Sine Rule, Cosine Rule & Triangle Area

Using the sine and cosine rules in non-right-angled triangles, and the formula Area = ½ab sin C.

For non-right-angled triangles you need the sine rule, cosine rule and the area formula involving sine. These are Higher-tier essentials.

Sine rule

a / sin A = b / sin B = c / sin C Use when you know: • two angles and any side, or • two sides and a non-included angle (ambiguous case possible)

Worked example

In triangle ABC, A = 40°, B = 60°, a = 8 cm. Find side b.

Solution

b / sin 60° = 8 / sin 40° b = 8 × sin 60° / sin 40° ≈ **10.7 cm** (3 s.f.)

Cosine rule

a² = b² + c² − 2bc cos A Or rearranged: cos A = (b² + c² − a²) / (2bc) Use when you know: • two sides and the included angle, or • all three sides (to find an angle)

Worked example

Sides b = 5, c = 7, included angle A = 60°. Find side a.

Solution

a² = 25 + 49 − 2×5×7×cos 60° a² = 74 − 70×0.5 = 74 − 35 = 39 a = √39 ≈ **6.24** (3 s.f.)

Area of a triangle

Area = ½ a b sin C where C is the included angle between sides a and b.

Practice questions

Try these without looking at the answers first.

Q1.Area of triangle with sides 6 cm, 8 cm and included angle 30°?

Hint: ½ × 6 × 8 × sin 30° = 24 × 0.5

Show answer

12 cm²

Q2.When do you use the cosine rule rather than sine rule?

Show answer

Two sides and included angle, or three sides

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