$sin (n+1)<sin (n+2) \to \sin(n+\frac{3}{2}-\frac{1}{2})<\sin(n+\frac{3}{2}+\frac{1}{2}) \to \sin \frac{1}{2} \cos(n+\frac{3}{2})>0 \to \cos (n+\frac{3}{2})>0$ So $\cos (n+i+\frac{1}{2})>0$ for $i=1,...,k-1$ but it is means that for some $j$ it is true: $2\pi *j-\frac{\pi}{2}<n+1+\frac{1}{2}< n+k-1+\frac{1}{2}<2\pi*j+\frac{\pi}{2} \to k \leq 5$ for $n=41$ it is true that $\sin(42) < \sin(43) < \ldots < \sin(46)$