75-Archetype SyllabusMathematics 2 • ARCH-MM-14 • Spec MM4

Full 6-Card Lesson 15-Question Timed Drill

UAT-UK SPEC CODE: MM4 (Trigonometry)ASSUMED MATHS 1 PREREQUISITE: M5.11FREQUENCY: HIGH

Sine Rule, Cosine Rule, Triangle Area (½ab sin C), The Ambiguous Case & 3D Pyramid Angles

Deconstructed from 22 past UAT-UK questions (5.37% of all historical Mathematics 2 papers). Students using standard A-Level classroom methods spend an average of 165s on this archetype—blowing past the 90s ESAT budget. Using the 2-line Sub-90 heuristic below compresses solve time to 45s (saving 120s per question).

Historical Frequency

22 Past Qs

5.37% of Mathematics 2

Classroom Trap Time

165s

+75s over 90s budget

Sub-90 Target Time

45s

Banks +120s on the clock

Dominant UI Format

5-Option A–E

Maths 1 Skill: M5.11

Card 1 & Card 2 Preview: The Classroom Time Trap vs. Sub-90 2-Line Rule

Excerpted from our interactive 6-Step Topic Lesson for ARCH-MM-14.

Launch All 6 Active-Recall Cards

CARD 1 OF 6 • THE CLASSROOM TIME TRAP (165s)

Why Students Bleed Time:

By the Cosine Rule,

AB^2 = AC^2 + BC^2 - 2(AC)(BC)\cos(120^\circ)

. Substituting

\cos(120^\circ) = -\frac{1}{2}

gives

AB^2 = 9 + 25 - 2(3)(5)\left(-\frac{1}{2}\right) = 9 + 25 + 15 = 49

, so

AB = 7\text{ cm}

(~120s).

In 22 past UAT-UK questions (5.37% of Mathematics 2), candidates lose 120s by when given ssa (two sides and a non-included acute angle), sin(b) = k yields two non-congruent triangles (b₁ acute and b₂ = 180° - b₁ obtuse) whenever b_opp < a_adj. Instead, apply MM4 with Maths 1 skill M5.11.

CARD 2 OF 6 • THE 2-LINE SUB-90 HEURISTIC (<45s)

**Line 1 (

120^\circ

Cosine Rule Identity):** Since

\cos 120^\circ = -\frac{1}{2}

c^2 = a^2 + ab + b^2

(the

+ab

Eisenstein identity).

Line 2 (Evaluate):

c = \sqrt{3^2 + (3)(5) + 5^2} = \sqrt{49} = 7\text{ cm}

(25s).

Sanity & Limit Check: Verify units/signs against fence_post_or_domain_trap (Finding only the acute angle theta from sin(theta) = k and missing the obtuse supplement 180° - theta (or pi - theta) in the Sine Rule Ambiguous Case) and sign_or_direction_flip (Using +2ab*cos(C) instead of -2ab*cos(C) in c² = a² + b² - 2ab*cos(C), or forgetting cos(C) < 0 for obtuse angles).

Examiner Distractor Blueprint for ARCH-MM-14

FOIL #1fence_post_or_domain_trap

Finding only the acute angle theta from sin(theta) = k and missing the obtuse supplement 180° - theta (or pi - theta) in the Sine Rule Ambiguous Case

When given SSA (two sides and a non-included acute angle), sin(B) = k yields two non-congruent triangles (B₁ acute and B₂ = 180° - B₁ obtuse) whenever b_opp < a_adj.

Open Fence-Post / Domain Trap Clinic

FOIL #2sign_or_direction_flip

Using +2ab*cos(C) instead of -2ab*cos(C) in c² = a² + b² - 2ab*cos(C), or forgetting cos(C) < 0 for obtuse angles

Sign error in the Cosine Rule when angle C > 90° (where -2ab*cos(C) becomes positive).

Open Sign / Direction Flip Clinic

FOIL #3fence_post_or_domain_trap

Projecting a 3D pyramid slant edge onto the side midpoint instead of the base centre (half the base diagonal)

Confusing slant height of a triangular face with the slant edge from apex to corner vertex.

Open Fence-Post / Domain Trap Clinic

FOIL #4factor_of_two_or_stoichiometry_trap

Omitting 1/2 in Area = (1/2)*a*b*sin(C)

Computing a*b*sin(C) without halving.

Open Factor-of-2 / Stoichiometry Clinic

Try 1 Live Sample Question Under the 90-Second Stopwatch

Press W to open the 2-line dry-wipe scratchpad, select A–E, and lock in to reveal the Dual-View solution.

Full 6-Card Lesson Launch All 15 Questions (ARCH-MM-14)

ARCH-MM-14: Sine Rule, Cosine Rule, Triangle Area (½ab sin C), The Ambiguous Case & 3D Pyramid Angles (Sample)Mathematics 2

00:00.0

SUB-90 ZONE (<45S)

0.0s45s Sub-9090s ESAT Budget150s Classroom

Question 1 of 1ARCH-MM-14 (6-Card Lesson)Format: Standard 5-Option

Right-click any option to cross out • Keys A–E + Enter

Question 1 of 1 • 90s Target

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The question stem, diagram, and answer options are hidden behind this pre-start gate. When you click below, the question will reveal and your 90-second split stopwatch will immediately begin ticking.

Mathematics 2•ARCH-MM-14•Standard 5-Option

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