Ta có: $\cos ^{2}\alpha +\sin ^{2}\alpha =1$.
=> $\sin ^{2}\alpha =1-\cos ^{2}\alpha$.
<=> $\sin ^{2}\alpha =1-(\frac{1}{3})^{2}=\frac{8}{9}$.
Mặt khác: $P = 3\sin ^{2}\alpha+\cos ^{2}\alpha = 2\sin ^{2}\alpha+\sin ^{2}\alpha+\cos ^{2}\alpha=2\sin ^{2}\alpha+1$
<=> $P=2.\frac{8}{9}+1=\frac{25}{9}$
Vậy $P=\frac{25}{9}$