Question

Find sin(x/2) if sin(x)= -0.4 and 3pi/2 < or equal to (x) < or equal to 2pi

Let's use cos 2A = 1 - 2sin² A

and we can match
cos x = 1 - 2sin² (x/2)

so we will need cos x

we know sin x = -.4 and x is in the fourth quadrant, so the cosine will be positive.

Drawing a right angled triangle with side 2 and hypotenuse 5 ( .4 = 4/10 = 2/5), and using Pythagoras it is easy to see that cos x = √21/5

then:
√21/5 = 1 - 2sin² (x/2)
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I got sin (x/2) = √(5-√21)/√10 or appr.2043