Question

Question #20640

Answer 1

`sin x = - sqrt(7 - 3sqrt5)/sqrt14 = - 0.54/3.74`

Explanation:

`sin 2x = - 2/7`. Find cos 2x.
`cos^2 2x = 1 - sin^2 2x = 1 - 4/49 = 45/49`
`cos 2x = 3sqrt5/7` (because 2x is in Quadrant 4, cos 2x > 0)
Find sin x by the trig identity:
`2sin^2 x = 1 - cos 2x = 1 - (3sqrt5)/7 = (7 - 3sqrt5)/7`
`sin^2 x = (7 - 3sqrt5)/14`.
`sin x = - sqrt(7 - 3sqrt5)/(sqrt14) = - 0.54/3.74`
(2x is in quadrant 4, so, x is in Quadrant 3, then sin x < 0)
Check by calculator
`sin x = - 0.14` --> `x = -8^@05` --> `2x = - 16^@10` -->
`sin (-16^@10) = - 0.28 = - 2/7`. Proved