If cot θ = 24/7 for an acute angle θ in trigonometry, what is the exact value of sin θ?

Introduction / Context:
This problem checks your understanding of basic trigonometric ratios in a right angled triangle, especially how to move between cot θ and sin θ. Questions like this appear frequently in aptitude tests because they combine ratio interpretation, Pythagoras theorem and the definition of the trigonometric functions.

Given Data / Assumptions:

• cot θ = 24/7 for an acute angle θ.
• θ lies in the first quadrant, so all trigonometric ratios are positive.
• We are asked to determine sin θ.

Concept / Approach:
The key concepts are:

• cot θ is defined as adjacent side divided by opposite side in a right triangle.
• sin θ is defined as opposite side divided by hypotenuse.
• We can assume a right triangle with sides proportional to the given ratio, then apply Pythagoras theorem to find the hypotenuse and finally compute sin θ.

Step-by-Step Solution:
Step 1: From cot θ = 24/7, interpret this as adjacent side = 24k and opposite side = 7k for some positive scale factor k. Step 2: Use Pythagoras theorem to find the hypotenuse h: h^2 = (24k)^2 + (7k)^2 = 576k^2 + 49k^2 = 625k^2. Step 3: Therefore h = √(625k^2) = 25k. Step 4: By definition, sin θ = opposite / hypotenuse = 7k / 25k. Step 5: Simplify the ratio: sin θ = 7/25.
Verification / Alternative check:
Step 1: Once the sides are 7k, 24k and 25k, notice that this is a scaled 7-24-25 Pythagorean triple, which is a standard right triangle. Step 2: Compute cot θ again from these sides: cot θ = adjacent / opposite = 24k / 7k = 24/7, which matches the given value. Step 3: This confirms that our interpretation of the triangle and the resulting sin θ value of 7/25 is consistent.
Why Other Options Are Wrong:
Option 24/25 corresponds to adjacent / hypotenuse, which is cos θ, not sin θ. Option 8/25 does not relate to any side of the 7-24-25 triangle and is arbitrary. Option 9/25 also does not match any correct ratio derived from the triangle. Option 25/24 is greater than 1, whereas the sine of an acute angle must always lie between 0 and 1.
Common Pitfalls:
A frequent mistake is to confuse cot θ with tan θ and invert the given ratio incorrectly. Some learners mistakenly treat 24 and 7 as hypotenuse and one leg rather than adjacent and opposite, which breaks the Pythagorean relationship. Another error is forgetting that trigonometric ratios for acute angles cannot exceed 1 in magnitude for sine and cosine.
Final Answer:
The required value of sin θ is 7/25.