# How do you find the exact value of csc[arctan(-5/12)]?

$- \frac{13}{5}$
Use definitions with respect to a right angled $\triangle$ABC.
The hypotenuse of the right angled $\triangle$ABC, with sides a and c including right angle B as 5 and 12, is 13.
$\arctan \left(- \frac{5}{12}\right) = - \arctan \left(\frac{5}{12}\right) = - a r c \csc \left(\frac{13}{5}\right)$.
csc arccsc (k) = k and csc(-theta)=-csc(theta)
So, the answer is $- \frac{13}{5}$.