Here,
#sintheta=-1/sqrt2 ,where,-pi < theta <=2pi#
We have,
#sintheta=-1/sqrt2 < 0 =>color(red)(III^(rd)Quadrant) orcolor(blue)( IV^(th)Quadrant)#
Now,
#-pi < theta <=2pi=>-pi < theta < 0 or 0 <= theta <=2pi#
We have two options :
#(i)-pi < theta < 0 ,then#
#sintheta=-1/sqrt2#
#=>theta=-pi+pi/4=-(3pi)/4tocolor(red)(III^(rd)Quadrant#
#or theta=0-pi/4=-pi/4tocolor(blue)(IV^(th)Quadrant)#
#(ii)0 <=theta <=2pi#
#sintheta=-1/sqrt2#
#=>theta=pi+pi/4=(5pi)/4tocolor(red)(III^(rd)Quadrant#
#or theta=2pi-pi/4=(7pi)/4tocolor(blue)(IV^(th)Quadrant)#
Hence,
#theta=-(3pi)/4,-pi/4, or (5pi)/4,(7pi)/4#