Final answer:
sin 2B is -120/169 using the double-angle formula, and sin B/2 is 3/√13 using the positive branch of the half-angle formula because B/2 is in the second quadrant.
Step-by-step explanation:
Since sin B is given as -12/13 and B is in the range between Π and 3π/2, B is in the third quadrant, where sine is negative and cosine is negative. We can find the cosine of B using the Pythagorean identity: cos2B = 1 - sin2B. This gives us:
cos2B = 1 - (-12/13)2
cos2B = 1 - 144/169
cos2B = 25/169
Since cosine is also negative in the third quadrant, cos B = -5/13.
For sin 2B, we use the double angle formula sin 2B = 2 sin B cos B.
sin 2B = 2(-12/13)(-5/13) = 24/13 * -5/13 = -120/169
To find sin B/2, we must use the half-angle formula. For sin B/2, we have two cases depending on the sign of the half angle. Since B/2 will be in the second quadrant, sin B/2 will be positive. We have:
sin B/2 = ±1 √(1 - cos B)/2
sin B/2 = √(1 - (-5/13))/2 = √(1 + 5/13)/2 = √(18/13)/2 = √(9/13) = 3/ √13