For an acute angle, cosec theta gives the ratio hypotenuse/opposite, which can be used to infer the Pythagorean triple.\nIf cosec θ = 25/7 for an acute angle θ, what is the value of cot θ?

Introduction / Context:
This question tests how to convert one trigonometric ratio into another for an acute angle. Given cosec θ, we can find sin θ. Then, because the angle is acute, all ratios are positive and we can use a right-triangle interpretation to compute cot θ = cos θ / sin θ.

Given Data / Assumptions:

• cosec θ = 25/7.
• θ is acute (0 < θ < 90 degrees), so sin θ and cos θ are positive.
• We need cot θ.

Concept / Approach:
Use cosec θ = 1/sin θ, so sin θ = 7/25. Interpret sin θ = opposite/hypotenuse. That gives opposite = 7 and hypotenuse = 25 (scaled). Then use Pythagoras to find adjacent = sqrt(25^2 - 7^2) = 24. Finally, cot θ = adjacent/opposite = 24/7.

Step-by-Step Solution:
cosec θ = 25/7 => sin θ = 1/(25/7) = 7/25 Let opposite = 7 and hypotenuse = 25 for angle θ Find adjacent using Pythagoras: adjacent^2 = 25^2 - 7^2 = 625 - 49 = 576 So adjacent = sqrt(576) = 24 cot θ = adjacent/opposite = 24/7

Verification / Alternative check:
cos θ would be adjacent/hypotenuse = 24/25. Then cot θ = cos θ / sin θ = (24/25)/(7/25) = 24/7, matching the result.

Why Other Options Are Wrong:
7/24 is tan θ, not cot θ (cot is the reciprocal).
24/25 is cos θ, not cot θ.
7/25 is sin θ, not cot θ.
25/24 is sec θ, not cot θ.

Common Pitfalls:
Confusing cot with tan, or inverting the ratio incorrectly. Another mistake is forgetting to use Pythagoras to find the missing side and assuming adjacent equals hypotenuse minus opposite, which is invalid.

Final Answer:
cot θ = 24/7