If tanθ = 4/3 for an acute angle θ, then what is the value of the expression (3 sinθ + 2 cosθ) / (3 sinθ − 2 cosθ)?

Introduction / Context:
This trigonometry problem uses a given tangent value to construct a right triangle and then evaluate a ratio involving linear combinations of sine and cosine. It is a standard method to interpret tanθ in terms of opposite and adjacent sides and then compute sinθ and cosθ from that triangle.

Given Data / Assumptions:

• tanθ = 4/3 for an acute angle θ.
• We must evaluate (3 sinθ + 2 cosθ) / (3 sinθ − 2 cosθ).

Concept / Approach:
We interpret tanθ = 4/3 as a right triangle where the side opposite θ is 4k and the adjacent side is 3k for some positive k. Then the hypotenuse is 5k by the Pythagorean theorem. Using these, we find sinθ and cosθ and substitute them directly into the given expression, simplifying to a simple numeric value.

Step-by-Step Solution:
Step 1: From tanθ = 4/3, set opposite side = 4k and adjacent side = 3k. Step 2: The hypotenuse is √((4k)^2 + (3k)^2) = √(16k^2 + 9k^2) = √(25k^2) = 5k. Step 3: Then sinθ = opposite/hypotenuse = 4k / 5k = 4/5. Step 4: Similarly, cosθ = adjacent/hypotenuse = 3k / 5k = 3/5. Step 5: Compute the numerator: 3 sinθ + 2 cosθ = 3(4/5) + 2(3/5) = 12/5 + 6/5 = 18/5. Step 6: Compute the denominator: 3 sinθ − 2 cosθ = 3(4/5) − 2(3/5) = 12/5 − 6/5 = 6/5. Step 7: Form the ratio: (3 sinθ + 2 cosθ) / (3 sinθ − 2 cosθ) = (18/5) / (6/5) = 18/5 × 5/6 = 18/6 = 3.

Verification / Alternative check:
We can test with an angle whose tangent is exactly 4/3. The triangle construction guarantees such an angle exists. Any consistent scale factor k cancels out when forming sine and cosine, so the final ratio is independent of k. The arithmetic within the ratio is straightforward and double checking confirms that the result is 3.

Why Other Options Are Wrong:
Options A (1/2) and B (3/2) suggest miscalculations when forming the numerator or denominator. Option D (−3) would require the denominator to be negative, which is not the case here. Option E (2) arises if 12/5 and 6/5 are mishandled in simplification. Only 3 is consistent with correct substitution and simplification.

Common Pitfalls:
Common errors include mixing up opposite and adjacent sides, miscomputing the hypotenuse, or incorrectly simplifying the fraction (18/5)/(6/5). Another frequent mistake is to think that tanθ alone is enough to find sinθ and cosθ without constructing the triangle, which can lead to algebraic mistakes if done recklessly.

Final Answer:

The value of the expression is 3.