Which of the following is not equivalent to csc ((3pi)/(8)) ? csc(-(13 pi)/(8)) csc ((19 pi)/(8)) csc ((21 pi)/(8)) csc ((11 pi)/(8))

Question

Which of the following is not equivalent to  csc ((3pi)/(8)) ?  csc(-(13 pi)/(8))  csc ((19 pi)/(8))  csc ((21 pi)/(8))  csc ((11 pi)/(8))

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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. We will use this property to determine if the given options are equivalent to csc((3pi)/(8)).Analyze Option 1: Analyze the first option: csc(-(13pi)/(8)). Using the periodicity of the cosecant function, we can add 2π to the angle to find an equivalent positive angle: csc(-(13pi)/(8) + 2π) = csc((3pi)/(8)). Since -(13pi)/(8) + 2π = (3pi)/(8), this option is equivalent to csc((3pi)/(8)).Analyze Option 2: Analyze the second option: csc((19pi)/(8)). Using the periodicity of the cosecant function, we can subtract 2π from the angle to find an equivalent angle: csc((19pi)/(8) - 2π) = csc((3pi)/(8)). Since (19pi)/(8) - 2π = (3pi)/(8), this option is also equivalent to csc((3pi)/(8)).Analyze Option 3: Analyze the third option: csc((21pi)/(8)). Using the periodicity of the cosecant function, we can subtract 2π from the angle to find an equivalent angle: csc((21pi)/(8) - 2π) = csc((5pi)/(8)). Since (21pi)/(8) - 2π = (5pi)/(8), this option is not equivalent to csc((3pi)/(8)). Therefore, this is the option that is not equivalent.Analyze Option 4: Analyze the fourth option: csc((11pi)/(8)). Using the periodicity of the cosecant function, we can subtract 2π from the angle to find an equivalent angle: csc((11pi)/(8) - 2π) = csc((3pi)/(8)). Since (11pi)/(8) - 2π = (3pi)/(8), this option is equivalent to csc((3pi)/(8)).

Which of the following is not equivalent to $\csc \frac{3 \pi}{8}$ ? $\newline$ $\csc \left(-\frac{13 \pi}{8}\right)$ $\newline$ $\csc \frac{19 \pi}{8}$ $\newline$ $\csc \frac{21 \pi}{8}$ $\newline$ $\csc \frac{11 \pi}{8}$

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