Which of the following is not equivalent to csc ((2pi)/(5))? csc ((7pi)/(5)) csc ((13 pi)/(5)) csc ((3pi)/(5)) csc(-(8pi)/(5))

Question

Which of the following is not equivalent to  csc ((2pi)/(5))?  csc ((7pi)/(5))  csc ((13 pi)/(5))  csc ((3pi)/(5))  csc(-(8pi)/(5))

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Determine Equivalent Options: We need to determine which of the given options is not equivalent to csc((2pi)/(5)). The cosecant function has a period of 2π, which means that csc(θ) = csc(θ + 2πk) for any integer k. We will use this property to check the equivalence of each option.Consider csc((7pi)/(5)): First, let's consider csc((7pi)/(5)). We can write (7pi)/(5) as (2pi)/(5) + (5pi)/(5), which simplifies to (2pi)/(5) + π. Since adding π to an angle results in the same cosecant value but with an opposite sign, and since the cosecant function is odd, csc((7pi)/(5)) is equivalent to -csc((2pi)/(5)). Therefore, csc((7pi)/(5)) is not equivalent to csc((2pi)/(5)) because of the sign difference.Consider csc((13pi)/(5)): Next, let's consider csc((13pi)/(5)). We can write (13pi)/(5) as (2pi)/(5) + (11pi)/(5), which simplifies to (2pi)/(5) + 2π + π. Since adding multiples of 2π does not change the value of the cosecant function, and adding π changes the sign, csc((13pi)/(5)) is equivalent to -csc((2pi)/(5)). Therefore, csc((13pi)/(5)) is not equivalent to csc((2pi)/(5)) because of the sign difference.Consider csc((3pi)/(5)): Now, let's consider csc((3pi)/(5)). We can write (3pi)/(5) as (2pi)/(5) + (pi)/(5). Since we are not adding a multiple of π, the cosecant value will be different. Therefore, csc((3pi)/(5)) is not equivalent to csc((2pi)/(5)).Consider csc(-(8pi)/(5)): Finally, let's consider csc(-(8pi)/(5)). We can write -(8pi)/(5) as -(2pi)/(5) - (6pi)/(5), which simplifies to -(2pi)/(5) - 2π + π. Since subtracting multiples of 2π does not change the value of the cosecant function, and subtracting π changes the sign, csc(-(8pi)/(5)) is equivalent to -csc((2pi)/(5)). Therefore, csc(-(8pi)/(5)) is not equivalent to csc((2pi)/(5)) because of the sign difference.

Which of the following is not equivalent to $\csc \frac{2 \pi}{5} ?$ $\newline$ $\csc \frac{7 \pi}{5}$ $\newline$ $\csc \frac{13 \pi}{5}$ $\newline$ $\csc \frac{3 \pi}{5}$ $\newline$ $\csc \left(-\frac{8 \pi}{5}\right)$

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