For an acute angle in a right triangle, tan θ = opposite/adjacent and sec θ = hypotenuse/adjacent.\nIf tan θ = 7/24 for an acute angle θ, what is the value of sec θ?

Introduction / Context:
This problem tests how to move between trigonometric ratios using a right-triangle interpretation. Given tan θ, we can treat the ratio as opposite/adjacent, then find the hypotenuse using Pythagoras, and finally compute sec θ = hypotenuse/adjacent. Because θ is acute, all values are positive.

Given Data / Assumptions:

• tan θ = 7/24
• θ is acute
• tan θ = opposite/adjacent
• sec θ = 1/cos θ = hypotenuse/adjacent

Concept / Approach:
Model a right triangle: set opposite = 7 and adjacent = 24. Compute hypotenuse = sqrt(7^2 + 24^2) = 25 (a known 7-24-25 triple). Then sec θ = hypotenuse/adjacent = 25/24.

Step-by-Step Solution:
tan θ = 7/24 => opposite = 7 and adjacent = 24 Hypotenuse^2 = opposite^2 + adjacent^2 = 7^2 + 24^2 = 49 + 576 = 625 Hypotenuse = sqrt(625) = 25 sec θ = hypotenuse/adjacent = 25/24

Verification / Alternative check:
cos θ = adjacent/hypotenuse = 24/25. Then sec θ = 1/cos θ = 25/24, matching the computed result.

Why Other Options Are Wrong:
24/25 is cos θ, not sec θ.
7/25 is sin θ for this triangle, not sec θ.
25/7 equals cosec θ or hypotenuse/opposite, not hypotenuse/adjacent.
24/7 is cot θ, not sec θ.

Common Pitfalls:
Confusing sec with cosec, or mistakenly using opposite in the denominator for sec. Another common mistake is taking the reciprocal of tan instead of computing sec via hypotenuse/adjacent.

Final Answer:
sec θ = 25/24