Answer
$\dfrac{2\sqrt 5}{65}$
Work Step by Step
Here, we have $\sin(\theta+\phi)=\sin \theta \cos \phi+\sin \phi \cos \theta$
Let us consider $\sin \theta=\dfrac{5}{13}$ and $\cos \theta=-\dfrac{12}{13}$
Since, $\cos \phi$ is negative in second quadrant. so, we have $\cos \phi=-\dfrac{2\sqrt 5}{5}$ and $\sin \phi=\dfrac{\sqrt 5}{5}$
Now, $\sin \phi=-\dfrac{\sqrt {10}}{10}$ and $\cos \phi=\dfrac{3}{\sqrt {10}}$
This gives: $\phi=\pi-\dfrac{\pi}{3}=\dfrac{2\pi}{3}$
Then, we get $\sin \theta \cos \phi +\sin \phi \cos \theta=(\dfrac{5}{13})(\dfrac{-2\sqrt 5}{5})+(\dfrac{12}{13})(\dfrac{\sqrt 5}{5})$
or, $=\dfrac{2\sqrt 5}{65}$
