Topic Brief: Trigonometric Equations

Provenance: written 2026-09-29, against research/veridian/WMA12-verified-facts.md (read in full, specifically §2 spec item 6, §5's archive-scope statement, §6.7, §7 and §8), content/veridian/curriculum/wma12/trigonometric-equations.ts (read in full, to confirm this brief does not duplicate its VERIDIAN-original worked exemplars), and this paper's own technique-hub documents — signal-frequency-and-boundaries.md, codex-formula-and-method-bank.md §6–7, and forge-exam-technique-guide.md — read or grepped and the relevant sections read in full. Cross-checked against content/veridian/question-bank/wma12.ts to confirm which real question text is already reproduced there. No new primary-source PDFs were opened for this document; every claim below traces to material this course had already verified, re-read directly rather than taken on trust from a prior summary.

#SIGNAL topic master brief — Trigonometric equations (WMA12, spec 6.1–6.2)

#What this document is, and isn't

This is the real exam-history layer for one spec topic: every series this course's own research has individually confirmed tested it, what those questions actually asked, what mark tariff they carried, and what pattern (if any) recurs across them. signal-frequency-and-boundaries.md already carries this topic as a single row; this document is that row expanded to its full citations. It is not a method reference (codex-formula-and-method-bank.md §7 already gives the worked method skeletons and mark-scheme conventions) and not a trap taxonomy (forge-exam-technique-guide.md already lists ten named traps for this lesson). Nothing below is a prediction, a percentage, or a grade boundary.

#Real exam history: seen in 4 of 20 series checked

The archive behind this course's WMA12 research has 20 series with a question paper and mark scheme read (June 2019–January 2026), 17 of those with a published examiner report. Trigonometric equations has a confirmed, named citation — a specific series and question, traced to a real mark scheme and/or examiner report — in 4 of those 20:

• October 2021, Q10(i), 5 marks (M1, dM1, A1, A1, A1) — tan²(2x + π/4) = 3 over a symmetric interval. The squared term must be square-rooted (both signs) before the compound-angle inverse-and-range-scale method runs on each sign separately. Per the real examiner report, most marks lost were the final two, from candidates who "ignored the negative root" or found some but not all four roots.
• October 2022, Q5 — a sin²θ + cos²θ = 1 substitution collapsing to a 3-term quadratic in cosθ. Verified, named slips: cos2θ replaced with 1 − sinθ instead of 1 − sin²θ, and a coefficient dropped when expanding 3(1 − sin²θ) to 3 − sin²θ. The report also states plainly that "very few sketched helpful graphs or CAST diagrams" and that work was "sometimes seen in degrees."
• October 2023, Q3 — a tanθ = sinθ/cosθ substitution question. The correct route factorises to sinθ(2 + 3cosθ) = 0; the verified error cancels the sinθ factor instead, which silently discards the entire sinθ = 0 solution branch — the same structural mistake as dividing a quadratic through by x instead of factorising it.
• June 2025, Q8, two parts, 3 + 5 = 8 marks — part (i) a single compound-angle tan equation in radians (tan(2x + π/5), dM1/A1), where candidates lost marks for not isolating the tan term first and for answering in degrees or decimals instead of exact kπ form; part (ii) a three-term quadratic in cosθ reached via a tanθ = sinθ/cosθ substitution (M1/dM1/A1/ddM1/A1), where the report records candidates multiplying through by cosθ on only one of two terms, and others losing the final accuracy mark to premature rounding of an exact cosθ value before finding θ.

Exact mark allocations are confirmed above for two of the four (October 2021 and June 2025); the mark schemes for October 2022 Q5 and October 2023 Q3 were not individually re-transcribed in the facts bank beyond the error citations quoted, so no specific tally is stated for those two here rather than guessed. All four citations come from series with a published examiner report — every trig-equation finding in this course's research to date is grounded in real examiner commentary, not the mark scheme alone.

#Mark tariff and position in the paper

Across the four confirmed instances, this topic has carried 3 to 8 marks, usually as a single self-contained question rather than split across a multi-topic part. Its position in the paper varies rather than anchoring one fixed slot: Q3, Q5, Q8, and Q10(i) across the four series checked — unlike Proof, which the facts bank confirms anchors Q1 in most series checked. Where the topic sits at the higher end of its own range (June 2025's 8 marks across two parts), it has been split into two related but distinct question shapes rather than one long chain.

#Recurring context and angle type

Two structural shapes account for all four citations, and each recurs more than once: a compound-angle argument requiring range-scaling before inversion (October 2021, June 2025 part (i)) and a collapse-to-one-function substitution producing a 3-term quadratic (October 2022, October 2023, June 2025 part (ii)) — using either sin²θ + cos²θ = 1 or tanθ = sinθ/cosθ. A degrees-instead-of-radians error recurs independently across two of the four series (October 2022 and June 2025), despite affecting different question shapes, which suggests it is a genuine cross-series habit rather than one series' fluke. Three of the four confirmed citations are October series and only one is June; no January series has a confirmed named citation for this topic in this course's research to date. That last point is a gap in what has been individually verified, not a claim that January papers omit the topic — the facts bank's own §8 already flags that examiner-report coverage depth is uneven, with the trigonometry sub-topic specifically named as one of the thinner-covered areas pending a further sweep.

#If this topic appears: what to expect

Expect a self-contained question worth 3–8 marks, most likely built around one of two shapes: a compound or squared-compound angle needing a range-scaled search (check the interval scales with any coefficient on the angle before inverting), or a mixed sin/cos/tan equation that collapses to a 3-term quadratic once the right identity is substituted (never divide out a common trig factor — factorise instead, or a whole solution branch is lost). Whichever shape appears, three checks recur across every verified series above: confirm degrees vs. radians against the interval as given before touching a calculator, keep the final angle in its required exact form rather than a rounded decimal, and — if a square root was taken anywhere — check that both signs were carried all the way through to the final answer set.

#Sources

• research/veridian/WMA12-verified-facts.md, §2 (spec item 6, verbatim identities), §5 (archive scope: 20/17 series), §6.7 (the four named series and questions), §7 and §8 (topic-list placement and coverage-depth caveat).
• content/veridian/technique-hub/wma12/signal-frequency-and-boundaries.md, Part 1 (this topic's existing one-line frequency-table entry, expanded here).
• content/veridian/technique-hub/wma12/codex-formula-and-method-bank.md, §6–7 (method skeletons and mark-scheme conventions, not repeated here).
• content/veridian/technique-hub/wma12/forge-exam-technique-guide.md (trap-taxonomy list and the degrees/radians cross-series note).
• content/veridian/curriculum/wma12/trigonometric-equations.ts and content/veridian/question-bank/wma12.ts (checked, not duplicated).