Sine rule and cosine rule — A/L Combined Mathematics

What must I remember about Sine rule and cosine rule?

• Standard triangle notation: In any triangle ABC, angles are A, B, C and opposite side lengths are a, b, c; semi-perimeter s = (a + b + c) / 2.
• The Sine Rule: a / sin A = b / sin B = c / sin C = 2R, where R is the circumradius of triangle ABC; applicable when two angles and a side, or two sides and an opposite angle are known.
• The Cosine Rule: a² = b² + c² - 2bc cos A; cos A = (b² + c² - a²) / (2bc); applicable when two sides and the included angle, or all three sides are known.
• Triangle Area formulas: Δ = (1/2)ab sin C = (1/2)bc sin A = (1/2)ca sin B = abc / (4R) = rs (where r is inradius); Heron's formula: Δ = √[s(s - a)(s - b)(s - c)].
• Projection Rule: a = b cos C + c cos B; b = c cos A + a cos C; c = a cos B + b cos A.

Test yourself (1 minute)

1. Ambiguous Case of Sine Rule (SSA)

   Given sides a, b and non-included angle A (with A acute): if a < b sin A, no triangle exists; if a = b sin A, exactly one right-angled triangle exists; if b sin A < a < b, TWO triangles exist; if a ≥ b, exactly one triangle exists.

2. Inradius and Circumradius formulas

   r = Δ/s = (s - a)tan(A/2); R = abc/(4Δ) = a/(2 sin A); connection: r = 4R sin(A/2) sin(B/2) sin(C/2).

3. Half-angle tangent formula (Napier's Analogies)

   tan((B - C)/2) = [(b - c)/(b + c)] cot(A/2).

Practise Sine rule and cosine rule questions today

Source: NIE G.C.E. (A/L) Combined Mathematics Resource Book - Pure Mathematics, Unit 11

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Where does Sine rule and cosine rule fit in the syllabus?

Year 1 · Term 2