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Sine Rule, Cosine Rule and Area of Triangles

Pythagoras and Trigonometry

Finding a side

  • The printed form is a ÷ sin A = b ÷ sin B = c ÷ sin C
  • Only two of the three parts are needed, so pick the pair that contains your unknown and the pair you know completely
  • Substitute, then multiply across to make the unknown side the subject
Worked example

A side by the sine rule

In triangle ABC, angle A = 52°, angle B = 71° and side a = 9.4 cm. Find side b, correct to 3 significant figures.

Triangle ABC with B at the bottom left, C at the bottom right and A above. The angle at A is marked 52 degrees, the angle at B is marked 71 degrees, and the side BC facing A is labelled 9.4 cm. The side b, facing B, is unlabelled. The figure is marked NOT TO SCALE.

Solution:

  • Side a and angle A are an opposite pair, and side b faces angle B, so use the sine rule
  • b ÷ sin 71° = 9.4 ÷ sin 52°
  • Multiply both sides by sin 71°: b = 9.4 × sin 71° ÷ sin 52°
  • b = 11.27887…
  • So b = 11.3 cm to 3 significant figures
  • Check: b faces the larger angle, so it should be the longer side, and it is

Finding an angle

  • Inverting all three fractions moves the angles into the numerators, giving sin A ÷ a = sin B ÷ b = sin C ÷ c
  • That flipped version is not printed, but it follows from the printed one, so quote the printed form first and then invert it
  • Substitute, make the sine of the unknown angle the subject, then apply the inverse sine
  • Give angles correct to one decimal place, as the syllabus requires

The ambiguous case

  • Inverse sine on a calculator only ever returns the acute answer, but an obtuse angle can have the same sine
  • Whenever the sine rule is used to find an angle, a second possibility exists: 180° minus the acute answer
  • The syllabus names this the ambiguous case and includes it explicitly, along with obtuse angles generally
  • Decide from the diagram and the other angles which of the two is intended, since both must still leave a total of 180°
Why two triangles fit the same three facts: with the angle theta and the side a fixed, a side of length b can reach the base in two places, giving one triangle with an acute angle alpha and another with the obtuse angle 180 − alpha at that corner
Source: Sine rule by Save My Exams
  • This never happens with the cosine rule, which returns the obtuse angle directly
Worked example

An angle, with the ambiguous case

In triangle ABC, side a = 7.6 cm, side b = 5.2 cm and angle B = 34°. Find angle A.

Solution:

  • Side b and angle B are an opposite pair, and angle A faces side a, so the sine rule applies — inverted, to put the angles in the numerators
  • sin A ÷ 7.6 = sin 34° ÷ 5.2
  • Multiply both sides by 7.6: sin A = 7.6 × sin 34° ÷ 5.2
  • sin A = 0.8172…
  • The calculator gives A = sin⁻¹(0.8172…) = 54.8136…, so A = 54.8° to 1 decimal place
  • The obtuse alternative is 180 − 54.8136… = 125.2°, and both are consistent with the given information
  • Either can be correct here, so the diagram decides which one the question wants