We can find the exact value of sin 157.5 like this:
sin²(157.5) = 1/2 (1 − cos(2*157.5))
= 1/2 (1 − cos(315))
= 1/2 (1 − √2/2)
= (2−√2)/4
Since 157.5 is in quadrant II, then sin(157.5) > 0, so we take positive square root.
sin(157.5) = √(2−√2)/2
I really hope this can help you
