Question

If csc x=-2 and cos x is greater than 0, find cos x/2?

Answer 1

Find `cos (x/2)`

Ans: `sqrt(2 + sqrt3)/2`

Explanation:

sin x = 1/csc x = -1/2 --> `x = ((7pi)/6)` and `x = ((11pi)/6)`.
Since cos x > 0, x should be in the Quadrant IV, and only the second number is accepted:
`x = ((11pi)/6)` --> `sin x = - 1/2` --> `cos x = sqrt3/2` (Trig Table)
Apply the trig identity: `cos 2a = 2cos^2 a - 1`
`cos x = sqrt3/2 = 2cos^2 (x/2) - 1`
`2cos^2 (x/2) = 1 + sqrt3/2 = (2 + sqrt3)/2`
`cos^2 (x/2) = (2 + sqrt3)/4`
`cos (x/2) = +- sqrt(2 + sqrt3)/2`
Since cos x > 0, then only the positive answer is accepted
`cos (x/2) = sqrt(2 + sqrt3)/2`
Check by calculator.
cos ((11pi)/6) = cos 330 = 0.83
sqrt(2 + sqrt3)/2 = 1.65/2 = 0.83. OK