Short Answer:
If the sinθ= √2/2 in the 2nd quadrant, then it is a reference angle of 3π/4 or 135°. Therefore 2θ = 270º. So, by using the unit circle, the sin 270º= -1, cos 270º=0, and the tan 270º= -1/0 or undefined.
Long Answer:
Using your trig identities:
sin(2x) = 2 sin(x) cos(x)
cos(2x) = cos2(x) – sin2(x)
tan(2x)=2tan(x)/(1−tan2(x))
If the sinθ=√2/2 then it is a reference angle of 3π/4 or 135°. Therefore the cosθ= -√2/2 because it is in the second quadrant, and the tanθ = -1.
Now we use this information and substitute them into the trig identities that were asked for:
sin 2θ=2•(√2/2)•( -√2/2)= -1
cos 2θ= (-√2/2)^2 - (√2/2) ˆ2 = 0
tan 2θ= (2•-1)/( (1-(-1)2)= 1/0 or undefined
