A Side From a Cosecant

A reciprocal ratio can find a side directly. Since csc θ is hypotenuse over opposite, the hypotenuse is the opposite side times csc θ: one multiplication.

The side opposite 40° is 6 and you want the hypotenuse x. $\csc\left(40^{\circ}\right) = \frac{x}{6}$, so $x = 6\csc\left(40^{\circ}\right)$, about 9.33. With the sine the same answer comes from $x = \frac{6}{\sin\left(40^{\circ}\right)}$: multiplying by csc is dividing by sin.