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
To find sin(x/2), we can use the half-angle formula for sine.
The half-angle formula states that sin(x/2) = ±√((1 - cos(x))/2), where the sign (+ or -) depends on the quadrant in which x/2 lies.
In this case, we know that sin(x) = -0.4 and 3π/2 ≤ x ≤ 2π, which means that x lies in the fourth quadrant.
To find cos(x), we can use the fact that sin(x) = -0.4. Since sin(x) is negative and x is in the fourth quadrant, cos(x) is positive.
Using the Pythagorean identity, cos^2(x) + sin^2(x) = 1, we can solve for cos(x):
cos^2(x) = 1 - sin^2(x)
cos^2(x) = 1 - (-0.4)^2
cos^2(x) = 1 - 0.16
cos^2(x) = 0.84
Taking the square root of both sides, we get:
cos(x) ≈ √(0.84)
cos(x) ≈ 0.917
Now that we have the value of cos(x), we can substitute it into the half-angle formula:
sin(x/2) = ±√((1 - cos(x))/2)
sin(x/2) = ±√((1 - 0.917)/2)
sin(x/2) = ±√(0.083/2)
sin(x/2) = ±√(0.0415)
Since x/2 lies in the fourth quadrant and sin(x/2) is positive in the fourth quadrant, we can discard the negative sign.
Thus, sin(x/2) = √(0.0415) ≈ 0.2043.
Therefore, sin(x/2) ≈ 0.2043.