In trigonometry, let θ be an acute angle such that Cosec θ = 17/8. Using a right triangle representation, find the exact value of Cot θ in simplest fractional form.

Introduction / Context:
This problem asks you to convert one trigonometric ratio, Cosec θ, into another, Cot θ, for an acute angle. The question is typical of those that test understanding of reciprocal trigonometric functions and right triangle geometry. Once you interpret the given ratio as the ratio of sides in a right triangle, you can use Pythagoras theorem to find the remaining side and then compute the required trigonometric function.

Given Data / Assumptions:

• θ is an acute angle, so all primary trigonometric ratios are positive.
• Cosec θ = 17/8.
• Cosec θ is defined as 1 / sin θ.
• We are asked to find Cot θ in simplest fractional form.

Concept / Approach:
Since Cosec θ = 1 / sin θ, knowing Cosec θ allows us to determine sin θ as its reciprocal. In a right triangle, sin θ = opposite / hypotenuse. Once sin θ is known, we can view this as a ratio of two sides and use Pythagoras theorem to find the adjacent side. Then cot θ, which equals adjacent / opposite or cos θ / sin θ, can be computed directly. The fact that θ is acute guarantees all side lengths and ratios are positive.

Step-by-Step Solution:
1) Start with Cosec θ = 17/8. 2) Use the reciprocal relationship sin θ = 1 / Cosec θ, so sin θ = 8 / 17. 3) Interpret sin θ = opposite / hypotenuse in a right triangle, so take the opposite side as 8 and the hypotenuse as 17. 4) Use Pythagoras theorem to find the adjacent side: adjacent^2 = hypotenuse^2 − opposite^2 = 17^2 − 8^2 = 289 − 64 = 225. 5) Therefore, the adjacent side has length √225 = 15. 6) Cot θ is defined as adjacent / opposite. 7) Using the triangle sides, Cot θ = 15 / 8.

Verification / Alternative check:
We can also compute cos θ and then use Cot θ = cos θ / sin θ. From the triangle, cos θ = adjacent / hypotenuse = 15 / 17 and sin θ = 8 / 17. Then Cot θ = cos θ / sin θ = (15 / 17) / (8 / 17) = 15 / 8, which matches the result obtained directly from adjacent / opposite. Additionally, check that sin^2 θ + cos^2 θ = (8/17)^2 + (15/17)^2 = 64/289 + 225/289 = 289/289 = 1, confirming that the side lengths are consistent with a right triangle.

Why Other Options Are Wrong:
Option b (17/15) is the reciprocal of cos θ, not Cot θ. Option c (8/15) is Tan θ, the reciprocal of Cot θ. Option d (17/8) repeats Cosec θ, the given value, rather than converting to Cot θ. Option e (15/17) represents cos θ. Only option a, 15/8, correctly gives the value of Cot θ for the specified Cosec θ and acute angle θ.

Common Pitfalls:
Learners sometimes invert the wrong ratio, confusing Cosec θ with Sec θ or mis identifying which side of the triangle is adjacent or opposite. Another common error is to mis apply Pythagoras theorem and compute 17^2 + 8^2 instead of 17^2 − 8^2 for the adjacent side. Careful labelling of the triangle and a clear understanding of sine, cosine, and their reciprocals help avoid these mistakes.

Final Answer:
For an acute angle θ with Cosec θ = 17/8, the exact value of Cot θ is 15/8.