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As θ varies over all real values, the expression 7sinθ + 24cosθ is greatest at the angles where tanθ equals:

Competition practiceTrigonometry
  • A24/7
  • B7/24
  • C7/25
  • D24/25

Step-by-Step Solution

The amplitude is = √ √625 = 25, and the greatest value 25 is reached exactly when sinθ and cosθ are proportional to their OWN coefficients — that is, sinθ = 7/25 and cosθ = 24/25, so tanθ = 7/24. Testing confirms it: , the full amplitude. Pairing each coefficient with the other ratio gives tanθ = 24/7, where the value is only — well short of the maximum. The remaining choices, 7/25 and 24/25, are the sine and the cosine at that maximum, not the tangent.
Answer: B) 7/24

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