Find the Exact Value sin(arccos( square root of 3))

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

Rewrite ${\displaystyle -2}$ as ${\displaystyle -1\left(2\right)}$.

Rewrite ${\displaystyle \sqrt{-1\left(2\right)}}$ as ${\displaystyle \sqrt{-1}\cdot \sqrt{2}}$.