In a right triangle with acute angle θ, suppose Cot θ = 21/20. Using the Pythagorean relationship between sides, what is the exact value of Cosec θ expressed as a simplified fraction?

Introduction / Context:
This trigonometry question links the Cotangent and Cosecant functions through the geometry of a right triangle. Such questions test whether you can interpret ratios as side lengths and then apply the Pythagorean theorem to find other trigonometric values.

Given Data / Assumptions:

• θ is an acute angle in a right triangle.
• Cot θ = 21 / 20.
• We must find Cosec θ.

Concept / Approach:
Cotangent is the ratio of the adjacent side to the opposite side: Cot θ = adjacent / opposite. We can assign adjacent = 21k and opposite = 20k for some positive scale factor k. Then the hypotenuse is found using the Pythagorean theorem. Cosec θ is defined as hypotenuse / opposite. After simplification the scale factor k cancels, leaving a simple fraction.

Step-by-Step Solution:
Cot θ = 21 / 20 means adjacent / opposite = 21 / 20. Let adjacent side = 21k and opposite side = 20k. Use the Pythagorean theorem to find the hypotenuse h: h^2 = (21k)^2 + (20k)^2. Compute: (21k)^2 = 441k^2, (20k)^2 = 400k^2, so h^2 = 441k^2 + 400k^2 = 841k^2. Thus h = √(841k^2) = 29k. Cosec θ is hypotenuse / opposite = h / opposite = 29k / 20k = 29 / 20.
Verification / Alternative check:
We can also find Sin θ first. Sin θ = opposite / hypotenuse = 20k / 29k = 20 / 29, hence Cosec θ is 1 / Sin θ = 29 / 20. This agrees with our geometric derivation.

Why Other Options Are Wrong:
Option a (21 / 29) corresponds to Cos θ, not Cosec θ. Option b (29 / 21) and option c (20 / 29) are other possible ratios but do not equal Cosec θ. Option e (41 / 29) is not related to the Pythagorean triple used here. Only 29 / 20 matches the definition of Cosec θ for the given triangle.

Common Pitfalls:
Many learners confuse the roles of adjacent and opposite, or mistakenly treat Cot θ as opposite / adjacent. Others forget to take the square root correctly when computing the hypotenuse. Drawing a quick sketch of the triangle with labelled sides often helps to avoid these errors.

Final Answer:
The required value of the Cosecant is Cosec θ = 29 / 20.