Question

Sin(x+pi/4)-sin(x-pi/4)=1 solve the equation

Answer 1

sin(α+β)=sin(α)cos(β)+cos(α)sin(β)
sin(α-β)=sin(α)cos(β)-cos(α)sin(β)



`sin(x+ ( \pi )/(4) )=sin(x)cos( \pi )/(4) +cos(x)sin( \pi )/(4) \\ sin(x- ( \pi )/(4) )=sin(x)cos( \pi )/(4) -cos(x)sin( \pi )/(4) \\ \\ \\sin(x+ ( \pi )/(4) )-sin(x- ( \pi )/(4) ) =1 \\ sin(x)cos( \pi )/(4) +cos(x)sin( \pi )/(4) -(sin(x)cos( \pi )/(4) -cos(x)sin( \pi )/(4) )=1 \\ sin(x)cos( \pi )/(4) +cos(x)sin( \pi )/(4) -sin(x)cos( \pi )/(4) +cos(x)sin( \pi )/(4) =1 \\ cos(x)sin( \pi )/(4) +cos(x)sin( \pi )/(4) =1 \\`

`2cos(x)sin( \pi )/(4) =1 \\ sin( \pi )/(4) = ( √(2) )/(2) \\ 2cos(x) ( √(2) )/(2) =1 \\ 2 \cdot ( √(2) )/(2)cos(x) =1 \\ √(2) cos(x)=1 \\`

`cos(x)= (1)/( √(2) ) \\ x=\pm arccos (1)/( √(2) )+2 \pi k , k \in Z \\ x=\pm ( \pi )/(4) +2 \pi k , k \in Z`