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TMUA 2018 · Paper 1 · Question 19 of 20

TMUA 2018 Paper 1 Question 19

Trigonometry — Cosine rule · the ambiguous case as a quadratic. Try it first; the answer and a full worked solution are below.

TMUA 2018 · Paper 1TrigonometryCosine rule · the ambiguous case as a quadratic6 options
A triangle ABC is to be drawn with AB = 10 cm, BC = 7 cm and the angle at A equal to θ, where θ is a certain specified angle.

Of the two possible triangles that could be drawn, the larger triangle has three times the area of the smaller one.

What is the value of cosθ?

  1. A57
  2. B151200
  3. C2√25
  4. D√175
  5. E√518
  6. F√348
Show the answer and worked solution
answer · D
  1. A57
  2. B151200
  3. C2√25
  4. D√175
  5. E√518
  6. F√348
The two possible triangles differ only in the third side AC = b, so treat the cosine rule as a quadratic in b: 72 = b2 + 102 − 20bcosθ, that is b2 − (20cosθ)b + 51 = 0. Its two roots are the two possible values of b, with product 51 and sum 20cosθ. Each area is 12 ⋅ 10 ⋅ bsinθ, proportional to b, so the areas being in ratio 3:1 means b1 = 3b2. Then 3b22 = 51 gives b2 = √17 and b1 = 3√17, so 20cosθ = 4√17 and cosθ = √175.