bearing in mind that sin Θ < 0, is just another way to say the sine of the angle is negative, and that only occurs on the III and IV quadrants.
Also let's keep in mind that the hypotenuse, is just a radius unit, thus is never negative.
[tex]\bf sec(\theta )=\cfrac{\stackrel{hypotenuse}{\sqrt{10}}}{\stackrel{adjacent}{3}}\impliedby \textit{now let's find the \underline{opposite side}}
\\\\\\
\textit{using the pythagorean theorem}\\\\
c^2=a^2+b^2\implies \pm\sqrt{c^2-a^2}=b\qquad
\begin{cases}
c=hypotenuse\\
a=adjacent\\
b=opposite\\
\end{cases}[/tex]
[tex]\bf \pm\sqrt{(\sqrt{10})^2-3^2}=b\implies \pm\sqrt{10-9}=b
\\\\\\
\pm 1=b\implies \stackrel{\textit{III or IV quadrant}}{-1=b}\\\\
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tan(\theta )=\cfrac{\stackrel{opposite}{-1}}{\stackrel{adjacent}{3}}\implies tan(\theta )=-\cfrac{1}{3}[/tex]
