Which of the following is not equivalent to cot ((pi)/(6)) ? cot ((13 pi)/(6)) cot ((11 pi)/(6)) cot ((7pi)/(6)) cot(-(11 pi)/(6))

Question

Which of the following is not equivalent to  cot ((pi)/(6)) ?  cot ((13 pi)/(6))  cot ((11 pi)/(6))  cot ((7pi)/(6))  cot(-(11 pi)/(6))

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Periodicity Understanding: Understand the cotangent function's periodicity. The cotangent function has a period of π, which means cot(θ) = cot(θ + kπ) for any integer k.Evaluate cot(pi/6): Evaluate cot((pi)/(6)). The cotangent of π/6 is the reciprocal of the tangent of π/6. Since tan(π/6) = √3/3, cot(π/6) = 3/√3 = √3.Evaluate cot(13pi/6): Evaluate cot((13pi)/(6)). Using the periodicity from Step 1, we can subtract π (6π/6) from 13π/6 to find an equivalent angle within the first period: cot((13pi)/(6)) = cot((13pi)/(6) - 2π) = cot((7pi)/(6)).Evaluate cot(11pi/6): Evaluate cot((11pi)/(6)). Using the periodicity from Step 1, we can subtract π (6π/6) from 11π/6 to find an equivalent angle within the first period: cot((11pi)/(6)) = cot((11pi)/(6) - π) = cot((5pi)/(6)).Evaluate cot(7pi/6): Evaluate cot((7pi)/(6)). Using the periodicity from Step 1, we can subtract π (6π/6) from 7π/6 to find an equivalent angle within the first period: cot((7pi)/(6)) = cot((7pi)/(6) - π) = cot((pi)/(6)).Evaluate cot(-(11pi/6): Evaluate cot(-(11pi)/(6)). Using the periodicity from Step 1, we can add π (6π/6) to -11π/6 to find an equivalent angle within the first period: cot(-(11pi)/(6)) = cot(-(11pi)/(6) + π) = cot(-(5pi)/(6)).Non-Equivalent Cotangent: Determine which cotangent value is not equivalent to cot((pi)/(6)). From the previous steps, we have found that cot((7pi)/(6)) is equivalent to cot((pi)/(6)). However, cot((5pi)/(6)) and cot(-(5pi)/(6)) are not equivalent to cot((pi)/(6)) because the tangent (and thus cotangent) of these angles would be negative, as they are in the second and third quadrants respectively, where the cotangent function is negative. Therefore, cot((11pi)/(6)) and cot(-(11pi)/(6)) are not equivalent to cot((pi)/(6)).Identify Incorrect Option: Identify the incorrect option. Since cot((11pi)/(6)) and cot(-(11pi)/(6)) are both not equivalent to cot((pi)/(6)), we need to choose one of them as the answer. However, the question asks for the one that is not equivalent, and since we have two options that are not equivalent, there is an issue with the question as it assumes there is only one correct answer. Therefore, we cannot determine a single final answer based on the information given.

Which of the following is not equivalent to $\cot \frac{\pi}{6}$ ? $\newline$ $\cot \frac{13 \pi}{6}$ $\newline$ $\cot \frac{11 \pi}{6}$ $\newline$ $\cot \frac{7 \pi}{6}$ $\newline$ $\cot \left(-\frac{11 \pi}{6}\right)$

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