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TMUA 2023 · Paper 1 · Question 8 of 20

TMUA 2023 Paper 1 Question 8

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

TMUA 2023 · Paper 1TrigonometrySine rule · the ambiguous case5 options
A triangle XYZ is called fun if it has the following properties: angle  YXZ = 30∘ XY = √3 a YZ = a where a is a constant.

For a given value of a, there are two distinct fun triangles S and T, where the area of S is greater than the area of T.

Find the ratio area of  S : area of  T.

  1. A1 : 1
  2. B2 : 1
  3. C2 : √3
  4. D√3 : 1
  5. E3 : 1
Show the answer and worked solution
answer · B
  1. A1 : 1
  2. B2 : 1
  3. C2 : √3
  4. D√3 : 1
  5. E3 : 1
YZ is opposite the 30∘ angle, so the sine rule gives asin 30∘ = √3 asin Z and sin Z = √32. Hence Z = 60∘ or Z = 120∘ — the ambiguous case — leaving Y = 90∘ or Y = 30∘. Using XZsin Y = asin 30∘ gives XZ = 2a in the first case and XZ = a in the second. The areas are 12(√3 a)(2a)sin 30∘ = √32a2 and 12(√3 a)(a)sin 30∘ = √34a2, a ratio of 2 : 1.