Question

If Tan(x) = 2, and 0 < x < 2pi, find the exact value of Sin(x + pi/4)?

Answer 1

`= (3sqrt2/2)cos x`

Explanation:

`tan x = sin x/cos x = 2`. t. could be in Quadrant 1 or Quadrant 3.
sin x = 2cos x
Hence,
`sin (x + pi/4) = sin (pi/4).cos x + sin x.cos (x/4) =`
`= sin (pi/4).cos x + 2cos x.cos pi/4 =`
`= (sqrt2/2)cos x + sqrt2cos x = (sqrt2/2)cos x + sqrt2cos x =`
`= ((3sqrt2)/2)cos x`. Find cos x
`cos^2 x = 1/(1 + tan^2 x) = 1/(1 + 4) = 1/5`
`cos x = +- sqrt5/5`
Therefor
`sin (x + pi/4) = +- ((3sqrt2)/2)(sqrt5/5) = +- (3sqrt10)/10`