Difficulty: Medium
Correct Answer: 7/25
Explanation:
Introduction / Context: This question tests how to convert one trigonometric ratio into another using right-triangle relationships. Given sec θ, you can immediately find cos θ because sec θ is the reciprocal of cos θ. Once cos θ is known, sin θ can be found using the Pythagorean identity sin^2 θ + cos^2 θ = 1. The “acute angle” condition matters because it ensures sin θ is positive, so you choose the positive square root. This is a standard aptitude pattern: derive triangle sides from a given ratio and then compute the required ratio exactly.
Given Data / Assumptions:
Concept / Approach: First convert sec to cos: cos θ = 1/sec θ. Then use sin^2 θ = 1 - cos^2 θ. Finally take the positive square root because θ is acute. You can also interpret the ratio as a right triangle: sec θ = hypotenuse/adjacent, so hypotenuse = 25 and adjacent = 24, and the opposite side becomes 7 by Pythagoras, giving sin θ = opposite/hypotenuse = 7/25.
Step-by-Step Solution: 1) Convert sec θ to cos θ: sec θ = 25/24 ⇒ cos θ = 24/25 2) Square cos θ: cos^2 θ = (24/25)^2 = 576/625 3) Use sin^2 θ = 1 - cos^2 θ: sin^2 θ = 1 - 576/625 = (625 - 576)/625 = 49/625 4) Take square root (acute angle ⇒ positive): sin θ = √(49/625) = 7/25
Verification / Alternative check: Triangle check: sec θ = hypotenuse/adjacent = 25/24 means take hypotenuse = 25 and adjacent = 24. Then opposite = √(25^2 - 24^2) = √(625 - 576) = √49 = 7. So sin θ = opposite/hypotenuse = 7/25. Both methods match, confirming accuracy.
Why Other Options Are Wrong: • 24/25 is cos θ, not sin θ. • 24/7 and 25/7 are incorrect reciprocals based on mixing up opposite and adjacent. • 1/25 has no basis in the correct triangle side relationships.
Common Pitfalls: • Forgetting sec is the reciprocal of cos. • Using sin^2 = 1 + cos^2 (wrong sign). • Taking the negative root even though θ is acute.
Final Answer: 7/25
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