Answer
$\frac{8\sqrt{39}}{39}$
Work Step by Step
Given: $\cos\theta=\frac{5}{8}$
$\csc\theta=\frac{1}{\sin\theta}$
$\sin^2\theta+\cos^2\theta=1$
$\sin\theta=\pm\sqrt{1-\cos^2\theta}=\pm\sqrt{1-\left(\frac{5}{8}\right)^2}=\pm\frac{\sqrt{39}}{8}$
In quadrant I sine is positive, so
$\sin\theta=\frac{\sqrt{39}}{8}$
$\csc\theta=\frac{1}{\frac{\sqrt{39}}{8}}=\frac{8}{\sqrt{39}}=\frac{8\sqrt{39}}{39}$
