x-1$ is the form that yields a quadratic.
A different sub-topic's identity (2.1's reciprocal identities) opening a question that then crosses fully into 2.3 territory once rewritten as $\sin2\theta=k$ (Jan 2022 Q2). Its distinctive error — reporting the doubled angle instead of dividing by 2 to recover $\theta$ — is not an ordinary sign/quadrant slip.
Harmonic form applied inside a real-world modelling context, not tested as bare algebra — the one located instance embeds it in a height model with a half-angle argument, adding an extra "double back" step once the shifted angle is recovered (Oct 2020 Q7).
A higher multi-angle identity derived by rewriting it as a compound angle ( $3\theta=2\theta+\theta$ ) — real, examined content despite the spec's own guidance never naming a multiple beyond "double angle" (Oct 2020 Q5).

Common thread: none of the four test the bare identities in isolation — every real anchor embeds the machinery as a step inside a larger equation, proof, or modelling question.

#If this topic appears — what to expect

A 2.3 question is unlikely to just ask for a memorised identity. Expect it dressed as: an equation needing a double-angle substitution before it becomes a solvable quadratic (check what the rest of the equation is already written in, to pick the matching form); a harmonic-form construction ( $R,\alpha$ exact, in the unit named) feeding a modelling or optimisation question, possibly with a half-angle argument needing an extra recovery step; or a "show that" multi-angle identity used immediately afterward, most often to integrate. Range-restricted final answers are the norm — every detailed anchor solves over an explicit range or asks for a specifically qualified solution, so an unfiltered solve rarely earns the final mark.

#Honest gaps

Three of the five series this topic is attested in (Jun 2022, Jun 2023, Jan 2024) are named only through a cross-topic error pattern in the facts bank, not through their own transcribed question text — this brief does not manufacture what those questions asked. No likelihood, percentage, or grade-boundary claim is made anywhere above; every count is a "seen in N of M series checked" citation traceable to an existing document in this folder, not an estimate of future probability.

#Sources

research/veridian/WMA13-verified-facts.md — §5 (Trigonometry, 2.3) for every anchor's question text, mark scheme, and examiner-report quote; §2 for the series-review provenance.
signal-frequency-and-boundaries.md (this folder) — the "5 of 13" frequency count, reused rather than recomputed.
codex-deep-dive-compound-and-double-angle-formulae.md (this folder) — the full mark-by-mark ladder for each anchor; consult it for workings, not this brief.
forge-exam-technique-guide.md (this folder) — the exact-answer convention on harmonic-form construction marks, and the divide-by-2 recovery finding.
content/veridian/curriculum/wma13/compound-and-double-angle-formulae.ts, harmonic-form.ts — the taught derivations and worked chains, not reproduced here.