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beks73 [17]
3 years ago
11

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

Mathematics
1 answer:
evablogger [386]3 years ago
6 0
Sin(α+β)=sin(α)cos(β)+cos(α)sin(β)
sin(α-β)=sin(α)cos(β)-cos(α)sin(β)


sin(x+ \frac{ \pi }{4} )=sin(x)cos\frac{ \pi }{4} +cos(x)sin\frac{ \pi }{4}  \\ sin(x- \frac{ \pi }{4} )=sin(x)cos\frac{ \pi }{4} -cos(x)sin\frac{ \pi }{4}  \\ \\   \\sin(x+ \frac{ \pi }{4} )-sin(x- \frac{ \pi }{4} ) =1 \\ sin(x)cos\frac{ \pi }{4} +cos(x)sin\frac{ \pi }{4}  -(sin(x)cos\frac{ \pi }{4} -cos(x)sin\frac{ \pi }{4}  )=1 \\  sin(x)cos\frac{ \pi }{4} +cos(x)sin\frac{ \pi }{4}  -sin(x)cos\frac{ \pi }{4} +cos(x)sin\frac{ \pi }{4}  =1 \\  cos(x)sin\frac{ \pi }{4} +cos(x)sin\frac{ \pi }{4}  =1 \\
2cos(x)sin\frac{ \pi }{4}  =1    \\ sin\frac{ \pi }{4} = \frac{ \sqrt{2} }{2}  \\ 2cos(x) \frac{ \sqrt{2} }{2}  =1 \\ 2 \cdot \frac{ \sqrt{2} }{2}cos(x)   =1 \\   \sqrt{2} cos(x)=1 \\
cos(x)= \frac{1}{ \sqrt{2} }  \\ x=\pm arccos \frac{1}{ \sqrt{2} }+2 \pi k , k \in Z \\ x=\pm \frac{ \pi }{4} +2 \pi k , k \in Z
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