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What is x in equation sin5xcos3x=sin9xcos7x?

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greenbel | Honors

Posted July 11, 2013 at 5:20 PM via web

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What is x in equation sin5xcos3x=sin9xcos7x?

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aruv | High School Teacher | Valedictorian

Posted July 11, 2013 at 5:44 PM (Answer #1)

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`sin(5x) cos(3x)=sin(9x)cos(7x)`

Multiply both side by 2

`2sin(5x)cos(3x)=2sin(9x)cos(7x)`  

we know

`2sin(a)cos(b)=sin(a+b)+sin(a-b)`  

Thus

`sin(5x+3x)+sin(5x-3x)=sin(9x+7x)+sin(9x-7x)`

`sin(8x)+sin(2x)=sin(16x)+sin(2x)`

`sin(8x)=sin(16x)`

`sin(16x)-sin(8x)=0`

`2sin(8x)cos(8x)-sin(8x)=0`

`sin(8x)(2cos(8x)-1)=0`

`either`

`sin(8x)=0`

`8x=npi`

`x=(npi)/8`

`or`

`2cos(8x)-1=0`

`cos(8x)=1/2`

`cos(8x)=cos(pi/3)`

`8x=2mpi+-(pi/3)`

`x=(mpi)/4+-pi/24`

`` where m and n are integers.

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