Which of the following is not equivalent to csc ((pi)/(4)) ? csc ((9pi)/(4)) csc(-(7pi)/(4)) csc ((11 pi)/(4)) csc(-(pi)/(4))

Question

Which of the following is not equivalent to  csc ((pi)/(4)) ?  csc ((9pi)/(4))  csc(-(7pi)/(4))  csc ((11 pi)/(4))  csc(-(pi)/(4))

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Determine Equivalent Options: We need to determine which of the given options is not equivalent to csc((pi)/(4)). 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) = csc(θ + 2πk) for any integer k. We will use this property to evaluate the equivalence of each option.Evaluate csc((9pi)/(4)): First, let's consider csc((9pi)/(4)). Since (9pi)/(4) = (2pi)/(4) + (7pi)/(4) = (pi)/(2) + 2π, we can see that (9pi)/(4) is (pi)/(2) plus a multiple of the period of the sine function. Therefore, csc((9pi)/(4)) is equivalent to csc((pi)/(2)).Evaluate csc(-(7pi)/(4)): Next, let's evaluate csc(-(7pi)/(4)). Since -(7pi)/(4) = -(2pi)/(4) - (5pi)/(4) = -(pi)/(2) - 2π, we can see that -(7pi)/(4) is -(pi)/(2) minus a multiple of the period of the sine function. Therefore, csc(-(7pi)/(4)) is equivalent to csc(-(pi)/(2)).Evaluate csc((11pi)/(4)): Now, let's look at csc((11pi)/(4)). Since (11pi)/(4) = (8pi)/(4) + (3pi)/(4) = 2π + (3pi)/(4), we can see that (11pi)/(4) is (3pi)/(4) plus a multiple of the period of the sine function. Therefore, csc((11pi)/(4)) is equivalent to csc((3pi)/(4)).Consider csc(-(pi)/(4)): Finally, let's consider csc(-(pi)/(4)). Since -(pi)/(4) is not a multiple of the period away from (pi)/(4), it is not equivalent to csc((pi)/(4)). Instead, csc(-(pi)/(4)) is equivalent to csc(2π - (pi)/(4)) = csc((7pi)/(4)), which is different from csc((pi)/(4)).Compare Values: Comparing the values, we see that csc((pi)/(4)), csc((9pi)/(4)), and csc((11pi)/(4)) are all equivalent because they differ by a multiple of the full period 2π. However, csc(-(pi)/(4)) is not equivalent to csc((pi)/(4)) because it represents a different angle in the unit circle that is not a full period away from (pi)/(4). Therefore, csc(-(pi)/(4)) is the correct answer.

Which of the following is not equivalent to $\csc \frac{\pi}{4}$ ? $\newline$ $\csc \frac{9 \pi}{4}$ $\newline$ $\csc \left(-\frac{7 \pi}{4}\right)$ $\newline$ $\csc \frac{11 \pi}{4}$ $\newline$ $\csc \left(-\frac{\pi}{4}\right)$

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