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

1 Answer
Nghi N.
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#

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