I take it you meant θ angle, anyway.

we know the tan(θ) = -4/7... alrite, we also know that 270° < θ < 360°, which is another to say that θ is in the IV quadrant, where the adjacent side or "x" value is positive whilst the opposite side or "y" value is negative.

$\bf tan(\theta )=\cfrac{\stackrel{opposite}{-4}}{\stackrel{adjacent}{7}}\impliedby \textit{now let's find the \underline{hypotenuse}}
\\\\\\
\textit{using the pythagorean theorem}
\\\\
c^2=a^2+b^2\implies c=\sqrt{a^2+b^2}
\qquad 
\begin{cases}
c=hypotenuse\\
a=adjacent\\
b=opposite\\
\end{cases}
\\\\\\
c=\sqrt{7^2+(-4)^2}\implies \implies c=\sqrt{65}\\\\
-------------------------------\\\\
sec(\theta )=\cfrac{\stackrel{hypotenuse}{\sqrt{65}}}{\stackrel{adjacent}{7}}$

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3 years ago