Answer:
We can use the half-angle formula for sine to find the exact value of sin(a/2) in terms of sin(a):
sin(a/2) = ±√[(1 - cos(a))/2]
where the ± sign depends on the quadrant of a/2.
To use this formula, we first need to find cos(a). We can do this using the identity:
sin^2(a) + cos^2(a) = 1
Substituting sin(a) = 7/9, we get:
(7/9)^2 + cos^2(a) = 1
Simplifying and solving for cos(a), we get:
cos(a) = ±4/9
Since a is in quadrant I, we take the positive value of cos(a):
cos(a) = 4/9
Now we can use the half-angle formula for sine to find sin(a/2):
sin(a/2) = ±√[(1 - cos(a))/2]
Substituting cos(a) = 4/9, we get:
sin(a/2) = ±√[(1 - 4/9)/2]
Simplifying, we get:
sin(a/2) = ±√(5/18)
Since a is in quadrant I, a/2 is also in quadrant I, so we take the positive value of sin(a/2):
sin(a/2) = √(5/18)
Therefore, the exact value of sin(a/2) is √(5/18)