If cot θ = 4 for an acute angle θ, then what is the value of the expression (5 sin θ + 3 cos θ) / (5 sin θ − 3 cos θ)?

Introduction / Context:
This trigonometry problem combines a given value of cot θ with an expression involving both sin θ and cos θ. The goal is to convert everything into one trigonometric ratio, use the given information, and then simplify the resulting fraction. Handling such expressions is a common skill tested in aptitude exams.

Given Data / Assumptions:

• cot θ = 4 for an acute angle θ (0° < θ < 90°).
• Expression to evaluate: (5 sin θ + 3 cos θ) / (5 sin θ − 3 cos θ).
• Since θ is acute, sin θ > 0 and cos θ > 0.
• Standard trigonometric relationships and right triangle interpretation may be used.

Concept / Approach:
We know that cot θ = cos θ / sin θ. If cot θ = 4, we can imagine a right triangle where the adjacent side is 4k and the opposite side is k, for some positive k. Then sin θ and cos θ can be expressed in terms of k and the hypotenuse. An even quicker method is to use the identity sin θ = 1 / √(1 + cot² θ) and cos θ = cot θ · sin θ, which directly give numerical values for sin θ and cos θ. Substituting these into the expression allows us to simplify the ratio fully.

Step-by-Step Solution:
Given cot θ = 4 = cos θ / sin θ. Compute sin θ using sin θ = 1 / √(1 + cot² θ). cot² θ = 16, so sin θ = 1 / √(1 + 16) = 1 / √17. Then cos θ = cot θ · sin θ = 4 · (1 / √17) = 4 / √17. Compute the numerator: 5 sin θ + 3 cos θ = 5(1 / √17) + 3(4 / √17). This is (5 + 12) / √17 = 17 / √17. Compute the denominator: 5 sin θ − 3 cos θ = 5(1 / √17) − 3(4 / √17). This is (5 − 12) / √17 = −7 / √17. Therefore, (5 sin θ + 3 cos θ) / (5 sin θ − 3 cos θ) = (17 / √17) / (−7 / √17) = −17/7.

Verification / Alternative check:
We can approximate numerically: sin θ = 1 / √17 ≈ 0.2425 and cos θ = 4 / √17 ≈ 0.9701. The numerator is about 5(0.2425) + 3(0.9701) ≈ 1.2125 + 2.9103 ≈ 4.1228. The denominator is about 5(0.2425) − 3(0.9701) ≈ 1.2125 − 2.9103 ≈ −1.6978. The ratio 4.1228 / (−1.6978) ≈ −2.428 which matches −17/7 ≈ −2.4286, confirming the exact value.

Why Other Options Are Wrong:
The options 1/9, 1/3, 3 and 9 are all positive, whereas the true value is negative because the denominator turns out to be negative while the numerator is positive. None of those values are close to −17/7 in magnitude either. Only −17/7 matches the result obtained from both exact and approximate calculations.

Common Pitfalls:
Common mistakes include mixing up cot θ with tan θ, miscomputing sin θ from cot θ, or ignoring the sign of the denominator. Some learners might incorrectly assume the answer must be positive because θ is acute, forgetting that the linear combination in the denominator can still be negative. Carefully computing sin θ and cos θ from cot θ and substituting them avoids these errors.

Final Answer:
Thus, the value of (5 sin θ + 3 cos θ) / (5 sin θ − 3 cos θ) is −17/7.