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Solve 2 cos 2𝑥° = 1 for 0 ≤ 𝑥 ≤ 180.

$\begin{array}{|rcll|} \hline 2 \cos (2x) &=& 1 \quad & | \quad : 2 \\ \cos (2x) &=& \frac{1} {2} \quad & | \quad \pm\arccos() \\ 2x &=& \pm\arccos( \frac{1} {2} ) +360\cdot k \qquad k \in \mathbb{Z} \\ 2x &=& \pm 60^{\circ} + 360^{\circ} \cdot k \quad & | \quad : 2 \\ x &=& \pm 30^{\circ} + 180^{\circ} \cdot k \\ \mathbf{x_1} & \mathbf{=} & \mathbf{30^{\circ}} \\\\ x_2 &=& -30^{\circ} + 180^{\circ} \\ \mathbf{x_2} & \mathbf{=} & \mathbf{150^{\circ}} \\ \hline \end{array}$