Question

Find csc if
cot = −28/45
and
sin > 0.

Answer 1

sine > 0 is another way to say that the sine of the angle is positive, well, that simply only happens on the I or II Quadrant, we also have a negative cotangent, so we can say that's on the II Quadrant since that'd give us a positive opposite side or sine and negative adjacent side, let's check some

`cot(\theta )=\cfrac{\stackrel{adjacent}{-28}}{\underset{opposite}{45}}\hspace{5em}\textit{let's find its \underline{hypotenuse}} \\\\\\ \textit{using the pythagorean theorem} \\\\ c^2=a^2+b^2\implies c=√(a^2 + b^2) \qquad \begin{cases} c=hypotenuse\\ a=adjacent\\ b=opposite\\ \end{cases} \\\\\\ c=√((-28)^2 + 45^2)\implies c=√(2809)\implies c=53 \\\\[-0.35em] ~\dotfill\\\\ ~\hfill csc(\theta )=\cfrac{\stackrel{hypotenuse}{53}}{\underset{opposite}{45}}~\hfill`