Question

How do you use the double-angle identities to find tan(2x) if cos x=8/17 and sin x is less than 0?

Answer 1

Find tan 2x knowing `cos x = 8/17`

Ans: 1.49

Explanation:

`cos x = 8/17`.
Apply the trig identity: `cos^2 x + sin^2 x = 1`
`sin^2 x = 1 - 64/289 = (289 - 64)/289 = 225/289`
`sin x = +-15/17.` Since sin x < 0, therefor:
`sin x = -15/17`
Apply the two trig identities: `sin 2a = 2sin a.cos a`
and `cos 2a = 2cos^2 a - 1`.
`sin 2x = 2sin x.cos x = 2(-15/17)(8/17) = -240/289`
`cos 2x = 2cos^2 x - 1 = 2(64/289) - 1 = 128/289 - 1 =`
`= (128 - 289)/289 = 161/289`.
`tan 2x = (sin 2x)/(cos 2x) = -240/161 = 1.49`

Check by calculator.
`cos x = 8/17` --> `x = 61^@92` --> 2x = 123^`85 -->`tan 2x = 1.49#. OK