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 ).
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A√3
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B1 √3
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C1
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D2
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ENone of these
Answer
Correct Answer: 2
Explanation
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:
tan θ = (r sin θ)/(r cos θ) = 1/√3S = √3 * tan θ + 1 = √3 * (1/√3) + 1 = 1 + 1 = 2Check: r^2 = (r sin θ)^2 + (r cos θ)^2 = 1 + 3 = 4 ⇒ r = 2 (consistent)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