Answer
$-\sqrt2$
Work Step by Step
RECALL:
$\sin{s} = y
\\\cos{s} = x
\\\tan{s} = \frac{y}{x}
\\\cot{s} = \frac{x}{y}
\\\sec{s} = \frac{1}{x}
\\\csc{s}=\frac{1}{y}$
(refer to Figure 11 , page 111 of the textbook)
The angle $\frac{5\pi}{4}$ intersects the unit circle at the point $(-\frac{\sqrt2}{2}, -\frac{\sqrt2}{2})$.
This point has:
$x= -\frac{\sqrt2}{2}$
$y=-\frac{\sqrt2}{2}$
Thus,
$\sec{\frac{5\pi}{4}}
\\= \frac{1}{x}
\\=\frac{1}{-\frac{\sqrt2}{2}}
\\=1 \cdot -\frac{2}{\sqrt2}
\\=-\frac{2}{\sqrt2}$
Rationalize the denominator by multiplying $\sqrt2$ to both the numerator and the denominator to obtain:
$=-\frac{2}{\sqrt2} \cdot \frac{\sqrt2}{\sqrt2}
\\=-\frac{2\sqrt2}{2}
\\=-\sqrt2
$