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Which of the following is equivalent to
sec ((pi)/(5)) ?

sec ((14 pi)/(5))

sec(-(pi)/(5))

sec ((6pi)/(5))

sec ((4pi)/(5))

Which of the following is equivalent to $\sec \frac{\pi}{5}$ ? $\newline$ $\sec \frac{14 \pi}{5}$ $\newline$ $\sec \left(-\frac{\pi}{5}\right)$ $\newline$ $\sec \frac{6 \pi}{5}$ $\newline$ $\sec \frac{4 \pi}{5}$

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Write a formula for an arithmetic sequence

Full solution

Q. Which of the following is equivalent to

\sec \frac{\pi}{5}

?

\newline

\sec \frac{14 \pi}{5}

\newline

\sec \left(-\frac{\pi}{5}\right)

\newline

\sec \frac{6 \pi}{5}

\newline

\sec \frac{4 \pi}{5}

Properties of Secant Function: Understand the properties of the secant function. The secant function, $\sec(\theta)$ , is periodic with a period of $2\pi$ . This means that $\sec(\theta) = \sec(\theta + 2\pi k)$ for any integer $k$ . Additionally, $\sec(-\theta) = \sec(\theta)$ because secant is an even function.
Evaluate First Option: Evaluate the first option, $\sec\left(\frac{14\pi}{5}\right)$ . Since the secant function has a period of $2\pi$ , we can subtract $2\pi$ from $\frac{14\pi}{5}$ until we are within the interval $[0, 2\pi)$ . $\frac{14\pi}{5} - 2\pi = \frac{14\pi}{5} - \frac{10\pi}{5} = \frac{4\pi}{5}$ . So, $\sec\left(\frac{14\pi}{5}\right)$ is equivalent to $\sec\left(\frac{4\pi}{5}\right)$ .
Evaluate Second Option: Evaluate the second option, $\sec\left(-\frac{\pi}{5}\right)$ . Since the secant function is even, $\sec\left(-\frac{\pi}{5}\right)$ is equivalent to $\sec\left(\frac{\pi}{5}\right)$ .
Evaluate Third Option: Evaluate the third option, $\sec\left(\frac{6\pi}{5}\right)$ . Subtracting $2\pi$ from $\frac{6\pi}{5}$ gives us $\frac{6\pi}{5} - 2\pi = \frac{6\pi}{5} - \frac{10\pi}{5} = -\frac{4\pi}{5}$ . Since secant is an even function, $\sec\left(-\frac{4\pi}{5}\right)$ is equivalent to $\sec\left(\frac{4\pi}{5}\right)$ , which is not equivalent to $\sec\left(\frac{\pi}{5}\right)$ .
Evaluate Fourth Option: Evaluate the fourth option, $\sec\left(\frac{4\pi}{5}\right)$ . This is not equivalent to $\sec\left(\frac{\pi}{5}\right)$ because $\frac{4\pi}{5}$ is not equal to $\frac{\pi}{5}$ plus any multiple of $2\pi$ .
Compare Results: Compare the results from Steps $2$ , $3$ , $4$ , and $5$ . The only equivalent expressions to $\sec\left(\frac{\pi}{5}\right)$ are $\sec\left(-\frac{\pi}{5}\right)$ from Step $3$ and $\sec\left(\frac{14\pi}{5}\right)$ from Step $2$ , which simplifies to $\sec\left(\frac{4\pi}{5}\right)$ . However, since $\sec\left(\frac{4\pi}{5}\right)$ is not equivalent to $\sec\left(\frac{\pi}{5}\right)$ , there must be a mistake in the previous steps.

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