Evaluate an expression from r sinθ and r cosθ Given r sin θ = 1 and r cos θ = √3, find the value of ( √3 * tan θ + 1 ).

Introduction / Context:This is a trig identity/evaluation problem. Using the given r-scaled sine and cosine helps find tan θ directly by division, and the Pythagorean identity can verify consistency of r and θ.

Given Data / Assumptions:

• r sin θ = 1
• r cos θ = √3
• We seek S = √3 * tan θ + 1

Concept / Approach:Compute tan θ = (r sin θ)/(r cos θ) = 1/√3. Then plug into S. Optionally, confirm r by squaring and summing to ensure a consistent setup.

Step-by-Step Solution:

Verification / Alternative check:If tan θ = 1/√3, then θ corresponds to 30° in the principal acute range; the expression evaluates to 2 as found.

Why Other Options Are Wrong:Values like √3 or 1 occur if tan θ is misread as √3 or 0.

Common Pitfalls:Dividing in the wrong order (cos/sin) or forgetting that √3 * (1/√3) simplifies to 1.

Final Answer:2