First of all, let's convert this angle from radians to degree, so applying rule of three:
[tex]\pi \ radians ---\ \textgreater \ 180^{\circ} \\ \frac{13\pi}{4} \ radians --\ \textgreater \ x \\ \\ x=\frac{13\pi}{4}\times \frac{180}{\pi}=585^{\circ}[/tex]
So we can represent this angle as follows:
[tex]585^{\circ}=360^{\circ}+225^{\circ}[/tex]
One complete turn equals 360°, so if a point is traveling in a circumference, the position of the point in 585° is the same as in the angle of 225°. In this way, we also know the following regarding quadrants:
[tex]Let \ x:An \ angle \\ \\ 0^{\circ} \ \textless \ x \ \textless \ 90^{\circ} \ I \ Quadrant \\ 90^{\circ} \ \textless \ x \ \textless \ 180^{\circ} \ II \ Quadrant \\ 180^{\circ} \ \textless \ x \ \textless \ 270^{\circ} \ III \ Quadrant \\ 270^{\circ} \ \textless \ x \ \textless \ 360^{\circ} \ IV \ Quadrant[/tex]
Given that 225° is between 180° and 270° then the conclusion is that this angle is in the III Quadrant and we can find the value as follows:
[tex]sec(\frac{13\pi}{4})=sec(585^{\circ})=sec(225^{\circ})= \frac{1}{cos(225^{\circ})}=- \sqrt{2} [/tex]
