Question

If sine of theta= 12/13 and theta is an acute angle, find cot theta.

Answer 1

if the angle is an acute angle, meaning is less than 90°, that simply means that θ is in the I quadrant, therefore, its adjacent side is positive, thus


`\bf sin(\theta )=\cfrac{\stackrel{opposite}{12}}{\stackrel{hypotenuse}{13}}\impliedby \textit{now, let's find the \underline{adjacent side}} \\\\\\ \textit{using the pythagorean theorem}\\\\ c^2=a^2+b^2\implies \pm√(c^2-b^2)=a \qquad \begin{cases} c=hypotenuse\\ a=adjacent\\ b=opposite\\ \end{cases}`


`\bf \pm√(13^2-12^2)=a\implies \pm√(169-144)=a\implies \pm√(25)=a \\\\\\ \pm 5=a\implies \stackrel{I~quadrant}{+5=a}\qquad therefore\qquad cot(\theta )=\cfrac{\stackrel{adjacent}{5}}{\stackrel{opposite}{12}}`