Find the exact value of sin(theta/2) if cos theta = 2/3 and 270 degrees<theta<360 degrees.
Answer: sqrt6/6
half angle between 135 and 180 which is quadrant 2
half angle formula in quad 2
sqrt [(1-cos t)/2 ]
sqrt [ (1/3) /2 ]
sqrt (1/6)
sqrt 1/sqrt 6
sqrt 6/6 agree
To find the value of sin(theta/2), we need to use the half-angle identity for sine.
The half-angle identity for sine states that sin(theta/2) = ±√[(1 - cos theta)/2].
Given that 270 degrees < theta < 360 degrees and cos theta = 2/3, we can determine the value of sin(theta/2) by substituting cos theta into the half-angle identity.
First, let's find the value of (1 - cos theta)/2:
(1 - cos theta)/2 = (1 - 2/3)/2
= (1/3)/2
= 1/6
Now, substitute (1 - cos theta)/2 into the half-angle identity:
sin(theta/2) = ±√(1/6)
= ±√(6/36)
= ±√(6)/√(36)
= ±√(6)/6
Since theta is between 270 degrees and 360 degrees, which is in the third quadrant, sin(theta/2) will be negative. Therefore, we can conclude that sin(theta/2) = -√(6)/6.