#### Answer

The two candidates for the value of angle $C$ are $87.8^{\circ}$ and $92.2^{\circ}$

#### Work Step by Step

We can use the law of sines to find candidates for angle $C$:
$\frac{a}{sin~A} = \frac{c}{sin~C}$
$sin~C = \frac{c~sin~A}{a}$
$C = arcsin(\frac{c~sin~A}{a})$
$C = arcsin(\frac{15~sin~60^{\circ}}{13})$
$C = arcsin(0.99926)$
$C = 87.8^{\circ}$
Another possible value for angle $C$ is $180^{\circ}-87.8^{\circ}$ which is $92.2^{\circ}$
The two candidates for the value of angle $C$ are $87.8^{\circ}$ and $92.2^{\circ}$