TITLE: Greatest value of $\frac{1}{4\sin \theta-3 \cos \theta+6}$
QUESTION [5 upvotes]: State the greatest value of $$\frac{1}{4\sin \theta-3 \cos \theta+6}$$
Can anyone give me some hints on this?

REPLY [3 votes]: $$4\sin\theta-3\cos\theta$$ is the dot product of the vector $(-3,4)$ with the unit vector in direction $\theta$. This dot product is minimized when the two vectors are antiparallel, and equals minus the product of the norms, i.e. $-5$.
The requested maximum is $$\frac1{-5+6}.$$