Sine Cosine Rule And Area Of Triangles

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 $= 180^{\circ}$ – acute angle** from calculator.

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 $= b^{2} + c^{2}$ – 2bc cos A**, where A is the angle between b and c.
• For an angle: **cos $A = (b^{2} + c^{2}$ – $a^{2}) /$ (2bc)**.
• No ambiguous case – the cosine rule gives a unique angle.
• Label sides carefully: side a is opposite angle A.

Area of a Triangle

• For any triangle: **Area = ½ ab sin C**, where C is the angle between sides a and b.
• If $C = 90^{\circ}, \sin 90^{\circ} = 1$, 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.

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.

Worked Example – Sine Rule

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

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 $= (4.2^{2} + 3.8^{2}$ – $7.1^{2}) / (2 \times 4.2 \times 3.8)$ → θ = cos⁻¹(...) ≈ 125.0°.

Worked Example – Area of a Triangle

• Given triangle ABC with $AB = 32$ cm, $AC = 1.1 m$, angle $BAC = 74^{\circ}$.
• Convert to same units: $AB = 0.32 m, AC = 1.1 m$.
• Area = ½ $\times 1.1 \times 0.32 \times \sin 74^{\circ}$ ≈ $0.169 m^{2} (3 s.f.)$.

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: $\cos ABC = (4.4^{2}$ – $7.4^{2}$ – $4.8^{2}) /$ (–2 $\times 7.4 \times 4.8)$ → ABC ≈ 34.65°.
• Then area = ½ $\times 4.8 \times 7.4 \times \sin 34.65^{\circ}$ ≈ $10.1 cm^{2} (3 s.f.)$.

Sine Rule – Triangle Labelling

Cosine Rule – Triangle Labelling

Area of a Triangle – Formula

Flowchart – Choosing the Rule