Difficulty: Medium
Correct Answer: sqrt3
Explanation:
Introduction / Context: This trigonometry question tests evaluating cotangent at an angle greater than 180° using reference angles and quadrant signs. The angle 210° lies in the third quadrant. In quadrant III, both sine and cosine are negative, which means tangent (sin/cos) is positive and therefore cotangent (cos/sin) is also positive. After determining the correct sign, the magnitude is found using the reference angle 30°, where tan 30° and cot 30° are standard exact values. The main challenge is sign control and choosing the right reference angle.
Given Data / Assumptions:
Concept / Approach: Write 210° as 180° + 30°. This shows the reference angle is 30° and the angle lies in quadrant III. In quadrant III, cot is positive. Also, tan(180° + α) = tan α, so tan 210° = tan 30°. Then cot 210° = 1 / tan 210° = 1 / tan 30° = cot 30° = √3.
Step-by-Step Solution: 1) Express 210° using a reference angle: 210° = 180° + 30° 2) Determine the quadrant: 210° is in quadrant III 3) Determine the sign of cot in quadrant III: cos is negative and sin is negative, so cos/sin is positive 4) Use tangent periodicity: tan(180° + 30°) = tan 30° 5) Use the standard value: tan 30° = 1/√3 6) Convert to cot: cot 210° = 1 / tan 210° = 1 / (1/√3) = √3
Verification / Alternative check: Using sine and cosine directly: sin 210° = -sin 30° = -1/2 and cos 210° = -cos 30° = -√3/2. Then cot 210° = cos 210° / sin 210° = (-√3/2)/(-1/2) = √3. Same result, confirming correctness.
Why Other Options Are Wrong: • -√3: wrong sign; cot is positive in quadrant III. • ±1/√3: those correspond to tan 30° magnitude, not cot 30°. • 0: cot is 0 only when cos is 0, which happens at 90° and 270°, not 210°.
Common Pitfalls: • Using the reference angle 60° instead of 30°. • Forgetting quadrant III makes cot positive. • Confusing tan and cot values for 30°.
Final Answer: √3
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