For any trigonometric function, its argument must be dimensionless. Therefore, the quantity $(\omega t - kx)$ must be dimensionless ($[M^0L^0T^0]$). This implies that both $\omega t$ and $kx$ must individually be dimensionless. Considering the term $kx$: $[kx] = [M^0L^0T^0]$ Since $x$ is distance, its dimension is $[L]$. So, $[k][L] = [M^0L^0T^0]$. Therefore, $[k] = [L^{-1}]$. (Here $k$ is the wave number).