If $$\theta$$ is an acute angle, and it is given that $$5 \sin \theta + 12 \cos \theta = 13$$, then what is the value of $$tan \theta$$ ?

we get $$5\sin\theta\ =13-12\cos\theta\ $$
squaring we get :
$$25\sin^2\theta\ =169+144\cos^2\theta-312\cos\theta\ \ $$
we get $$25-25\cos^2\theta\ =169+144\cos^2\theta-312\cos\theta\ \ $$ $$\left(1-\sin\ ^2\theta\ =\ \cos\ ^2\theta\right)$$
we get $$169\cos^2\theta-312\cos\theta\ \ +144\ =0$$
we get $$\cos\theta\ \ =\frac{12}{13}$$
and so $$\sin\theta\ =\sqrt{\ 1-\frac{144}{169}}=\sqrt{\ \frac{25}{169}}=\frac{5}{13}$$
we get $$\tan\theta\ =\frac{5}{12}$$