Question

How do you find the exact values of `sin2u, cos2u, tan2u` using the double angle values given `cosu=-2/3, pi/2<u<pi`?

Answer 1

`cos 2u = - 1/9`
`sin 2u = - 4sqrt5/9`
`tan 2u = 2sqrt5/3`

Explanation:

`cos u = - 2/3`
Find cos 2u by the identity:
`cos 2u = 2cos^2 u - 1`
`cos 2u = 2(4/9) - 1 = 8/9 - 1 = - 1/9`
`sin^2 2u = 1 - cos^2 2u = 1 - 1/81 = 80/81`
`sin 2u = - (4sqrt5)/9` (because 2u is in Quadrant III)
`tan 2u = sin/(cos) = - ((4sqrt5)/9)(- 3/2) = (2sqrt5)/3`