For an acute angle θ, it is given that sin θ = 12/13. Using exact trigonometric relationships (no decimals), find the value of cot θ.

Introduction / Context:
This question tests converting one trigonometric ratio into another using a right-triangle interpretation and the Pythagorean identity. When sin θ is given for an acute angle, we can treat sin θ as opposite/hypotenuse in a right triangle. Because the angle is acute, all basic trig ratios (sin, cos, tan, cot) are positive, which removes sign ambiguity and makes the computation straightforward.

Given Data / Assumptions:

• θ is acute (0° < θ < 90°) • sin θ = 12/13 • Required: cot θ

Concept / Approach:
Interpret sin θ = opposite/hypotenuse: Opposite side = 12, hypotenuse = 13 (in proportional units). Find the adjacent side using: adjacent = √(hypotenuse^2 - opposite^2). Then compute: cot θ = adjacent / opposite. Because θ is acute, cot θ will be positive.

Step-by-Step Solution:
1) Use sin definition: sin θ = opposite/hypotenuse = 12/13 2) Let opposite = 12 and hypotenuse = 13 3) Find adjacent using Pythagoras: adjacent^2 = 13^2 - 12^2 4) Compute squares: 13^2 = 169, 12^2 = 144 5) Subtract: adjacent^2 = 169 - 144 = 25 6) Take square root (positive for acute angle): adjacent = 5 7) Compute cot θ: cot θ = adjacent/opposite = 5/12

Verification / Alternative check:
Compute cos θ = adjacent/hypotenuse = 5/13. Then tan θ = sin θ / cos θ = (12/13)/(5/13) = 12/5. Therefore cot θ = 1/tan θ = 5/12, confirming the same result using identities instead of triangle sides.

Why Other Options Are Wrong:
• 12/5 is tan θ, not cot θ. • 13/12 and 13/5 are not ratios built from the 5-12-13 triangle in the correct positions for cot. • 5/13 is cos θ, not cot θ.

Common Pitfalls:
• Confusing cot θ with tan θ (they are reciprocals). • Forgetting the angle is acute and mistakenly using a negative adjacent value.

Final Answer:
5/12