Question

If cos A = -4/5 how do you find tan 2A?

Answer 1

`­±40/7 * 3/5 = ± 24/7`

Explanation:

`cos x = -4/5`

`tan 2x = frac{sin 2x}{cos 2x} = frac{2 sin x cos x}{cos^2 x - underbrace{sin^2 x}_{1 - cos^2 x}} = frac{-2 * sin x * 4/5}{2 * 16/25 - 1}`

`± sin x = sqrt {1 - 16/25} = sqrt{{25-16}/25} = 3/5`

`tan 2x = -8/5 * sin x * 25/{32-25} = -40/7 sin x`

Answer 2

24/7

Explanation:

`cos A = -4/5`.
A could be in Quadrant 2 or Quadrant 3.
`sin^2 A = 1 - cos^2 A = 1 - 16/25 = 9/25`
`sin A = +- 3/5`
`sin 2A = 2sin A.cos A = 2(+- 3/5)(- 4/5) = +- 24/25`
`cos ^ 2A = 1 - sin^2 2A = 1 - (576/625) = 49/625`
`cos 2A = +- 7/25`
`tan 2A = (sin 2A)/(cos 2A) = (+- 24/25)(+- 25/7) = +- 24/7`
If A lies in Quadrant 2, then, 2A lies in Quadrant 3, and tan 2A is positive
If A lies in Q. 3, then, 2A lies in Q. 1, then, tan 2A is positive.
Finally
`tan 2A = 24/7`