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

Full 15-Q Timed Drill

Step 1 of 6 • The Core Idea

ARCH-MM-14: Why the 165s Classroom Method Fails Under the 90s Clock

Standard Classroom Method (Too Slow)~165s (Blows 90s Clock)

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).

The Fast ShortcutTarget: 45s (Saves 120s)

**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).

Where Students Lose Time: 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.

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