In trigonometry, if tan θ = 9/40 for an acute angle θ, model the situation with a right triangle, use the Pythagoras theorem to find the hypotenuse, and then determine the exact value of sec θ.

Introduction / Context:
This question tests your ability to convert from one trigonometric ratio to another by using right triangle relationships. You are given tan θ for an acute angle θ and asked to find sec θ. Understanding how tangent relates to opposite and adjacent sides, and how secant relates to hypotenuse and adjacent sides, allows you to solve such questions quickly in aptitude and competitive exams.

Given Data / Assumptions:

• tan θ = 9/40.
• θ is an acute angle, so all standard trigonometric ratios are positive.
• tan θ = opposite side / adjacent side.
• sec θ = hypotenuse / adjacent side.
• Pythagoras theorem: hypotenuse^2 = opposite^2 + adjacent^2.

Concept / Approach:
Interpret tan θ = 9/40 in terms of side lengths of a right triangle. Take the opposite side to θ as 9 units and the adjacent side as 40 units. Then, compute the hypotenuse using the Pythagoras theorem. Once all three sides are known, sec θ is simply the ratio of hypotenuse to adjacent side. This approach is far more efficient than trying to determine the angle in degrees or radians.

Step-by-Step Solution:
Let opposite side to θ be 9 and adjacent side be 40, so tan θ = 9/40. Apply Pythagoras theorem: hypotenuse^2 = 9^2 + 40^2. Compute squares: hypotenuse^2 = 81 + 1600 = 1681. Take the square root: hypotenuse = 41 since 41^2 = 1681. Now sec θ = hypotenuse / adjacent = 41 / 40.
Verification / Alternative check:
From the triangle, cos θ = adjacent / hypotenuse = 40/41. Since sec θ is the reciprocal of cos θ, sec θ = 1 / cos θ = 41/40. This confirms the direct computation. You may also note that 9, 40, and 41 form a well known Pythagorean triple, which further supports that the side lengths are consistent.

Why Other Options Are Wrong:
Option b (40/41) is cos θ rather than sec θ. Option c (41/9) incorrectly uses the hypotenuse over the opposite side and corresponds more closely to cosec θ if the roles of opposite and adjacent were mixed. Option d (9/41) is the sine of θ. Option e (40/9) is the cotangent of θ. None of these equals sec θ, which must be the reciprocal of cos θ in this right triangle model.

Common Pitfalls:
Learners sometimes confuse which side is opposite and which is adjacent when using tan θ, leading to wrong assignments of 9 and 40. Another common issue is forgetting that sec θ is 1/cos θ, not 1/tan θ. Carefully drawing a right triangle, labelling opposite and adjacent sides, and then applying Pythagoras theorem will usually prevent these mistakes.

Final Answer:
The exact value of sec θ when tan θ = 9/40 and θ is acute is 41/40.