In trigonometry for an acute angle θ, you are given cos θ = 35/37. Using a right triangle model and the Pythagoras theorem, first determine the remaining side length and then calculate the exact value of cot θ from basic trigonometric ratios.

Introduction / Context:
This aptitude question checks your understanding of basic trigonometric ratios and how to move from one ratio to another by using a right triangle. You are given the value of cos θ for an acute angle θ and asked to find cot θ. This is a very common pattern in exams, where one trigonometric ratio is given and another is required, without the need to find the actual angle in degrees or radians.

Given Data / Assumptions:

• cos θ = 35/37 for an acute angle θ.
• Since θ is acute, all standard trigonometric ratios are positive.
• In a right triangle, cos θ = adjacent side / hypotenuse.
• cot θ is defined as adjacent side / opposite side.
• Pythagoras theorem holds: hypotenuse^2 = adjacent^2 + opposite^2.

Concept / Approach:
We interpret cos θ = 35/37 in geometric terms. Think of a right triangle where the side adjacent to θ has length 35 units and the hypotenuse has length 37 units. Once these two sides are known, the third side can be obtained from Pythagoras theorem. With all three side lengths available, any trigonometric ratio, including cot θ, can be computed as a simple ratio of sides. This approach avoids any need for calculators or angle tables.

Step-by-Step Solution:
From cos θ = 35/37, let adjacent side = 35 and hypotenuse = 37. Use Pythagoras theorem: hypotenuse^2 = adjacent^2 + opposite^2. Substitute the values: 37^2 = 35^2 + opposite^2. Compute squares: 1369 = 1225 + opposite^2, so opposite^2 = 1369 - 1225 = 144. Therefore opposite side = 12 (taking the positive root because θ is acute). Now cot θ = adjacent / opposite = 35 / 12 = 35/12.
Verification / Alternative check:
You can confirm the triple 12, 35, 37 as a valid Pythagorean triple because 12^2 + 35^2 = 144 + 1225 = 1369 = 37^2. Once this is verified, all ratios follow consistently. For example, sin θ would be 12/37 and tan θ would be 12/35, which agree with the usual relationships tan θ = 1 / cot θ and sin^2 θ + cos^2 θ = 1.

Why Other Options Are Wrong:
Option b (12/35) is actually tan θ, not cot θ. Option c (37/12) incorrectly uses the hypotenuse as numerator and mixes up the ratio definition. Option d (12/37) is sin θ. Option e (5/12) does not correspond to any main ratio in this triangle and likely comes from confusing 12^2 = 144 with 12 times 12 or 12 times some other number without correct reasoning.

Common Pitfalls:
A common mistake is to reverse the ratio and treat cos θ as hypotenuse over adjacent, which leads to incorrect side assignments. Another error is miscomputing 37^2 or 35^2, which then damages the calculation of the opposite side. Always compute the squares carefully and remember that cos θ is adjacent over hypotenuse, while cot θ is adjacent over opposite.

Final Answer:
The exact value of cot θ for cos θ = 35/37 with θ acute is 35/12.