Respuesta :
well, the hypotenuse is always a positive value, since it's just the distance from the center to the arc in a circle or between two points, so is never negative, hmmm we know the sine is -1/5, and we also know the cosine is >0, which is another way to say is positive, hmmm let's use that, keeping in mind that the sine is negative and the cosine positive only in the IV Quadrant.
[tex]sin(\theta )=\cfrac{\stackrel{opposite}{-1}}{\underset{hypotenuse}{5}}\impliedby \qquad \textit{let's find the \underline{adjacent side}} \\\\\\ \textit{using the pythagorean theorem} \\\\ c^2=a^2+b^2\implies \sqrt{c^2 - b^2}=a \qquad \begin{cases} c=hypotenuse\\ a=adjacent\\ b=opposite\\ \end{cases}[/tex]
[tex]\pm\sqrt{5^2 - (-1)^2}=a\implies \pm\sqrt{24}=a\implies \stackrel{IV~Quadrant}{+\sqrt{24}=a} \\\\[-0.35em] ~\dotfill\\\\ cot(\theta )=\cfrac{\stackrel{adjacent}{\sqrt{24}}}{\underset{opposite}{-1}}\implies cot(\theta )=-\sqrt{24}[/tex]
