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Sine Cosine Rule And Area Of Triangles

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Notes de leçon

The Sine Rule

  • Used in non-right-angled triangles to find missing side lengths or angles.
  • States: a / sin A = b / sin B = c / sin C, where a is opposite A, etc.
  • To find a missing length, use two equal parts of the rule and solve.
  • To find a missing angle, rearrange to sin A / a = sin B / b = sin C / c.
  • Ambiguous case: given two sides and a non-included angle, there may be two possible triangles (acute and obtuse).
  • If the required angle is obtuse, use: obtuse angle = 180° – acute angle from calculator.

Triangle ABC

Triangle ABCcabABC

The Cosine Rule

  • Used in non-right-angled triangles when you have two sides and the included angle (to find the third side) or all three sides (to find an angle).
  • For a side: a² = b² + c² – 2bc cos A, where A is the angle between b and c.
  • For an angle: cos A = (b² + c² – a²) / (2bc).
  • No ambiguous case – the cosine rule gives a unique angle.
  • Label sides carefully: side a is opposite angle A.

Triangle ABC

Triangle ABCcabABC

Area of a Triangle

  • For any triangle: Area = ½ ab sin C, where C is the angle between sides a and b.
  • If C = 90°, sin 90° = 1, so Area = ½ × base × height (right-angled triangle).
  • Ensure all lengths are in the same units before calculating area.
  • If the included angle is not given, use sine or cosine rule first to find it.

Triangle ABC

Triangle ABCcabABC

Deciding the Trig Rule

  • Sine rule: use when you have an opposite pair (side and angle) and need another side or angle.
  • Cosine rule: use when you have two sides and the included angle (to find the third side) or all three sides (to find an angle).
  • Area rule: use when you have two sides and the included angle (to find area).
  • If no rule fits directly, use angles in a triangle sum to 180° to find a missing angle.
  • Harder questions may require multiple trig rules in sequence.

Deciding the Trig Rule

Deciding the Trig Rule

Worked Example – Sine Rule

  • Given triangle ABC with AB = 8.1 cm, BC = 12.3 cm, angle BCA = 27°.
  • Find angle x (at A): use sin x / 12.3 = sin 27° / 8.1 → x = sin⁻¹(12.3 sin 27° / 8.1) ≈ 43.6°.
  • Find side y (AC): first find angle ABC = 180° – 27° – 43.6° = 109.4°, then y / sin 109.4° = 8.1 / sin 27° → y ≈ 16.8 cm.

Triangle ABC

Triangle ABC8.1 cm12.3 cmy cmxAB27°C

Worked Example – Cosine Rule

  • Given triangle ABC with AB = 4.2 km, BC = 3.8 km, AC = 7.1 km.
  • Find angle ABC: use cos θ = (4.2² + 3.8² – 7.1²) / (2 × 4.2 × 3.8) → θ = cos⁻¹(...) ≈ 125.0°.

Triangle ABC

Triangle ABC4.2 km3.8 km7.1 kmAθBC

Worked Example – Area of a Triangle

  • Given triangle ABC with AB = 32 cm, AC = 1.1 m, angle BAC = 74°.
  • Convert to same units: AB = 0.32 m, AC = 1.1 m.
  • Area = ½ × 1.1 × 0.32 × sin 74° ≈ 0.169 m² (3 s.f.).

Triangle ABC

Triangle ABC0.32 m1.1 m74°ABC

Worked Example – Multiple Rules

  • Find area of triangle with sides 4.4 cm, 7.4 cm, 4.8 cm.
  • Use cosine rule to find an angle: cos ABC = (4.4² – 7.4² – 4.8²) / (–2 × 7.4 × 4.8) → ABC ≈ 34.65°.
  • Then area = ½ × 4.8 × 7.4 × sin 34.65° ≈ 10.1 cm² (3 s.f.).

Triangle ABC

Triangle ABC7.4 cm4.8 cm4.4 cmA34.7°BC

Diapos

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  1. 1.What is the formula for the area of a triangle given two sides and the included angle?

    Easy
    • AArea = 1/2 ab sin C
    • BArea = 1/2 ab cos C
    • CArea = ab sin C
    • DArea = 1/2 ab tan C

  2. 2.The sine rule states that for any triangle ABC:

    Easy
    • Aa/sin A = b/sin B = c/sin C
    • Ba/sin B = b/sin C = c/sin A
    • Csin A/a = sin B/b = sin C/c
    • Da sin A = b sin B = c sin C

  3. 3.Which rule should be used to find a side when you know two sides and the included angle?

    Easy
    • ACosine rule
    • BSine rule
    • CArea rule
    • DPythagoras' theorem

  4. 4.In the cosine rule a² = b² + c² - 2bc cos A, what does angle A represent?

    Easy
    • AThe angle opposite side a
    • BThe angle between sides b and c
    • CThe angle opposite side b
    • DThe angle between sides a and c

  5. 5.Triangle ABC has AB = 8 cm, AC = 5 cm and angle BAC = 30°. Find the area of the triangle.

    Easy
    • A10 cm²
    • B20 cm²
    • C40 cm²
    • D34.6 cm²

  6. 6.In triangle XYZ, XY = 7 cm, XZ = 9 cm and angle YXZ = 60°. Use the cosine rule to find YZ.

    Medium
    • A√(49 + 81 - 2×7×9×cos 60°) = √(130 - 63) = √67 ≈ 8.19 cm
    • B√(49 + 81 - 2×7×9×sin 60°) = √(130 - 108.9) = √21.1 ≈ 4.59 cm
    • C√(49 + 81 - 2×7×9×cos 30°) = √(130 - 108.9) = √21.1 ≈ 4.59 cm
    • D√(49 + 81 - 2×7×9×tan 60°) = √(130 - 218.2) = negative

  7. 7.In triangle PQR, PQ = 10 cm, PR = 14 cm and angle PQR = 40°. Which rule can be used to find angle PRQ?

    Easy
    • ASine rule
    • BCosine rule
    • CArea rule
    • DPythagoras' theorem

  8. 8.A triangle has sides of lengths 5 cm, 6 cm and 7 cm. Find the angle opposite the side of length 7 cm.

    Medium
    • A78.5°
    • B44.4°
    • C57.1°
    • D101.5°

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