Solution: The angle between the diagonal of a square and its side is $45^\circ$, but here the angle is $150^\circ$, which lies in the second quadrant. Using the reference angle $30^\circ$, $\cos 150^\circ = -\cos 30^\circ = -\frac{\sqrt{3}}{2}$. $\boxed{-\dfrac{\sqrt{3}}{2}}$

["Understanding the Unique Angle: Why the Cosine of 150° is $-\frac{\sqrt{3}}{2}$", "When exploring geometric relationships, one of the foundational concepts is the angle formed between a square’s side and its diagonal. In a regular square, this angle is always $45^\circ$, as the diagonal bisects the $90^\circ$ corner into two equal parts. However, in this intriguing scenario, we examine a different angular configuration—where the angle between the diagonal and a side of a square measures $150^\circ$, a value that lies in the second quadrant of the unit circle.", "At first glance, an angle of $150^\circ$ may seem unusual for a square’s geometry, as perfect $90^\circ$ or $45^\circ$ angles dominate traditional square properties. Yet, mathematics reveals elegant patterns through reference angles. The reference angle for $150^\circ$, measured from the x-axis, is $180^\circ - 150^\circ = 30^\circ$. In the second quadrant, cosine values are negative, so:", "$$\n\cos 150^\circ = -\cos 30^\circ\n$$", "Since we know that $\cos 30^\circ = \frac{\sqrt{3}}{2}$, substituting gives:", "$$\n\cos 150^\circ = -\frac{\sqrt{3}}{2}\n$$", "This negative cosine reflects the position of $150^\circ$ in the second quadrant, where the x-coordinate (and thus the cosine value) is negative. Understanding this shift reinforces how reference angles connect familiar geometric angles to broader trigonometric principles.", "In practical applications, this calculation matters in fields such as architecture, engineering, and computer graphics, where angular measurements define structural integrity and object orientation. Recognizing the precise value of $\cos 150^\circ$ ensures accurate computations when analyzing forces, lighting, or dimensions influenced by such angular offsets.", "Key Takeaway:\nEven when angles deviate from classical square geometry, foundational principles like reference angles maintain mathematical consistency. With $\cos 150^\circ = -\frac{\sqrt{3}}{2}$, we embrace precision and clarity—essential tools for solving geometric challenges with confidence.", "---", "\boxed{-\dfrac{\sqrt{3}}{2}}\nThe exact cosine value of 150°, reflecting its position in the second quadrant through a negative reference angle cosine."]