Question

How could I turn 2sin5xcos3x into a sum of trig functions?

Answer 1

`2sin5xcos3x=color(red)(sin(8x)+sin(2x)`

Explanation:

We know `"the "color(blue)"Product Identity"`

`color(blue)(2sinAcosB=sin(A+B)+sin(A-B)`

Substitute , `A=5x and B=3x`

`2sin5xcos3x=sin(5x+3x)+sin(5x-3x)`

`2sin5xcos3x=color(red)(sin(8x)+sin(2x)larr"sum of trig. functions."`

Answer 2

`sin8x+sin2x`

Explanation:

`"using the "color(blue)"product to sum formula"`

`•color(white)(x)2sinxcosy=sin(x+y)+sin(x-y)`

`"here "x=5x" and "y=3x`

`2sin5xcos3x`

`=sin(5x+3x)+sin(5x-3x)=sin8x+sin2x`