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TMUA 2020 · Paper 2 · Question 2 of 20

TMUA 2020 Paper 2 Question 2

Trigonometry — Trigonometric ratios · choosing the right quadrant. Try it first; the answer and a full worked solution are below.

TMUA 2020 · Paper 2TrigonometryTrigonometric ratios · choosing the right quadrant8 options
Given that tanθ = 2 and 180∘ < θ < 360∘, find the value of cosθ.
  1. A√3
  2. B−√3
  3. C√32
  4. D−√32
  5. E√55
  6. F−√55
  7. G2√55
  8. H−2√55
Show the answer and worked solution
answer · F
  1. A√3
  2. B−√3
  3. C√32
  4. D−√32
  5. E√55
  6. F−√55
  7. G2√55
  8. H−2√55
A right-angled triangle with opposite 2 and adjacent 1 has hypotenuse √5, so cosθ = ±1√5. Now fix the sign: tanθ is positive, and in the range 180∘ < θ < 360∘ that only happens in the third quadrant, where cosine is negative. So cosθ = −1√5 = −√55. Taking 2√5 instead comes from using the opposite side rather than the adjacent one.