1. The problem is to sketch the graph of the function $y = \cos x$ for $0 \leq x \leq 360^\circ$.\n\n2. The cosine function is periodic with period $360^\circ$, and it oscillates between $-1$ and $1$. The key points to plot are at $0^\circ$, $90^\circ$, $180^\circ$, $270^\circ$, and $360^\circ$.\n\n3. Evaluate $y = \cos x$ at these points:\n- $\cos 0^\circ = 1$\n- $\cos 90^\circ = 0$\n- $\cos 180^\circ = -1$\n- $\cos 270^\circ = 0$\n- $\cos 360^\circ = 1$\n\n4. Plot these points on the coordinate system and connect them smoothly to form the wave shape of the cosine function. The graph starts at $1$, decreases to $0$ at $90^\circ$, reaches $-1$ at $180^\circ$, goes back to $0$ at $270^\circ$, and returns to $1$ at $360^\circ$.\n\n5. The graph is continuous and smooth, with maxima at $0^\circ$ and $360^\circ$, minima at $180^\circ$, and zeros at $90^\circ$ and $270^\circ$.\n\nFinal answer: The graph of $y = \cos x$ over $0 \leq x \leq 360^\circ$ is a wave starting at $1$, dipping to $-1$ at $180^\circ$, and returning to $1$ at $360^\circ$.