Which of the following is not equivalent to csc ((2pi)/(7)) ? csc(-(12 pi)/(7)) csc ((16 pi)/(7)) csc ((12 pi)/(7)) csc ((19 pi)/(7))

Question

Which of the following is not equivalent to  csc ((2pi)/(7)) ?  csc(-(12 pi)/(7))  csc ((16 pi)/(7))  csc ((12 pi)/(7))  csc ((19 pi)/(7))

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Properties and Periodicity: Understand the properties of the cosecant function and periodicity. The cosecant function, csc(θ), is the reciprocal of the sine function, sin(θ), and has a period of 2π. This means that csc(θ + 2πk) = csc(θ) for any integer k.Analyze Option 1: Analyze the first option csc(-(12 pi)/(7)). Using the property that csc(-θ) = -csc(θ) and the periodicity of the cosecant function, we can find an equivalent angle in the positive direction by adding 2π until we get a positive angle. csc(-(12 pi)/(7)) = csc((2π - (12π/7)) + 2πk) for k = 1, since we want the smallest positive coterminal angle. csc(-(12 pi)/(7)) = csc((14π/7 - 12π/7) + 14π/7) = csc((2π/7) + 14π/7) = csc((16π/7))Analyze Option 2: Analyze the second option csc((16 pi)/(7)). Using the periodicity of the cosecant function, we can subtract 2π to find an equivalent angle. csc((16 pi)/(7)) = csc((16π/7) - 2π) = csc((16π/7) - 14π/7) = csc((2π/7)) This is equivalent to the original expression csc((2π)/(7)).Analyze Option 3: Analyze the third option csc((12 pi)/(7)). Using the periodicity of the cosecant function, we can subtract 2π to find an equivalent angle. csc((12 pi)/(7)) = csc((12π/7) - 2π) = csc((12π/7) - 14π/7) = csc((-2π/7)) Since csc(-θ) = -csc(θ), this is not equivalent to csc((2π)/(7)) because the cosecant function is not an even function.Analyze Option 4: Analyze the fourth option csc((19 pi)/(7)). Using the periodicity of the cosecant function, we can subtract 2π multiple times to find an equivalent angle. csc((19 pi)/(7)) = csc((19π/7) - 2π) = csc((19π/7) - 14π/7) = csc((5π/7)) This is not equivalent to csc((2π)/(7)) because 5π/7 is not a coterminal angle of 2π/7.Determine Non-equivalent Option: Determine which option is not equivalent to csc((2pi)/(7)). From the previous steps, we have determined that csc((16 pi)/(7)) and csc(-(12 pi)/(7)) are equivalent to csc((2pi)/(7)). However, csc((12 pi)/(7)) simplifies to csc((-2π/7)), which is not equivalent to csc((2π)/(7)). Therefore, csc((12 pi)/(7)) is not equivalent to csc((2pi)/(7)).

Which of the following is not equivalent to $\csc \frac{2 \pi}{7}$ ? $\newline$ $\csc \left(-\frac{12 \pi}{7}\right)$ $\newline$ $\csc \frac{16 \pi}{7}$ $\newline$ $\csc \frac{12 \pi}{7}$ $\newline$ $\csc \frac{19 \pi}{7}$

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