Step 1: Determine the exact value of the given trigonometric ratio..

                        GIVEN: \sin 405\degree

                               \theta = 405

                       The given angle is more then 360\degree

                       Since, 405\degree - 360\degree = 45\degree

                       45\degree is between 0\degree and 90\degree it will lie in Quadrant I.

Step 2: Calculate the value of the ratio.

                   \sin (2\pi + 45\degree) = \sin (45\degree)

              All the ratios are positive. Since we are in Quadrant I

                   So, \sin(45\degree)=\frac{1}{\sqrt{2}}

                 Therefore,  \sin405\degree=\frac{1}{\sqrt{2}}