Question

How do you find the exact value of `sin (x/2)` and `cos (x/2)` `if cos x= - 1/4` and `pi< x<2pi`?

Answer 1

Use the trig identity: `cos x = 1 - 2sin^2 (x/2)`

`-1/4 = 1 - 2sin^2 (x/2) -> sin^2 (x/2) = 5/8 = 0.625`

sin x/2 = -0.79

Next, use trig identity: `cos^2 (x/2) - sin^2 (x/2) = cos x`

`cos^2 (x/2) = -1/4 + 5/8 = 3/8 -> **cos x/2 = -0.61**`

Check:
`sin^2 (x/2) + cos^2 (x/2) = 5/8 + 3/8 = 8/8 = 1` Correct.