If tan θ = 7/24 for an acute angle θ in a right-angled triangle, find the exact value of cosec θ. (Answer should be an exact fraction.)

Introduction / Context:This problem tests converting between trigonometric ratios using a right-triangle interpretation. When tan θ is given as a fraction, you can treat it as opposite/adjacent, find the hypotenuse using Pythagoras, and then compute sin θ and cosec θ.

Given Data / Assumptions:

• tan θ = 7/24
• θ is acute (0° < θ < 90°), so all trig ratios are positive.
• Use a right triangle model.

Concept / Approach:Interpret tan θ = opposite/adjacent = 7/24. Take opposite = 7 and adjacent = 24. Then hypotenuse = sqrt(7^2 + 24^2). Once you have hypotenuse, compute sin θ = opposite/hypotenuse, and cosec θ = 1/sin θ.

Step-by-Step Solution:

Verification / Alternative check:Check consistency: tan θ = 7/24 matches the triangle. Also, since hypotenuse is the largest side and θ is acute, sin θ must be less than 1 (7/25), so cosec θ must be greater than 1 (25/7), which is reasonable.

Why Other Options Are Wrong:

Common Pitfalls:Swapping opposite and adjacent, or using hypotenuse incorrectly. Remember: cosec is the reciprocal of sin, not the reciprocal of tan.

Final Answer:cosec θ = 25/7