Find the exact value of cos (2theta) given csc theta = 8/3 and 0 < theta < pi/2
cscx = 8/3
sinx = 3/8
cosx = sqrt [1 - sin^2x] = (sqrt55)/8
cos(2x) = 2 cos^2x -1 = 46/64 = 23/32
To find the exact value of cos(2θ), we can use the identity:
cos(2θ) = 1 - 2sin²(θ)
Given that csc(θ) = 8/3, we know that sin(θ) = 1/csc(θ) which is equal to 3/8.
Now, we can substitute this value into the identity:
cos(2θ) = 1 - 2 (3/8)²
To simplify further:
cos(2θ) = 1 - 2 (9/64)
cos(2θ) = 1 - 18/64
To simplify the fraction, we can find the common denominator:
cos(2θ) = 1 - 9/32
Finally:
cos(2θ) = 23/32
Therefore, the exact value of cos(2θ) is 23/32.