Use a half measure identity to find the exact value..
Cos t= 1/3 and 0 <t<1/2 pie find sin 1/2 t..
To find the value of sin(1/2t) using a half-angle identity, we need to start with the given value of cos(t) = 1/3 and determine the corresponding value of sin(t).
First, let's find sin(t) using the Pythagorean identity:
sin^2(t) + cos^2(t) = 1
Plugging in the given value of cos(t) = 1/3:
sin^2(t) + (1/3)^2 = 1
Simplifying the equation:
sin^2(t) + 1/9 = 1
sin^2(t) = 1 - 1/9
sin^2(t) = 8/9
Taking the square root of both sides:
sin(t) = ±sqrt(8/9)
However, since t is between 0 and π/2 (0 < t < π/2), we can eliminate the negative square root, so:
sin(t) = sqrt(8/9)
Now, using the half-angle identity for sin, we can find sin(1/2t):
sin(1/2t) = ±sqrt((1 - cos(t))/2)
Plugging in the given value of cos(t) = 1/3:
sin(1/2t) = ±sqrt((1 - 1/3)/2)
sin(1/2t) = ±sqrt(2/3)
Since t is between 0 and π/2 (0 < t < π/2), we can eliminate the negative sign in front of the square root, so:
sin(1/2t) = sqrt(2/3)
Therefore, the exact value of sin(1/2t) is sqrt(2/3).