In the diagram, A is an obtuse angle such that $\sin A = \tfrac{5}{13}$. Leaving your answer as a fraction, find the value of

1. Apply the sine identity for a supplementary angle.

Since $\sin(180^\circ-A)=\sin A$, and $\sin A=\dfrac{5}{13}$,

$\sin(180^\circ-A)=\dfrac{5}{13}.$

1. Find the magnitude of $\cos A$ using $\sin^2 A+\cos^2 A=1$.

Since $\sin A=\dfrac{5}{13}$,
$\cos^2 A=1-\sin^2 A=1-\left(\dfrac{5}{13}\right)^2=1-\dfrac{25}{169}=\dfrac{144}{169}.$

2. Choose the correct sign of $\cos A$ for an obtuse angle.

$\cos A=\pm\dfrac{12}{13}.$
Since $A$ is obtuse, $\cos A<0$.
$\cos A=-\dfrac{12}{13}.$