Answer:
- We can use the identity sin(2a) = 2sin(a)cos(a). Since cos(2a)>=-1/8, we know that cos^2(2a) <= 1/64. Using the identity cos^2(2a) + sin^2(2a) = 1, we can solve for sin(2a):
sin^2(2a) = 1 - cos^2(2a)
sin^2(2a) = 1 - 1/64
sin^2(2a) = 63/64
sin(2a) = ± √(63/64) = ± 3√(7)/4
However, we also know that sin(a) >= 3/4, which means that a is in the first or second quadrant. In these quadrants, sin(a) and sin(2a) have the same sign. Therefore, we can conclude that:
sin(2a) = 3√(7)/4
2. We can use the identity sin(a/2) = ± √[(1 - cos(a))/2]. Since cos(2a) <= -7/8, we know that cos(a) <= ±√[(1 + cos(2a))/2]. Since cos(a) <= -1/4, we can conclude that cos(a) is negative. Therefore, we can choose the negative sign in the formula for sin(a/2):
sin(a/2) = -√[(1 - cos(a))/2]
sin(a/2) = -√[(1 + |cos(a)|)/2]
sin(a/2) = -√[(1 - cos^2(a))/(2cos(a))]
sin(a/2) = -√[(sin^2(a))/(2cos(a))]
sin(a/2) = -sin(a)/√(2(1 - cos(a)))
Now, we need to find a lower bound for cos(a). From the inequality cos(2a) <= -7/8, we can deduce that cos(a) is negative and its absolute value is greater than or equal to √(2/3). Therefore, we can write:
sin(a/2) = -sin(a)/√(2(1 - cos(a)))
sin(a/2) = -sin(a)/√(2(1 + |cos(a)|))
sin(a/2) >= -sin(a)/√(2(1 + √(2/3)))
sin(a/2) >= -3√(2)/4
Therefore, we can conclude that:
sin(a/2) >= -3√(2)/4