If tan A = 1/2 and tan B = 1/3 for two acute angles A and B, what is the exact value of the compound trigonometric expression tan(2A + B)?

Introduction / Context:
This question tests the use of trigonometric angle addition and double angle formulas. You are given tan A and tan B and must compute tan(2A + B). This requires first finding tan 2A using the double angle formula and then applying the general tangent addition formula for tan(2A + B).

Given Data / Assumptions:

• tan A = 1/2.
• tan B = 1/3.
• A and B are chosen so that tangent is defined and likely acute for simplicity.
• We must find tan(2A + B).

Concept / Approach:
Use the double angle formula for tangent: tan 2A = 2 tan A / (1 − tan^2 A). Then use the angle addition formula for tangent: tan(2A + B) = (tan 2A + tan B) / (1 − tan 2A tan B). By substituting the given values, the expression simplifies to a rational number. It is important to handle fractions carefully to avoid arithmetic errors.

Step-by-Step Solution:
First compute tan 2A using tan A = 1/2. The formula is tan 2A = 2 tan A / (1 − tan^2 A).Substitute: tan 2A = 2 * (1/2) / (1 − (1/2)^2) = 1 / (1 − 1/4) = 1 / (3/4) = 4/3.We know tan B = 1/3. Now use the angle addition formula: tan(2A + B) = (tan 2A + tan B) / (1 − tan 2A tan B).Compute the numerator: tan 2A + tan B = 4/3 + 1/3 = 5/3.Compute the denominator: 1 − tan 2A tan B = 1 − (4/3)*(1/3) = 1 − 4/9 = 5/9.Therefore tan(2A + B) = (5/3) / (5/9) = (5/3) * (9/5) = 3.

Verification / Alternative check:
As a consistency check, we can approximate angles A and B. If tan A = 1/2, then A is approximately 26.565 degrees. If tan B = 1/3, then B is approximately 18.435 degrees. Then 2A + B is roughly 2 * 26.565 + 18.435 ≈ 71.565 degrees. The tangent of about 71.565 degrees is approximately 3, which matches the exact result obtained by algebraic formulas.

Why Other Options Are Wrong:

• 1 and 5 are too small or too large compared with the approximate tangent of about 71.565 degrees.
• 9 is much larger and would correspond to an angle closer to 84 degrees, which is not consistent with 2A + B.
• 7 is another large value that does not match the calculated result from the correct formulas.

Common Pitfalls:

• Misapplying the double angle formula, especially the denominator 1 − tan^2 A which is easy to miswrite.
• Forgetting the correct form of the tangent addition formula or reversing the sign in the denominator.
• Making arithmetic mistakes when dealing with fractions like 4/3 and 1/3.

Final Answer:
3