Which of the following is equivalent to sec ((2pi)/(7)) ? sec ((9pi)/(7)) sec ((19 pi)/(7)) sec ((12 pi)/(7)) sec ((5pi)/(7))

Question

Which of the following is equivalent to  sec ((2pi)/(7)) ?  sec ((9pi)/(7))  sec ((19 pi)/(7))  sec ((12 pi)/(7))  sec ((5pi)/(7))

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Understand Secant Function Properties: Understand the properties of the secant function. The secant function, sec(θ), is periodic with a period of 2π. This means that sec(θ) = sec(θ + 2πk) for any integer k. We will use this property to find an equivalent expression for sec((2π)/7).Compare with Options: Compare the given options with the original expression sec((2π)/7) by adding multiples of 2π to the original angle and see which option matches. We will start with sec((9π)/7) and check if it is equivalent to sec((2π)/7).Check sec((9π)/7): Add 2π to the original angle (2π)/7 to see if it matches sec((9π)/7). The calculation is (2π)/7 + 2π = (2π + 14π)/7 = (16π)/7. This does not match sec((9π)/7), so sec((9π)/7) is not equivalent to sec((2π)/7).Check sec((19π)/7): Check sec((19π)/7) by adding 2π to the angle (2π)/7 twice. The calculation is (2π)/7 + 2×2π = (2π)/7 + (28π)/7 = (30π)/7. This simplifies to (2×14π + 2π)/7 = (28π + 2π)/7 = (30π)/7. This does not match sec((19π)/7), so sec((19π)/7) is not equivalent to sec((2π)/7).Check sec((12π)/7): Check sec((12π)/7) by subtracting 2π from the angle (2π)/7. The calculation is (2π)/7 - 2π = (2π - 14π)/7 = (-12π)/7. This does not match sec((12π)/7), so sec((12π)/7) is not equivalent to sec((2π)/7).Check sec((5π)/7): Check sec((5π)/7) by adding 2π to the angle (2π)/7 three times. The calculation is (2π)/7 + 3×2π = (2π)/7 + (42π)/7 = (44π)/7. This simplifies to (6×7π + 2π)/7 = (42π + 2π)/7 = (44π)/7. This does not match sec((5π)/7), so sec((5π)/7) is not equivalent to sec((2π)/7).Correct Approach for Finding k: Realize that there was a mistake in the previous steps. We need to find a k such that (2π)/7 + 2πk = one of the given options. We should have been looking for a k that makes the expression (2π)/7 + 2πk/7 equal to one of the given options. We will re-evaluate the options with this corrected approach.

Which of the following is equivalent to $\sec \frac{2 \pi}{7}$ ? $\newline$ $\sec \frac{9 \pi}{7}$ $\newline$ $\sec \frac{19 \pi}{7}$ $\newline$ $\sec \frac{12 \pi}{7}$ $\newline$ $\sec \frac{5 \pi}{7}$

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Which of the following is equivalent to $\sec \frac{2 \pi}{7}$ ? $\newline$ $\sec \frac{9 \pi}{7}$ $\newline$ $\sec \frac{19 \pi}{7}$ $\newline$ $\sec \frac{12 \pi}{7}$ $\newline$ $\sec \frac{5 \pi}{7}$