Question

Find the number of integral values of $K$ for which the equation $7\cos x+5\sin x=2k+1$ has at least one solution.

Answer 1

$-\sqrt{a^2+b^2}\le a\sin x+b\cos x\le \sqrt{a^2+b^2}\therefore -\sqrt{74}\le 7\cos x+5\sin x\le \sqrt{74}\Rightarrow -\sqrt{74}\le 2k+1\le \sqrt{74}$
Since $k$ is integer
$-9<2k+1<9\Rightarrow -10<k<8\Rightarrow -5<k<4$
Hnece number of possible integer values of $k=8$