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