Find the maximum value of the expression $3\sin x+4\cos x+5$.

Solution

1. The expression is of the form $a\sin x+b\cos x+c$.
2. The range of $a\sin x+b\cos x$ is $[-\sqrt{a^2+b^2},\sqrt{a^2+b^2}]$.
3. Here $a=3$ and $b=4$, so $\sqrt{3^2+4^2}=5$.
4. The maximum value is $5 + 5 = 10$.
5. Hence the answer is (B).