The maximum value of the sine function is 1, so at \(\theta = 0^\circ\), \(y = k \sin(\alpha) = 2\). Therefore, \(k \sin(\alpha) = 2\).

At \(\theta = 150^\circ\), the graph crosses the \(\theta\)-axis, so \(y = k \sin(150^\circ + \alpha) = 0\). This implies \(150^\circ + \alpha = 180^\circ\), so \(\alpha = 30^\circ\).

Substituting \(\alpha = 30^\circ\) into \(k \sin(\alpha) = 2\), we have \(k \sin(30^\circ) = 2\).

Since \(\sin(30^\circ) = \frac{1}{2}\), we get \(k \cdot \frac{1}{2} = 2\).

Therefore, \(k = 4\).