\( \left. \begin{array} { l } { \operatorname { simplify } } \\ { \sin 775 ^ { \circ } } \end{array} \right. \)

To simplify \( \sin 775^\circ \), we can use the periodic nature of the sine function. Since the sine function has a period of \( 360^\circ \), we can subtract \( 360^\circ \) as many times as needed. If we subtract \( 720^\circ \) (which is \( 2 \times 360^\circ \)), we get: \[ 775^\circ - 720^\circ = 55^\circ \] Thus, \( \sin 775^\circ = \sin 55^\circ \). This means that instead of dealing with \( 775^\circ \), we can just work with the simpler angle of \( 55^\circ \). The value \( \sin 55^\circ \) is approximately \( 0.8192 \).