If $\sin \theta =\frac{5}{13}$, then what is the value of $\tan \theta +\sec \theta$?

Given:
$$\sin \theta =\frac{5}{13}=\frac{\text{Perpendicular}}{\text{Hypotenuse}}$$

Using the Pythagorean triplet $(5,12,13)$, the base is:

$$\text{Base}=\sqrt{13^2-5^2}=\sqrt{169-25}=\sqrt{144}=12$$

Now, find $\tan \theta$ and $\sec \theta$:

$$\tan \theta =\frac{\text{Perpendicular}}{\text{Base}}=\frac{5}{12}$$

$$\sec \theta =\frac{\text{Hypotenuse}}{\text{Base}}=\frac{13}{12}$$

Therefore:

$$\tan \theta +\sec \theta =\frac{5}{12}+\frac{13}{12}=\frac{18}{12}=\frac{3}{2}=1.5$$