sin theta= -sqrroot -5/5, find sin theta/2

Question

sinθ=-√5/5, 3pi/2 <θ<2pi  Find sin θ/2

Answer 1

Since theta is between (3pi/2, 2pi) in fourth quadrant, theta/2 must between ( 3pi/4, pi)  in second quadrant. 

Sin( theta) =-sqroot(5)/5 , then cos (theta)=2sqroot(5)/5 using Pythagorean identity (sin(theta))^2+(cos(theta))^2=1

 

then using half angle formula sin(theta/2)=sqroot((1-cos(theta))/2)

=sqroot((5-2sqroot(5))/10)=.2297