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A student is told that sinθ·cosθ = 3/5 for some angle θ and concludes that sinθ + cosθ = √(11/5). What is wrong with this?

Competition practiceTrigonometry
  • AThe sum should be √(1/5)
  • BNo angle θ has sinθ·cosθ = 3/5
  • Csinθ + cosθ can never be irrational
  • DNothing — the conclusion is correct

Step-by-Step Solution

sinθ·cosθ = (1/2)sin2θ, so the product never exceeds 1/2 = 0.5, while 3/5 = 0.6. The starting fact is impossible, so nothing follows from it. The student' algebra is fine — — the error lies in accepting the given value at all.
Answer: B) No angle θ has sinθ·cosθ = 3/5

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