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Sine Rule, Cosine Rule and Area of Triangles

Pythagoras and Trigonometry

Finding a side

  • The printed form is a² = b² + c² − 2bc cos A
  • Angle A must be the one between sides b and c, and side a must be the one facing it
  • Substitute, work out the right-hand side, then square root
  • Keep the subtraction intact: the whole 2bc cos A is subtracted, and where A is obtuse its cosine is negative, so the term is added in effect
Triangle ABC marked not to scale with all three sides given, AB 4.2 km, BC 3.8 km and AC 7.1 km, and the angle theta at B between the two shorter sides, which is the shape the cosine rule solves for an angle
Source: Cosine rule by Save My Exams
Worked example

A side by the cosine rule

In triangle ABC, b = 6.3 cm, c = 9.1 cm and their included angle is A = 47°. Find side a, correct to 3 significant figures.

Triangle ABC with A at the bottom left. The side AB along the bottom is labelled 9.1 cm and the side AC rising to the right is labelled 6.3 cm. The angle at A, between those two labelled sides, is marked 47 degrees. The side BC is unlabelled. The figure is marked NOT TO SCALE.

Solution:

  • Two sides with their included angle, and the third side wanted, so the cosine rule applies
  • a² = 6.3² + 9.1² − 2 × 6.3 × 9.1 × cos 47°
  • a² = 39.69 + 82.81 − 114.66 × cos 47°
  • a² = 122.5 − 78.1979… = 44.3020…
  • a = √44.3020… = 6.6559…
  • So a = 6.66 cm to 3 significant figures

Finding an angle

  • With all three sides known, rearrange to cos A = (b² + c² − a²) ÷ 2bc
  • That rearrangement is not printed, so either derive it or substitute into the printed form and solve
  • Apply the inverse cosine at the end
  • A negative value for cos A simply means the angle is obtuse, and the calculator returns it directly with no ambiguity
Worked example

An obtuse angle from three sides

A triangle has sides 5 cm, 7 cm and 10 cm. Find the angle opposite the 10 cm side, correct to 1 decimal place.

A triangle drawn with its longest side horizontal along the bottom and labelled 10 cm, the left-hand side labelled 5 cm and the right-hand side labelled 7 cm. The angle facing the 10 cm side is visibly obtuse and is not marked with a value.

Solution:

  • All three sides are known, so use the cosine rule for an angle, with a = 10 as the side facing the unknown angle
  • cos A = (5² + 7² − 10²) ÷ (2 × 5 × 7)
  • cos A = (25 + 49 − 100) ÷ 70 = −26 ÷ 70 = −0.3714…
  • The cosine is negative, so the angle is obtuse
  • A = cos⁻¹(−0.3714…) = 111.8037…
  • So A = 111.8° to 1 decimal place