Trigonometric Equations

The deciding fact is the bounded range of the cosine function, since for any real number $x$ the value of $\cos x$ always lies within the closed interval $[-1,1]$ and never outside it. Consequently the equation $\cos x=\frac{k}{3}$ can possess a real solution only when its right-hand side also lies inside that band, that is when $-1\le \frac{k}{3}\le 1$. Multiplying through by three gives the parameter restriction $-3\le k\le 3$, an inclusive interval because cosine actually attains both endpoint values. The integers satisfying this are $-3,-2,-1,0,1,2,3$, which number exactly seven. Option 6 wrongly excludes one endpoint, treating one bound as strict. Option 5 drops both endpoints. Option 4 undercounts still further. This follows the fundamental range-restriction principle for cosine equations emphasized in JEE Advanced. As a final boundary check, at $k=3$ the equation $\cos x=1$ is solvable at $x=0$, and at $k=-3$ the equation $\cos x=-1$ is solvable at $x=\pi$, so both extreme values are genuinely attainable and must be counted, confirming the total of seven admissible integers.