Practice Maths

The Cosine Rule and Area Formula

Key Terms

Use the cosine rule when the sine rule doesn't apply: SAS (two sides + included angle) or SSS (all three sides)
Finding a side (SAS): c² = a² + b² − 2ab cos C
Finding an angle (SSS): cos C = (a² + b² − c²)/(2ab)
The cosine rule always gives a unique answer for angle — no ambiguous case
Heron's formula
: A = √(s(s−a)(s−b)(s−c)) where s = (a+b+c)/2
Cosine Rule — Finding a Side (SAS):
c² = a² + b² − 2ab cos C

Cosine Rule — Finding an Angle (SSS):
cos C = (a² + b² − c²) / (2ab)

Area Formulae:
Area = ½ab sin C (when two sides + included angle known)
Area = √(s(s−a)(s−b)(s−c))   where s = (a+b+c)/2   (Heron's formula, SSS)

Which rule to use?
• SAS → cosine rule (find side), then sine rule (find angles)
• SSS → cosine rule (find largest angle first to detect obtuse)
Worked Example 1 (SAS — Find a side): In triangle ABC, a = 5 cm, b = 7 cm, C = 60°. Find side c.

c² = a² + b² − 2ab cos C
c² = 25 + 49 − 2(5)(7) cos 60°
c² = 74 − 70 × 0.5
c² = 74 − 35 = 39
c = √39 ≈ 6.24 cm
Worked Example 2 (SSS — Find an angle): In triangle ABC, a = 3 cm, b = 5 cm, c = 7 cm. Find angle B.

cos B = (a² + c² − b²) / (2ac)
cos B = (9 + 49 − 25) / (2 × 3 × 7)
cos B = 33 / 42 ≈ 0.7857
B = cos−1(0.7857) ≈ 38.2°
Hot Tip: When finding angles from SSS, always find the largest angle first (opposite the longest side). If it is obtuse, the cosine rule will correctly give a negative cosine value, and cos−1 of a negative number gives an obtuse angle. The sine rule cannot detect this, since sin of an obtuse angle is positive — making the ambiguous case a risk if you switch to the sine rule too early.

Why We Need the Cosine Rule

The sine rule is powerful but has a limitation: it always requires a known side-angle pair. When given SAS (two sides and the included angle, where the angle is between the two known sides) or SSS (all three sides), no such pair is immediately available. The cosine rule fills this gap — it is a generalisation of Pythagoras' theorem to any triangle.

For a right-angled triangle, C = 90° and cos C = 0, so the cosine rule reduces to c² = a² + b² exactly. The extra term −2ab cos C adjusts for the angle being more or less than 90°.

The Cosine Rule: Derivation and Forms

In any triangle, drop a perpendicular from vertex C to side c. Label the foot of the perpendicular D. Then in the right triangle formed, a cos B gives the horizontal distance, and the Pythagorean theorem on both sub-triangles eventually yields:

c² = a² + b² − 2ab cos C

By symmetry, the same formula applies with any vertex labelled as C. The rearranged form for finding an angle from three known sides is:

cos C = (a² + b² − c²) / (2ab)

This always yields a unique answer. Unlike the sine rule, there is no ambiguous case because the range of cosine over [0°, 180°] is one-to-one: negative cosine values correspond directly to obtuse angles.

Choosing Between Sine and Cosine Rules

A clear decision framework helps avoid errors under exam pressure:

Use the sine rule when you have: AAS (two angles, any side), ASA (two angles, included side), or SSA (two sides, non-included angle — with care for the ambiguous case).

Use the cosine rule when you have: SAS (two sides and the included angle) to find the third side, or SSS (all three sides) to find any angle.

A common strategy for SAS triangles is: (1) use the cosine rule to find the missing side, then (2) switch to the sine rule to find one of the remaining angles, then (3) subtract from 180° to get the third angle. This is typically faster than applying the cosine rule twice.

Heron's Formula

When all three sides are known but no angle is given, the area can be found using Heron's formula, named after Hero of Alexandria (c. 60 AD):

s = (a + b + c)/2,    Area = √(s(s−a)(s−b)(s−c))

where s is the semi-perimeter. This formula is elegant because it requires no angles whatsoever. It is equivalent to computing an angle via the cosine rule and then using Area = ½ab sin C, but avoids rounding errors by keeping everything in one calculation.

Navigational and Composite Problems

Many practical problems involve finding a distance or angle in a triangle where two sides and the included angle are known — exactly SAS. Examples include finding the distance between two ships given their bearings and distances from port, or determining the diagonal brace needed in a framework. The cosine rule is the standard tool for these situations.

In composite problems, a triangle may be divided into sub-triangles. Solving each sub-triangle in sequence — sometimes alternating between sine and cosine rules — builds toward the final answer. Drawing a large, clear diagram and labelling all known information at the outset is essential.

Exam technique: After finding an angle using cos−1, note whether it is acute or obtuse (check the sign of the numerator in the cosine formula). A negative numerator means an obtuse angle — your calculator will confirm this automatically, but awareness prevents checking errors.

Mastery Practice

  1. In triangle ABC, a = 6 cm, b = 8 cm, and C = 45°. Find side c using the cosine rule.
  2. In triangle PQR, p = 4 m, q = 7 m, r = 9 m. Find angle R.
  3. Find the area of a triangle with sides a = 5 m, b = 8 m, c = 11 m using Heron's formula.
  4. In triangle ABC, b = 10 cm, c = 13 cm, A = 72°. Find side a.
  5. A triangle has sides 7 cm, 10 cm, and 14 cm. Determine whether this triangle is acute or obtuse, and find its largest angle.
  6. In a quadrilateral ABCD, the diagonal AC = 12 m. In triangle ABC: AB = 9 m, BC = 15 m. In triangle ACD: CD = 7 m, DA = 10 m. Find the area of the quadrilateral.
  7. In triangle ABC, a = 12 cm, C = 110°, c = 20 cm. Explain why you should use the cosine rule rather than the sine rule to find side a, then find angle A.
  8. Two ships leave a port P simultaneously. Ship A travels 20 km on a bearing of 050°. Ship B travels 15 km on a bearing of 150°. Find the distance between the two ships.
  9. A triangular paddock has sides 120 m, 180 m, and 210 m. Find the largest angle of the paddock (to the nearest degree) and the area of the paddock (to the nearest square metre).
  10. In triangle ABC, the area is 30 cm², a = 8 cm, and b = 10 cm. Find all possible values of angle C and the corresponding lengths of side c. Explain why both values of C can give the same area.
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