Difficulty: Medium
Correct Answer: 9/25
Explanation:
Introduction / Context:
This question tests converting between trigonometric ratios using a right-triangle model and then simplifying an algebraic trig expression. When tan θ is given as a fraction, we can interpret it as opposite/adjacent in a right triangle. From that triangle, we find the hypotenuse and therefore sin θ. After that, the expression (1 - sin θ)/(1 + sin θ) becomes a pure fraction. The “acute angle” assumption matters because it guarantees sin θ is positive, removing sign ambiguity.
Given Data / Assumptions:
Concept / Approach:
Use tan θ = opposite/adjacent. Take opposite = 8 and adjacent = 15. Then hypotenuse = √(8^2 + 15^2) = √(64 + 225) = √289 = 17. So sin θ = opposite/hypotenuse = 8/17. Substitute into (1 - sin θ)/(1 + sin θ) and simplify using a common denominator.
Step-by-Step Solution:
1) Interpret tan θ = 8/15 as a right triangle ratio:
opposite = 8, adjacent = 15
2) Find the hypotenuse:
hypotenuse = √(8^2 + 15^2) = √(64 + 225) = √289 = 17
3) Compute sin θ:
sin θ = opposite/hypotenuse = 8/17
4) Substitute into the expression:
(1 - sin θ)/(1 + sin θ) = (1 - 8/17)/(1 + 8/17)
5) Convert 1 to 17/17 and simplify:
= ((17/17 - 8/17) / (17/17 + 8/17))
= (9/17) / (25/17)
6) Divide the fractions:
= (9/17) * (17/25) = 9/25
Verification / Alternative check:
Numerical sanity check: sin θ = 8/17 ≈ 0.4706. Then (1 - sin θ)/(1 + sin θ) ≈ 0.5294/1.4706 ≈ 0.36. And 9/25 = 0.36 exactly, so the simplified fraction matches the expected magnitude.
Why Other Options Are Wrong:
• 25/9 is the reciprocal of the correct value.
• 3/5 and 5/3 come from confusing sine with tangent or mixing triangle sides incorrectly.
• 1/5 is too small and does not match the computed ratio from sin θ = 8/17.
Common Pitfalls:
• Using sin θ = 8/15 (wrong; that is tan θ, not sin θ).
• Forgetting to compute the hypotenuse before finding sin θ.
• Losing the common denominator when simplifying (1 - 8/17)/(1 + 8/17).
Final Answer:
9/25
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