Find the exact value of $\sec \frac{7 \pi}{4}$ in simplest form with a rational denominator.

Full solution

Q. Find the exact value of

\sec \frac{7 \pi}{4}

in simplest form with a rational denominator.

Understand Secant Definition: To find the exact value of $\sec\left(\frac{7\pi}{4}\right)$ , we need to understand that secant is the reciprocal of cosine. So, we first find the cosine of $\frac{7\pi}{4}$ and then take its reciprocal.
Identify Quadrant of Angle: The angle $(7\pi)/4$ is an angle that lies in the fourth quadrant of the unit circle, where cosine values are positive. Since the unit circle is periodic with a period of $2\pi$ , we can subtract $2\pi$ from $(7\pi)/4$ to find a coterminal angle that is easier to evaluate. $\newline$ $(7\pi)/4 - 2\pi = (7\pi - 8\pi)/4 = (-\pi)/4$
Calculate Cosine Value: The cosine of $(-\pi)/4$ is the same as the cosine of $\pi/4$ because cosine is an even function, which means $\cos(-x) = \cos(x)$ . The cosine of $\pi/4$ is well-known and equals $\sqrt{2}/2$ .
Find Reciprocal for Secant: Now, we take the reciprocal of $\cos\left(\frac{7\pi}{4}\right)$ , which is the same as the reciprocal of $\cos\left(\frac{\pi}{4}\right)$ , to find $\sec\left(\frac{7\pi}{4}\right)$ . $\newline$ $\sec\left(\frac{7\pi}{4}\right) = \frac{1}{\left(\frac{\sqrt{2}}{2}\right)}$
Simplify and Finalize: To simplify the expression and eliminate the radical from the denominator, we multiply the numerator and denominator by $\sqrt{2}$ . $\sec\left(\frac{7\pi}{4}\right) = \left(1 \times \sqrt{2}\right) / \left(\frac{\sqrt{2}}{2} \times \sqrt{2}\right) = \sqrt{2} / \left(\frac{2}{2}\right) = \sqrt{2}$