Question

If `sin theta = sqrt5/3` then what is `cos 2 theta`?

Answer 1

`\cos(2\theta)=-1/9`. It does not depend on the quadrant `\theta` is in.

Explanation:

Double angle identities for cosine:

`cos(2\theta)`
`=2cos^2(\theta)-1`
`=cos^2(\theta)-sin^2(\theta)`
`=1-2sin^2(\theta)`

Note that there are only even powers of `sin(\theta)` and `cos(\theta)`. So any sign differences between quadrants cancels out, and if you know either `sin(\theta)` or `cos(\theta)` you have only one value for `cos(2\theta)`.