Question

How do you use the double-angle identities to find tan(2x) if sec x=root65 and sin x is less than 0?

Answer 1

Find tan 2x, knowing `sec x = sqrt65` ans sin x < 0.

Ans: 0.25

Explanation:

`sec x = 1/cos x = sqrt65 = 8.06`
`cos x = 1/sec x = 1/8.06 = 0.124`
Calculator --> cos x = 0.124 --> `x = +- 82.87`
Since sin x < 0, then x = - 82.87 (Quadrant IV)
Calculator --> sin x = -99.
tan x = -.(0.99)/0.124 = - 8
Apply the trig identity: `tan 2x = (2tan x)/(1 - tan^2 x)`

`tan 2x = (2(-8))/(1 - 64) = (-16)/-63 = 0.25`

Check by calculator. x = -82.87 --> 2x = -165.74 --> tan 2x = tan (-165.74) = 0.25. OK