Given that tan(θ + 15°) = √3, find the value of θ (in degrees), considering principal angle values where 0° ≤ θ ≤ 90°. Choose the correct θ.

Introduction / Context: This question tests knowledge of special angles and the tangent function. When tan of an angle equals √3, the angle is a well-known standard value in degrees.

Given Data / Assumptions:

• tan(θ + 15°) = √3 • Principal values: 0° ≤ θ ≤ 90° • Required: θ in degrees

Concept / Approach: Recall special-angle tangent values: tan 60° = √3. So if tan(θ + 15°) = √3, then (θ + 15°) should equal 60° within the principal angle range. Then solve by subtracting 15°.

Step-by-Step Solution: 1) Identify the angle whose tangent is √3: tan 60° = √3 2) Therefore, set: θ + 15° = 60° 3) Subtract 15° from both sides: θ = 60° − 15° = 45°

Verification / Alternative check: Plug back: θ = 45° gives θ + 15° = 60°. Then tan(60°) = √3, matching the condition exactly. Also 45° lies in 0° to 90°, so it is valid.

Why Other Options Are Wrong: • 15°: would give tan(30°) = 1/√3, not √3. • 30°: would give tan(45°) = 1. • 65° or 75°: would make θ + 15° = 80° or 90°, neither has tangent √3 (tan 90° is undefined).

Common Pitfalls: • Confusing tan 30° and tan 60°. • Forgetting to subtract 15° after identifying the correct angle.

Final Answer: 45°