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

1 Answer
Jun 16, 2018

= (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