Question

If sin=-1/5 and cos > 0 find cot

Answer 1

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.

`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 √(c^2 - b^2)=a \qquad \begin{cases} c=hypotenuse\\ a=adjacent\\ b=opposite\\ \end{cases}`

`\pm√(5^2 - (-1)^2)=a\implies \pm√(24)=a\implies \stackrel{IV~Quadrant}{+√(24)=a} \\\\[-0.35em] ~\dotfill\\\\ cot(\theta )=\cfrac{\stackrel{adjacent}{√(24)}}{\underset{opposite}{-1}}\implies cot(\theta )=-√(24)`