1. **State the problem:** We need to find all angles $x$ between 0 and 360 degrees such that $$\cos(2x) = \frac{\sqrt{3}}{2}.$$ 2. **Recall the cosine values:** The value $\frac{\sqrt{3}}{2}$ corresponds to cosine values at angles 30° and 330° (since cosine is positive in the first and fourth quadrants). 3. **Set up the general solutions for $2x$:** Since $$\cos(\theta) = \frac{\sqrt{3}}{2}$$ at $$\theta = 30^\circ + 360^\circ k$$ or $$\theta = 330^\circ + 360^\circ k,$$ where $k$ is any integer. Thus, $$2x = 30^\circ + 360^\circ k$$ or $$2x = 330^\circ + 360^\circ k.$$ 4. **Solve for $x$:** Divide both sides of each equation by 2: $$x = 15^\circ + 180^\circ k$$ or $$x = 165^\circ + 180^\circ k.$$ 5. **Find values of $x$ between 0° and 360°:** - For $$x = 15^\circ + 180^\circ k$$: - If $k=0$, $x = 15^\circ$ - If $k=1$, $x = 195^\circ$ - If $k=2$, $x = 375^\circ$ (outside range) - If $k=-1$, $x = -165^\circ$ (outside range) So possible $x$ are $15^\circ$ and $195^\circ$. - For $$x = 165^\circ + 180^\circ k$$: - If $k=0$, $x = 165^\circ$ - If $k=1$, $x = 345^\circ$ - If $k=2$, $x = 525^\circ$ (outside range) - If $k=-1$, $x = -15^\circ$ (outside range) So possible $x$ are $165^\circ$ and $345^\circ$. 6. **Final solution:** All possible values for $x$ in $[0^\circ, 360^\circ]$ are: $$\boxed{15^\circ, 165^\circ, 195^\circ, 345^\circ}.$$