Solution: Start by solving $ 2\cos(2\theta) + 1 = 0 \Rightarrow \cos(2\theta) = -\frac{1}{2} $. The general solutions for $ 2\theta $ are $ 2\theta = 120^\circ + 360^\circ k $ or $ 2\theta = 240^\circ + 360^\circ k $, where $ k \in \mathbb{Z} $. Dividing by 2, $ \theta = 60^\circ + 180^\circ k $ or $ \theta = 120^\circ + 180^\circ k $. For $ \theta \in [0^\circ, 360^\circ] $, the solutions are $ 60^\circ, 120^\circ, 240^\circ, 300^\circ $.

["How to Solve $ 2\cos(2\ heta) + 1 = 0 $: Complete Step-by-Step Guide", "Solving trigonometric equations can sometimes feel challenging, but with a clear method, even complex expressions become manageable. This article walks you through solving the equation $ 2\cos(2\ heta) + 1 = 0 $, explaining each step and deriving the general and specific solutions for $ \ heta $.", "---", "### Step 1: Simplify the Equation", "Start with the original equation:\n$$\n2\cos(2\ heta) + 1 = 0\n$$\nSubtract 1 from both sides:\n$$\n2\cos(2\ heta) = -1\n$$\nDivide both sides by 2:\n$$\n\cos(2\ heta) = -\frac{1}{2}\n$$\nThis is the key trigonometric equation we now solve.", "---", "### Step 2: Solve for $ 2\ heta $", "We know from trigonometric identities that:\n$$\n\cos x = -\frac{1}{2}\n$$\noccurs at angles where cosine equals negative one-half. Within one full rotation ($0^\circ \leq x < 360^\circ$), the solutions are:\n- $ x = 120^\circ $\n- $ x = 240^\circ $", "Since the cosine function is periodic with period $360^\circ$, the general solutions are:\n$$\n2\ heta = 120^\circ + 360^\circ k \quad \ ext{or} \quad 2\ heta = 240^\circ + 360^\circ k \quad \ ext{where } k \in \mathbb{Z}\n$$", "---", "### Step 3: Solve for $ \ heta $", "Divide each part by 2 to isolate $ \ heta $:\n$$\n\ heta = \frac{120^\circ + 360^\circ k}{2} = 60^\circ + 180^\circ k\n$$\n$$\n\ heta = \frac{240^\circ + 360^\circ k}{2} = 120^\circ + 180^\circ k\n$$", "These represent the complete set of general solutions for $ \ heta $:\n$$\n\ heta = 60^\circ + 180^\circ k \quad \ ext{or} \quad \ heta = 120^\circ + 180^\circ k\n$$", "---", "### Step 4: Find Solutions in $ [0^\circ, 360^\circ] $", "Plug in integer values of $ k $ to find all solutions within the interval from $ 0^\circ $ to $ 360^\circ $.", "- For $ \ heta = 60^\circ + 180^\circ k $:\n - $ k = 0 \Rightarrow \ heta = 60^\circ $\n - $ k = 1 \Rightarrow \ heta = 240^\circ $\n - $ k = 2 \Rightarrow \ heta = 420^\circ $ (outside interval)", "- For $ \ heta = 120^\circ + 180^\circ k $:\n - $ k = 0 \Rightarrow \ heta = 120^\circ $\n - $ k = 1 \Rightarrow \ heta = 300^\circ $\n - $ k = 2 \Rightarrow \ heta = 480^\circ $ (outside interval)", "Therefore, the full solution set in $ [0^\circ, 360^\circ] $ is:\n$$\n\ heta = 60^\circ, \quad 120^\circ, \quad 240^\circ, \quad 300^\circ\n$$", "---", "### Summary", "Solving $ 2\cos(2\ heta) + 1 = 0 $ boils down to recognizing the cosine value, using periodicity, and systematically finding all angles in the desired interval. The general solutions are:\n$$\n\ heta = 60^\circ + 180^\circ k, \quad \ heta = 120^\circ + 180^\circ k \quad (k \in \mathbb{Z})\n$$\nFor $ \ heta \in [0^\circ, 360^\circ] $, the full solution set remains:\n$$\n\boxed{60^\circ, \ 120^\circ, \ 240^\circ, \ 300^\circ}\n$$", "---", "Key Takeaways:", "- Always isolate the trigonometric function first.\n- Use known cosine values and periodicity to find all general solutions.\n- Plug back values of integer parameters to find specific solutions in the target interval.\n- Breaking the problem step-by-step ensures accuracy and clarity.", "Perfect for students learning trigonometry or anyone wanting a reliable method to solve equations of the form $ \cos(nx + \phi) = c $."]