Topic Brief: Sine and Cosine Rules Ambiguous Case

Provenance: compiled 2026-09-29. Source is research/veridian/WMA11-verified-facts.md §4.3 and its own "Lesson-audit log" entry for sine-and-cosine-rules-ambiguous-case.ts — no fresh PDF extraction was done for this document; every claim below traces to that file's own primary-source re-extractions (Jan 2019 QP/MS/ER, cross-checked pdftotext -layout against -raw) or to its June 2022 targeted content-gap audit. Checked against content/veridian/curriculum/wma11/sine-and-cosine-rules-ambiguous-case.ts's own header comment and content/veridian/technique-hub/wma11/ for overlap — this document does not repeat that lesson's worked exemplars or codex-formula-and-method-bank.md's formula listing (§1.3.1, §2.3.1); it only assembles the cross-series frequency picture, which neither existing document states. Not affiliated with or endorsed by Pearson Edexcel.

#SIGNAL Topic Master Brief — Sine and Cosine Rules: The Ambiguous Case (spec 3.1)

#Frequency: seen in 2 of 9 series checked

The facts bank has touched nine WMA11 series in total, but at very uneven depth: five have a fully-read, question-by-question examiner report (Jan 2019, Oct 2020, Oct 2021, Jan 2023, Jan 2025); two (Oct 2022, Jan 2024) were confirmed as genuine mark schemes but never mined question-by-question; two June series (June 2022, June 2023) were read for specific target questions only, not the whole paper.

Within the five fully-mined series, the ambiguous case of the sine rule is confirmed in exactly one: Jan 2019 Q7(a). A second, genuine sighting comes from outside that group — June 2022 Q2, found by a later targeted content-gap audit and recorded only as a one-line note (a sine-rule ambiguous case "disambiguated by a stated side-length comparison"), not mined into a full mark-scheme breakdown. So the honest count is 2 confirmed appearances, at very unequal levels of documentation, out of 9 series touched — not 2 of 5, because the second hit didn't come from one of the five full reads, and not a claim that the other 7 series are clean, since most of them were never checked for this topic specifically.

#The one fully-documented appearance: Jan 2019 Q7(a)

Real question (WMA11_01_1901_QP/MS/ER, re-extracted and cross-checked -layout against -raw): a roof-truss context, AB = 6.5 m, BC = BD = 4.7 m, angle BAC = 35°, "find, to one decimal place, the size of angle ACB" — 3 marks, the opening part of a larger 6-mark question whose part (b) (total length of wood) sits outside this spec point.

Verbatim mark scheme, 3 marks, M1, A1, A1 (no dM1 or ddM1 anywhere in the real scheme):

• M1 — attempts sin∠ACB/6.5 = sin35/4.7 (sine rule, sides and angles correctly paired).
• A1 — ∠ACB awrt(52 or 53)° or awrt(127 or 128)° — either candidate value alone earns this mark; the scheme does not require both.
• A1 — ∠ACB = 127.5° only — cao, not follow-through-able.

A mark-scheme-bullet coverage audit re-extracted this paper fresh rather than trusting an earlier transcription, and found the earlier build of this course's own lesson had mislabelled the middle mark as a dependent method mark ("finding BOTH candidate angles") — the real scheme types it as an accuracy mark earned by either candidate on its own. That correction is now reflected in the lesson; it is recorded here because it changes what "getting this far" is actually worth: a script that reaches the correct acute angle and stops has already banked 2 of the 3 marks, not 1.

The mark scheme also credits a genuine alternative route for this same SSA data (Alt1(a)): forming a quadratic in AC via the cosine rule, then a second cosine-rule step for the angle itself. This sits in tension with a common teaching shorthand that the sine rule is "the only tool that fits" SSA — it isn't the only mark-scheme-credited one, just the intended default.

#The second, thinly-documented appearance: June 2022 Q2

Recorded only as: a sine-rule ambiguous-case question resolved by a stated side-length comparison in the question itself, rather than by judging validity from a diagram or context. No mark tariff, exact wording, or M/A/B breakdown survives in the current research pass — this was a spot-check for a different audit's purposes, not a full read, and inventing those details here would break this course's own no-fabrication rule. Treat this as confirmation that the pattern below (a second real disambiguation route) exists, not as a second fully worked exemplar.

#The recurring angle: two different real ways a paper resolves the ambiguity

Across the two confirmed sightings, the ambiguity gets resolved by two structurally different mechanisms, not the same one twice:

1. Diagram/context-embedded validity (Jan 2019 Q7): the roof-truss setup constrains which of the two candidate angles is geometrically possible; a candidate has to reason about the shape, not just the algebra, to discard the acute answer.
2. Stated side-length comparison (June 2022 Q2, per its one-line record): the question itself supplies enough information — which side is longer — for the standard triangle fact that the larger angle sits opposite the longer side to pick the valid candidate directly, without needing the diagram.

Both are real, examiner-confirmed routes to the same underlying skill (reject the sine-derived candidate that doesn't fit), but they reward different habits: checking a diagram carefully versus applying the side/angle-size relationship as a filter. Revision that only drills the diagram route is missing half of what has actually been asked.

#Mark-tariff pattern

Only one full tariff survives in the record: 3 marks (M1, A1, A1), and even that sits inside a larger 6-mark question whose second part tests something else. With a single fully-documented data point, don't generalise a fixed length for this topic from past-paper pattern alone — treat 3 marks, M1/A1/A1, no dependent marks, as the one confirmed shape, and expect the ambiguous-case step itself to be a short component of a longer triangle question rather than a standalone multi-part item.

#Confirmed trap set (Jan 2019 Q7, verbatim-sourced)

• Stops at the acute candidate. The examiner report, verbatim: "The majority used the sine rule and achieved the correct acute angle. However, many failed to find the required obtuse angle." This is the named spec trap (3.1: "including the ambiguous case of the sine rule") landing on real scripts, not a textbook warning.
• Wrong calculator mode. A second, independent finding from the same examiner-report paragraph: "Some candidates worked in radians; candidates should be advised to work in the same angle measure which was degrees in this question." A degrees/radians slip, distinct from the ambiguous-case trap itself.

#If this topic appears: what to expect

Expect a short, 3-mark sine-rule step (M1 for the correctly-paired ratio, A1 for either candidate angle, A1 for the specific correct one) rather than a long standalone question — most likely as the opening part of a larger triangle or composite-shape problem. Expect the trap to be reaching the acute candidate and stopping, worth 2 of 3 marks rather than 0; and expect degrees/radians calculator-mode to be a live, separately-marked risk in the same question. Be ready for the ambiguity to be resolved either by a diagram/context clue (Jan 2019) or by a stated side-length comparison (June 2022) — check both, since the record now shows real papers using each.