Answer
$\dfrac{\sqrt{2}}{2}$
Work Step by Step
$\because \sin{(\theta+ 360^{\circ} k)} = \sin{\theta}$
$\therefore \sin{405^{\circ}} = \sin{(45^{\circ}+360^{\circ})} = \sin{45^{\circ}}$
$\sin{45^{\circ}} = \boxed{\dfrac{\sqrt{2}}{2}}$
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