Evaluate sin(arccos(( square root of 3)/2))

Draw a triangle in the plane with vertices ${\displaystyle (\frac{\sqrt{3}}{2},\sqrt{1^2-{\left(\frac{\sqrt{3}}{2}\right)}^2})}$, ${\displaystyle (\frac{\sqrt{3}}{2},0)}$, and the origin. Then ${\displaystyle \arccos \left(\frac{\sqrt{3}}{2}\right)}$ is the angle between the positive x-axis and the ray beginning at the origin and passing through ${\displaystyle (\frac{\sqrt{3}}{2},\sqrt{1^2-{\left(\frac{\sqrt{3}}{2}\right)}^2})}$. Therefore, ${\displaystyle \sin \left(\arccos \left(\frac{\sqrt{3}}{2}\right)\right)}$ is ${\displaystyle \sqrt{\frac{1}{4}}}$.

Rewrite ${\displaystyle \sqrt{\frac{1}{4}}}$ as ${\displaystyle \frac{\sqrt{1}}{\sqrt{4}}}$.

Any root of ${\displaystyle 1}$ is ${\displaystyle 1}$.