In trigonometry, let θ be an acute angle with 0 < θ < π/2 such that the relation 3 * tan θ = cot θ holds exactly. Using this condition, find the exact value of θ expressed in radians.

Introduction / Context:
This aptitude question tests understanding of basic trigonometric functions and how tangent and cotangent are related as reciprocals. By giving a simple equation involving tan θ and cot θ for an acute angle, the problem asks you to determine the exact radian measure of θ. Such questions are common in competitive exams because they check both algebraic manipulation and familiarity with standard special angles.

Given Data / Assumptions:

• θ is an acute angle, so 0 < θ < π/2.
• The trigonometric relation 3 * tan θ = cot θ is given.
• cot θ is the reciprocal of tan θ for all angles where these functions are defined.
• We need to find θ in exact radian form, not as a decimal approximation.

Concept / Approach:
The key concept is that cot θ = 1 / tan θ. Substituting this into the given equation transforms it into an algebraic equation in tan θ alone. Solving this equation gives the value of tan θ. Once tan θ is known, we match it with standard tangent values of special angles in the first quadrant, such as π/6, π/4, and π/3, to determine θ. Since θ is acute, we only consider positive tangent values.

Step-by-Step Solution:
1) Start from the given equation: 3 * tan θ = cot θ. 2) Replace cot θ with 1 / tan θ to obtain 3 * tan θ = 1 / tan θ. 3) Multiply both sides by tan θ (tan θ is non zero for an acute angle except at 0, which is excluded): 3 * tan^2 θ = 1. 4) Solve for tan^2 θ: tan^2 θ = 1 / 3. 5) Take the positive square root because θ is acute and tangent is positive in the first quadrant: tan θ = 1 / √3. 6) Recall that tan(π/6) = 1 / √3, tan(π/4) = 1, and tan(π/3) = √3. 7) Therefore θ must be π/6, since this is the only acute angle in the standard set with tan θ = 1 / √3.

Verification / Alternative check:
We can substitute θ = π/6 directly into the original equation. First compute tan(π/6) = 1 / √3. Then cot(π/6) = √3. The left side becomes 3 * tan θ = 3 * (1 / √3) = 3 / √3 = √3. The right side is cot θ = √3. Both sides are equal, so the equation is satisfied. No other standard acute angle produces tan θ = 1 / √3, which confirms that θ = π/6 is the unique solution within 0 < θ < π/2.

Why Other Options Are Wrong:
Option b (π/4) would require tan θ = 1, which does not satisfy 3 * tan θ = cot θ. Option c (π/3) gives tan θ = √3, leading to 3 * √3 on the left and 1 / √3 on the right, which do not match. Option d (π/2) is not valid because tan θ and cot θ are not both defined in a useful way at π/2. Option e (π/12) does not have the correct tangent value. Only π/6 makes the given identity true.

Common Pitfalls:
A frequent mistake is to forget that cot θ is 1 / tan θ and instead treat cot θ as an independent variable. Another common issue is taking both positive and negative square roots without considering the restriction that θ is acute, which forces tan θ to be positive. Some learners also guess from the options without checking the equation carefully. Systematically substituting cot θ = 1 / tan θ and solving the resulting quadratic in tan θ avoids these errors.

Final Answer:
The acute angle θ that satisfies 3 * tan θ = cot θ is π/6 radians.